SYNTOMIC COHOMOLOGY AND p-ADIC REGULATORS FOR VARIETIES OVER p-ADIC FIELDS

JAN NEKOVÁŘ, WIESŁAWA NIZIOŁ

Abstract. We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over p-adic rings extends uniquely to a cohomology theory for varieties over p-adic fields that satisfies h-descent. This new cohomology - syntomic cohomology - is a Bloch-Ogus cohomology theory, admits period map to étale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild-Serre spectral sequence on the étale side and is related to the Bloch-Kato exponential map. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soulé’s étale regulators land in the potentially semistable Selmer groups.

Our construction of syntomic cohomology is based on new ideas and techniques developed by Beilinson and Bhatt in their recent work on p-adic comparison theorems.

Contents

1. Introduction  2
1.1. Statement of the main result  2
1.2. Construction of syntomic cohomology  4
1.3. Syntomic period maps  5
1.4. p-adic regulators  7
1.5. Notation and Conventions  8
2. Preliminaries  8
2.1. The rings of periods  9
2.2. Derived log de Rham complex  9
2.3. h-topology  11
2.4. Galois cohomology  11
2.5. Comparison of spectral sequences  15
3. Syntomic cohomology  19
3.1. Hyodo-Kato morphism revisited  19
3.2. Log-syntomic cohomology  25
3.3. Arithmetic syntomic cohomology  31
3.4. Geometric syntomic cohomology  37
4. Relation between syntomic cohomology and étale cohomology  43
4.1. Geometric case  43
4.2. Arithmetic case  49
5. Syntomic regulators  54
5.1. Construction of syntomic Chern classes  54
5.2. Image of étale regulators  59
Appendix A. Vanishing of H^{2}(G_{K}, V) by Laurent Berger  59
Appendix B. The Syntomic ring spectrum by Frédéric Déglise  61
References  64

Date: August 26, 2016. The authors’ research was supported in part by the grant ANR-BLAN-0114 and by the NSF grant DMS0703696,

respectively.

1. Introduction

In this article we define syntomic cohomology for varieties over p-adic fields, relate it to the Bloch-Kato exponential map, and use it to study the images of Soulé’s étale regulators. Contrary to all the previous constructions of syntomic cohomology (see below for a brief review) we do not restrict ourselves to varieties coming with a nice model over the integers. Hence our syntomic regulators make no integrality assumptions on the K-theory classes in the domain.

1.1. Statement of the main result. Recall that, for varieties proper and smooth over a p-adic ring of mixed characteristic, syntomic cohomology (or its non-proper variant: syntomic-étale cohomology) was introduced by Fontaine and Messing [31] in their proof of the Crystalline Comparison Theorem as a natural bridge between crystalline cohomology and étale cohomology. It was generalized to log-syntomic cohomology for semistable varieties by Kato [42]. For a log-smooth scheme 𝒳 over a complete discrete valuation ring V of mixed characteristic (0, p) and a perfect residue field, and for any r ≥ 0, rational log-syntomic cohomology of 𝒳 can be defined as the ”filtered Frobenius eigenspace” in log-crystalline cohomology, i.e., as the following mapping fiber

(1)  RΓ_{syn}(𝒳 , r) := Cone (RΓ_{cr}(𝒳 , 𝒥^{[r]})⟶^{1−φ_{r}}RΓ_{cr}(𝒳 ))[−1],

where RΓ_{cr}(·, 𝒥^{[r]}) denotes the absolute rational log-crystalline cohomology (i.e., over Z_{p}) of the r’th Hodge filtration sheaf 𝒥^{[r]} and φ_{r} is the crystalline Frobenius divided by p^{r}. This definition suggested that the log-syntomic cohomology could be the sought for p-adic analog of Deligne-Beilinson cohomology. Recall that, for a complex manifold X, the latter can be defined as the cohomology RΓ(X, Z(r)_{𝒟} ) of Deligne complex Z(r)_{𝒟} :

0 → Z(r) → Ω_{X}^{1} → Ω_{X}^{2} → … → Ω_{X}^{r−1} → 0

And, indeed, since its introduction, log-syntomic cohomology has been used with some success in the study of special values of p-adic L-functions and in formulating p-adic Beilinson conjectures (cf. [9] for a review).

The syntomic cohomology theory with Q_{p}-coefficients RΓ_{syn}(X_{h}, r) (r ≥ 0) for arbitrary varieties – more generally, for arbitrary essentially finite diagrams of varieties – over the p-adic field K (the fraction field of V) that we construct in this article is a generalization of Fontaine-Messing(-Kato) log-syntomic cohomology. That is, for a semistable scheme ^{1} 𝒳 over V we have RΓ_{syn}(𝒳 , r) ≃ RΓ_{syn}(X_{h}, r), where X is the largest subvariety of 𝒳_{K} with trivial log-structure. An analogous theory RΓ_{syn}(X_{\overline{K},h}, r) (r ≥ 0) exists for (diagrams of) varieties over \overline{K}, where \overline{K} is an algebraic closure of K.

Our main result can be stated as follows.

Theorem A. For any variety X over K, there is a canonical graded commutative dg Q_{p}-algebra RΓ_{syn}(X_{h}, ∗) such that

(1) it is the unique extension of log-syntomic cohomology to varieties over K that satisfies h-descent, i.e., for any hypercovering π : Y_{•} → X in h-topology, we have a quasi-isomorphism

π^{∗} : RΓ_{syn}(X_{h}, ∗) →^{∼} RΓ_{syn}(Y_{•,h}, ∗).

(2) it is a Bloch-Ogus cohomology theory [12].

(3) for X = Spec(K), H_{syn}^{∗}(X_{h}, r) ≃ H_{st}^{∗}(G_{K}, Q_{p}(r)), where H_{st}^{i}(G_{K}, −) denotes the Ext-group Ext^{i}(Q_{p}, −) in the category of (potentially) semistable representations of G_{K} = Gal(\overline{K}/K).

(4) There are functorial syntomic period morphisms

ρ_{syn} : RΓ_{syn}(X_{h}, r) → RΓ(X_{ét}, Q_{p}(r)), ρ_{syn} : RΓ_{syn}(X_{\overline{K},h}, r) → RΓ(X_{\overline{K},ét}, Q_{p}(r))

compatible with products which induce quasi-isomorphisms

τ_{≤r}RΓ_{syn}(X_{h}, r) →^{∼} τ_{≤r}RΓ(X_{ét}, Q_{p}(r)), τ_{≤r}RΓ_{syn}(X_{\overline{K},h}, r) →^{∼} τ_{≤r}RΓ(X_{\overline{K},ét}, Q_{p}(r)).

1Throughout the Introduction, the divisors at infinity of semistable schemes have no multiplicities.

(5) The Hochschild-Serre spectral sequence for étale cohomology

étE_{2}^{i,j} = H^{i}(G_{K}, H^{j}(X_{\overline{K},ét}, Q_{p}(r))) ⟹ H^{i+j}(X_{ét}, Q_{p}(r))

has a syntomic analog

synE_{2}^{i,j} = H_{st}^{i}(G_{K}, H^{j}(X_{\overline{K},ét}, Q_{p}(r))) ⟹ H_{syn}^{i+j}(X_{h}, r).

(6) There is a canonical morphism of spectral sequences ^{syn}E_{t} → ^{ét}E_{t} compatible with the syntomic period map.

(7) There are syntomic Chern classes

c_{i,j}^{syn} : K_{j}(X) → H_{syn}^{2i−j}(X_{h}, i)

compatible with étale Chern classes via the syntomic period map.

As is shown in [25], syntomic cohomology RΓ_{syn}(X_{h}, ∗) can be interpreted as an absolute p-adic Hodge cohomology. That is, it is a derived Hom in the category of admissible (φ, N, G_{K})-modules between the trivial module and a complex of such modules canonically associated to a variety. Alternatively, it is a derived Hom in the category of potentially semistable representations between the trivial representation and a complex of such representations canonically associated to a variety. A particularly simple construction of such a complex, using Beilinson’s Basic Lemma, was proposed by Beilinson (and is presented in [25]). The category of modules over the syntomic cohomology algebra RΓ_{syn}(X_{h}, ∗) (taken in a motivic sense) yields a category of p-adic Galois representations that better approximates the category of geometric representations than the category of potentially semistable representations [25]. For further applications of syntomic cohomology algebra we refer the interested reader to Op. cit.

Similarly, as is shown in [55], geometric syntomic cohomology RΓ_{syn}(X_{\overline{K},h}, ∗) is a derived Hom in the category of effective φ-gauges (with one paw) [27] between the trivial gauge and a complex of such gauges canonically associated to a variety. In particular, geometric syntomic cohomology group is a finite dimensional Banach-Colmez Space [19], hence has a very rigid structure.

The syntomic descent spectral sequence and its compatibility with the Hochschild-Serre spectral sequence in étale cohomology imply the following proposition.

Proposition 1.1. Let i ≥ 0. The composition

H_{dR}^{i−1}(X)/F^{r} →^{∂} H_{syn}^{i}(X_{h}, r) ⟶^{ρ_{syn}} H_{ét}^{i}(X, Q_{p}(r)) → H_{ét}^{i}(X\overline{K}, Q_{p}(r))

is the zero map. The induced (from the syntomic descent spectral sequnce) map

H_{dR}^{i−1}(X)/F^{r} → H^{1}(G_{K}, H_{ét}^{i−1}(X\overline{K}, Q_{p}(r)))

is equal to the Bloch-Kato exponential associated with the Galois representation H_{ét}^{i−1}(X\overline{K}, Q_{p}(r)).

This yields a comparison between p-adic étale regulators, syntomic regulators, and the Bloch-Kato exponential (which was proved in the good reduction case in [48] and [51, Thm. 5.2]^{2}) that is of fundamental importance for theory of special values of L-functions, both complex valued and p-adic. The point is that syntomic regulators can be thought of as an abstract p-adic integration theory. The comparison results stated above then relate certain p-adic integrals to the values of the p-adic étale regulator via the Bloch-Kato exponential map. A modification of syntomic cohomology developed in [8] in the good reduction case (resp. in [10] – using the techniques of the present article – in the case of arbitrary reduction) can be used to perform explicit computations. For example, the formulas from [10, §3] were applied to a calculation of certain p-adic regulators in [7] and in [22].

2The Bloch-Kato exponential is called l there.

1.2. Construction of syntomic cohomology. We will now sketch the proof of Theorem A. Recall first that a little bit after log-syntomic cohomology had appeared on the scene, Selmer groups of Galois representations – describing extensions in certain categories of Galois representations – were introduced by Bloch and Kato [13] and linked to special values of L-functions. And a syntomic cohomology (in the good reduction case), a priori different than that of Fontaine-Messing, was defined in [51] and by Besser in [8] as a higher dimensional analog of the complexes computing these groups. The guiding idea here was that just as Selmer groups classify extensions in certain categories of ”geometric” Galois representations their higher dimensional analogs – syntomic cohomology groups – should classify extensions in a category of ”p-adic motivic sheaves”. This was shown to be the case for H^{1} by Bannai [1] who has also shown that Besser’s (rigid) syntomic cohomology is a p-adic analog of Beilinson’s absolute Hodge cohomology [2].

Complexes computing the semistable and potentially semistable Selmer groups were introduced in [47] and [32]. For a semistable scheme 𝒳 over V, their higher dimensional analog can be written as the following homotopy limit^{3}

(2)  RΓ′_{syn}(𝒳 , r) := [ \xymatrix{RΓ_{HK}(𝒳_{0}) \ar[d]^{N} \ar[r]^{(1−φ_{r},ι_{dR})} & RΓ_{HK}(𝒳_{0}) ⊕ RΓ_{dR}(𝒳_{K})/F^{r} \ar[d]^{(N,0)} \\ RΓ_{HK}(𝒳_{0}) \ar[r]^{1−φ_{r−1}} & RΓ_{HK}(𝒳_{0})} ]

where 𝒳_{0} is the special fiber of 𝒳 , RΓ_{HK}(·) is the Hyodo-Kato cohomology, N denotes the Hyodo-Kato monodromy, and RΓ_{dR}(·) is the logarithmic de Rham cohomology. The map ι_{dR} is the Hyodo-Kato morphism that induces a quasi-isomorphism ι_{dR} : RΓ_{HK}(𝒳_{0}) ⊗_{K_{0}} K →^{∼} RΓ_{dR}(𝒳_{K}), for K_{0} - the fraction field of Witt vectors of the residue field of V.

Using Dwork’s trick, we prove (cf. Proposition 3.8) that the two definitions of log-syntomic cohomology are the same, i.e., that there is a quasi-isomorphism

α_{syn} : RΓ_{syn}(𝒳 , r) →^{∼} RΓ′_{syn}(𝒳 , r).

It follows that log-syntomic cohomology groups vanish in degrees strictly higher than 2 dim X_{K} + 2 and that, if 𝒳 = Spec(V), then H^{i}RΓ_{syn}(𝒳 , r) ≃ H_{st}^{i}(G_{K}, Q_{p}(r)).

The syntomic cohomology for varieties over p-adic fields that we introduce in this article is a generalization of the log-syntomic cohomology of Fontaine and Messing. Observe that it is clear how one can try to use log-syntomic cohomology to define syntomic cohomology for varieties over fields that satisfies h-descent. Namely, for a variety X over K, consider the h-topology of X and recall that (using alterations) one can show that it has a basis consisting of semistable models over finite extensions of V [3]. By h-sheafifying the complexes Y ↦ RΓ_{syn}(Y, r) (for a semistable model Y) we get syntomic complexes 𝒮 (r). We define the (arithmetic) syntomic cohomology as

RΓ_{syn}(X_{h}, r) := RΓ(X_{h}, 𝒮 (r)).

A priori it is not clear that the so defined syntomic cohomology behaves well: the finite ramified field extensions introduced by alterations are in general a problem for log-crystalline cohomology. For example, the related complexes RΓ_{cr}(X_{h}, 𝒥^{[r]}) are huge. However, taking Frobenius eigenspaces cuts off the ”noise” and the resulting syntomic complexes do indeed behave well. To get an idea why this is the case, h-sheafify the complexes Y ↦ RΓ′_{syn}(Y, r) and imagine that you can sheafify the maps α_{syn} as well. We get sheaves 𝒮′(r) and quasi-isomorphisms α_{syn} : 𝒮 (r) →^{∼} 𝒮′(r). Setting RΓ′_{syn}(X_{h}, r) := RΓ(X_{h}, 𝒮′(r)) we obtain the following quasi-isomorphisms

3See section 1.5 for an explanation of the notation we use for certain homotopy limits.

(3)  RΓ_{syn}(X_{h}, r) ≃ RΓ′_{syn}(X_{h}, r) ≃ [ \xymatrix{RΓ_{HK}(X_{h}) \ar[d]^{N} \ar[r]^{(1−φ_{r},ι_{dR})} & RΓ_{HK}(X_{h}) ⊕ RΓ_{dR}(X_{K})/F^{r} \ar[d]^{(N,0)} \\ RΓ_{HK}(X_{h}) \ar[r]^{1−φ_{r−1}} & RΓ_{HK}(X_{h})} ]

where RΓ_{HK}(X_{h}) denotes the Hyodo-Kato cohomology (defined as h-cohomology of the presheaf: Y ↦ RΓ_{HK}(Y_{0})) and RΓ_{dR}(·) is Deligne’s de Rham cohomology [26]. The Hyodo-Kato map ι_{dR} is the hsheafification of the logarithmic Hyodo-Kato map. It is well-known that Deligne’s de Rham cohomology groups are finite rank K-vector spaces; it turns out that the Hyodo-Kato cohomology groups are finite rank K_{0}-vector spaces: we have a quasi-isomorphism RΓ_{HK}(X_{h}) →^{∼} RΓ_{HK}(X_{\overline{K},h})^{G_{K}} and the geometric Hyodo-Kato groups H^{∗}RΓ_{HK}(X_{\overline{K},h}) are finite rank K_{0}^{nr}-vector spaces, where K_{0}^{nr} is the maximal unramified extension of K_{0} (see (4) below).

It follows that syntomic cohomology groups vanish in degrees higher than 2 dim X_{K} + 2 and that syntomic cohomology is, in fact, a generalization of the classical log-syntomic cohomology, i.e., for a semistable scheme 𝒳 over V we have RΓ_{syn}(𝒳 , r) ≃ RΓ_{syn}(X_{h}, r), where X is the largest subvariety of 𝒳_{K} with trivial log-structure. This follows from the quasi-isomorphism α_{syn}: logarithmic Hyodo-Kato and de Rham cohomologies (over a fixed base) satisfy proper descent and the finite fields extensions that appear as the ”noise” in alterations do not destroy anything since logarithmic Hyodo-Kato and de Rham cohomologies satisfy finite Galois descent.

Alas, we were not able to sheafify the map α_{syn}. The reason for that is that the construction of α_{syn} uses a twist by a high power of Frobenius – a power depending on the field K. And alterations are going to introduce a finite extension of K – hence a need for higher and higher powers of Frobenius. So instead we construct directly the map α_{syn} : RΓ_{syn}(X_{h}, r) → RΓ′_{syn}(X_{h}, r). To do that we show first that the syntomic cohomological dimension of X is finite. Then we take a semistable h-hypercovering of X, truncate it at an appropriate level, extend the base field K to K′, and base-change everything to K′. There we can work with one field and use the map α_{syn} defined earlier. Finally, we show that we can descend.

1.3. Syntomic period maps. We pass now to the construction of the period maps from syntomic to étale cohomology that appear in Theorem A. They are easier to define over \overline{K}, i.e., from the geometric syntomic cohomology. In this setting, things go smoother with h-sheafification since going all the way up to \overline{K} before completing kills a lot of ”noise” in log-crystalline cohomology. More precisely, for a semistable scheme 𝒳 over V, we have the following canonical quasi-isomorphisms [4]

(4)  ι_{cr} : RΓ_{HK}(𝒳\overline{V})_{B_{cr}^{+}}^{τ} →^{∼} RΓ_{cr}(𝒳\overline{V}), ι_{dR} : RΓ_{HK}(𝒳\overline{V})τ ; \overline{K} →^{∼} RΓ_{dR}(𝒳\overline{K}),

where \overline{V} is the integral closure of V in \overline{K}, B_{cr}^{+} is the crystalline period ring, and τ denotes certain twist. These quasi-isomorphisms h-sheafify well: for a variety Xover K, they induce the following quasiisomorphisms [4]

(5)  ι_{cr} : RΓ_{HK}(X_{\overline{K},h})_{B_{cr}^{+}}^{τ} →^{∼} RΓ_{cr}(X_{\overline{K},h}), ι_{dR} : RΓ_{HK}(X_{\overline{K},h})τ ; \overline{K} →^{∼} RΓ_{dR}(X\overline{K}),

where the terms have obvious meaning. Since Deligne’s de Rham cohomology has proper descent (by definition), it follows that h-crystalline cohomology behaves well. That is, if we define crystalline sheaves 𝒥_{cr}^{[r]} and 𝒜_{cr} on X_{\overline{K},h} by h-sheafifying the complexes Y ↦ RΓ_{cr}(Y, 𝒥^{[r]}) and Y ↦ RΓ_{cr}(Y), respectively, for Y which are a base change to \overline{V} of a semistable scheme over a finite extension of V (such schemes Y form a basis of X_{\overline{K},h}) then the complexes RΓ(X_{\overline{K},h}, 𝒥^{[r]}) and RΓ_{cr}(X_{\overline{V} ,h}) := RΓ(X_{\overline{K},h}, 𝒜_{cr}) generalize log-crystalline cohomology (in the sense described above) and the latter one is a perfect complex of B_{cr}^{+}-modules.

We obtain syntomic complexes 𝒮 (r) on X_{\overline{K},h} by h-sheafifying the complexes Y ↦ RΓ_{syn}(Y, r) and (geometric) syntomic cohomology by setting RΓ_{syn}(X_{\overline{K},h}, r) := RΓ(X_{\overline{K},h}, 𝒮 (r)). They fit into an analog of the exact sequence (1) and, by the above, generalize log-syntomic cohomology.

To construct the syntomic period maps

(6)  ρ_{syn} : RΓ_{syn}(X_{\overline{K},h}, r) → RΓ(X_{\overline{K},ét}, Q_{p}(r)), ρ_{syn} : RΓ_{syn}(X_{h}, r) → RΓ(X_{ét}, Q_{p}(r))

consider the syntomic complexes 𝒮_{n}(r): the mod-p^{n} version of the syntomic complexes 𝒮 (r) on X_{\overline{K},h}. We have the distinguished triangle

𝒮_{n}(r) → 𝒥_{cr,n}^{[r]}⟶^{p^{r}−φ}𝒜_{cr,n}

Recall that the filtered Poincaré Lemma of Beilinson and Bhatt [4], [11] yields a quasi-isomorphism ρ_{cr} : J_{cr,n}^{[r]} →^{∼} 𝒥_{cr,n}^{[r]}, where J_{cr}^{[r]} ⊂ A_{cr} is the r’th filtration level of the period ring A_{cr}. Using the fundamental sequence of p-adic Hodge Theory

0 → Z/p^{n}(r)′ → J_{cr,n}^{<r>}⟶^{1−φ_{r}}A_{cr,n} → 0,

where Z/p^{n}(r)′ := (1/(p^{a}a!)Z_{p}(r)) ⊗ Z/p^{n} and a denotes the largest integer ≤ r/(p − 1), we obtain the syntomic period map ρ_{syn} : 𝒮_{n}(r) → Z/p^{n}(r)′. It is a quasi-isomorphism modulo a universal constant. It induces the geometric syntomic period map in (6), and, by Galois descent, its arithmetic analog.

To study the descent spectral sequences from Theorem A, we need to consider the other version of syntomic cohomology, i.e., the complexes (7)

RΓ′_{syn}(X_{\overline{K},h}, r) := [ \xymatrix{RΓ_{HK}(X_{\overline{K},h}) ⊗_{K_{0}^{nr}} B_{st}^{+} \ar[r]^{(1−φ_{r},ι_{dR})} \ar[d]^{N} & RΓ_{HK}(X_{\overline{K},h}) ⊗_{K_{0}^{nr}} B_{st}^{+} ⊕ (RΓ_{dR}(X\overline{K}) ⊗\overline{K} B_{dR}^{+})/F^{r} \ar[d]^{(N,0)} \\ RΓ_{HK}(X_{\overline{K},h}) ⊗_{K_{0}^{nr}} B_{st}^{+} \ar[r]^{1−φ_{r−1}} & RΓ_{HK}(X_{\overline{K},h}) ⊗_{K_{0}^{nr}} B_{st}^{+}} ]

where B_{st}^{+} and B_{dR}^{+} are the semistable and de Rham p-adic period rings, respectively. We deduce a quasi-isomorphism RΓ_{syn}(X_{\overline{K},h}, r) →^{∼} RΓ′_{syn}(X_{\overline{K},h}, r).

Remark 1.2. This quasi-isomorphism yields, for a semistable scheme 𝒳 over V, the following exact sequence

→ H_{syn}^{i}(𝒳\overline{K}, r) → (H_{HK}^{i}(𝒳 )_{Q} ⊗_{K_{0}} B_{st}^{+})^{φ=p^{r},N=0} → (H_{dR}^{i}(𝒳_{K}) ⊗_{K} B_{dR}^{+})/F^{r} → H_{syn}^{i+1}(𝒳\overline{K}, r) →

It is a sequence of finite dimensional Banach-Colmez Spaces [19] and as such is a key in the proof of semistable comparison theorem for formal schemes in [21].

We also have a syntomic period map

(8)  ρ′_{syn} : RΓ′_{syn}(X_{\overline{K},h}, r) → RΓ(X_{\overline{K},ét}, Q_{p}(r))

that is compatible with the map ρ_{syn} via α_{syn}. To describe how it is constructed, recall that the crystalline period map of Beilinson induces compatible Hyodo-Kato and de Rham period maps [4] (9) ρ_{HK} : RΓ_{HK}(X_{\overline{K},h}) ⊗_{K_{0}^{nr}} B_{st}^{+}→RΓ(X_{\overline{K},ét}, Q_{p}) ⊗ B_{st}^{+}, ρ_{dR} : RΓ_{dR}(X_{K}) ⊗_{K} B_{dR}^{+}→RΓ(X_{\overline{K},ét}, Q_{p}) ⊗ B_{dR}^{+}

Applying them to the above homotopy limit, removing all the pluses from the period rings, reduces the homotopy limit to the complex

(10) [ \xymatrix{RΓ(X_{\overline{K},ét}, Q_{p}(r)) ⊗ B_{st} \ar[r]^{(1−φ_{r},ι_{dR})} \ar[d]^{N} & RΓ(X_{\overline{K},ét}, Q_{p}(r)) ⊗ B_{st} ⊕ (RΓ(X_{\overline{K},ét}, Q_{p}(r)) ⊗ B_{dR})/F^{r} \ar[d]^{(N,0)} \\ RΓ(X_{\overline{K},ét}, Q_{p}(r)) ⊗ B_{st} \ar[r]^{1−φ_{r−1}} & RΓ(X_{\overline{K},ét}, Q_{p}(r)) ⊗ B_{st}} ]

By the familiar fundamental exact sequence

0 → Q_{p}(r) → B_{st} ⟶^{(N,1−φ_{r},ι)}B_{st} ⊕ B_{st} ⊕ B_{dR}/F^{r} ⟶^{(1−φ_{r−1})−N}B_{st} → 0

the above complex is quasi-isomorphic to RΓ(X_{\overline{K},ét}, Q_{p}(r)). This yields the syntomic period morphism from (8). We like to think of geometric syntomic cohomology as being represented by the complex from (7) and of geometric étale cohomology as represented by the complex (10).

From the above constructions we derive several of the properties mentioned in Theorem A. The quasiisomorphisms (9) give that

H_{HK}^{i}(X_{\overline{K},h}) ≃ D_{pst}(H^{i}(X_{\overline{K},ét}, Q_{p}(r))), H_{HK}^{i}(X_{h}) ≃ D_{st}(H^{i}(X_{\overline{K},ét}, Q_{p}(r))),

where D_{pst} and D_{st} are the functors from [32]. This combined with the diagram (3) immediately yields the spectral sequence ^{syn}E_{t} since the cohomology groups of the total complex of

[ \xymatrix{H_{HK}^{j}(X_{h}) \ar[d]^{N} \ar[r]^{(1−φ_{r},ι_{dR})} & H_{HK}^{j}(X_{h}) ⊕ H_{dR}^{j}(X_{K})/F^{r} \ar[d]^{(N,0)} \\ H_{HK}^{j}(X_{h}) \ar[r]^{1−φ_{r−1}} & H_{HK}^{j}(X_{h})} ]

are equal to H_{st}^{∗}(G_{K}, H^{j}(X_{\overline{K},ét}, Q_{p}(r))). Moreover, the sequence of natural maps of diagrams (3) →

(7) →^{ρ_{syn}} (10) yields a compatibility of the syntomic descent spectral sequence with the Hochschild-Serre spectral sequence in étale cohomology (via the period maps). We remark that, in the case of proper varieties with semistable reduction, this fact was announced in [49].

Looking again at the period map ρ_{syn} : (7)→(10) we see that truncating all the complexes at level r will allow us to drop + from the first diagram. Hence we have

ρ_{syn} : τ_{≤r}RΓ_{syn}(X_{\overline{K},h}, r) →^{∼} τ_{≤r}RΓ(X_{\overline{K},ét}, Q_{p}(r))

To conclude that we have

ρ_{syn} : τ_{≤r}RΓ_{syn}(X_{h}, r) →^{∼} τ_{≤r}RΓ(X_{ét}, Q_{p}(r))

as well, we look at the map of spectral sequences ^{syn}E → ^{ét}E and observe that, in the stated ranges of the Hodge-Tate filtration we have H_{st}^{∗}(G_{K}, ·) = H^{∗}(G_{K}, ·) (a fact that follows, for example, from the work of Berger [6]).

1.4. p-adic regulators. As an application of Theorem A, we look at the question of the image of Soulé’s étale regulators

r_{r,i}^{ét} : K_{2r−i−1}(X)_{0} → H^{1}(G_{K}, H^{i}(X_{\overline{K},ét}, Q_{p}(r))),

where K_{2r−i−1}(X)_{0} := ker(c_{r,i+1}^{ét} : K_{2r−i−1}(X) → H^{i+1}(X_{\overline{K},ét}, Q_{p}(r))), inside the Galois cohomology group. We prove that

Theorem B. The regulators r_{r,i}^{ét} factor through the group H_{st}^{1}(G_{K}, H^{i}(X_{\overline{K},ét}, Q_{p}(r))).

As we explain in the article, this fact is known to follow from the work of Scholl [57] on ”geometric” extensions associated to K-theory classes. In our approach, this is a simple consequence of good properties of syntomic cohomology and the existence of the syntomic descent spectral sequence. Namely, as can be easily derived from the presentation (3), syntomic cohomology has projective space theorem and homotopy property ^{4} hence admits Chern classes from higher K-theory. It can be easily shown that they are compatible with the étale Chern classes via the syntomic period maps. The factorization we want in the above theorem follows then from the compatibility of the two descent spectral sequences.

4As explained in Appendix B, it follows that it is a Bloch-Ogus cohomology theory.

1.5. Notation and Conventions. Let V be a complete discrete valuation ring with fraction field K of characteristic 0, with perfect residue field k of characteristic p, and with maximal ideal 𝔪_{K}. Let v be the valuation on K normalized so that v(p) = 1. Let \overline{K} be an algebraic closure of K and let \overline{V} denote the integral closure of V in \overline{K}. Let W (k) be the ring of Witt vectors of k with fraction field K_{0} and denote by K_{0}^{nr} the maximal unramified extension of K_{0}. Denote by e_{K} the absolute ramification index of K, i.e., the degree of K over K_{0}. Set G_{K} = Gal(\overline{K}/K) and let I_{K} denote its inertia subgroup. Let φ be the absolute Frobenius on W (\overline{k}). We will denote by V, V^{×}, and V^{0} the scheme Spec(V) with the trivial, canonical (i.e., associated to the closed point), and (N → V, 1 ↦ 0) log-structure respectively. For a log-scheme X over 𝒪_{K}, X_{n} will denote its reduction mod p^{n}, X_{0} will denote its special fiber.

Unless otherwise stated, we work in the category of integral quasi-coherent log-schemes. In general, we will not distinguish between simplicial abelian groups and complexes of abelian groups.

Let A be an abelian category with enough projective objects. In this paper A will be the category of abelian groups or Z_{p}-, Z/p^{n}-, or Q_{p}-modules. Unless otherwise stated, we work in the (stable) ∞-category 𝒟(A), i.e., stable ∞-category whose objects are (left-bounded) chain complexes of projective objects of A. For a readable introduction to such categories the reader may consult [38], [46, 1]. The ∞-derived category is essential to us for two reasons: first, it allows us to work simply with the Beilinson-Hyodo-Kato complexes; second, it supplies functorial homotopy limits.

Many of our constructions will involve sheaves of objects from 𝒟(A). The reader may consult the notes of Illusie [41] and Zheng [64] for a brief introduction to the subject and [45], [46] for a thorough treatment.

We will use a shorthand for certain homotopy limits. Namely, if f : C → C′ is a map in the dg derived category of abelian groups, we set

[ \xymatrix{C \ar[r]^{f} & C′} ] := holim(C → C′ ← 0).

We also set

[ \xymatrix{C_{1} \ar[d] \ar[r]^{f} & C_{2} \ar[d] \\ C_{3} \ar[r]^{g} & C_{4}}] := [[C_{1} →^{f} C_{2}] → [C_{3} →^{g} C_{4}]],

where the diagram in the brackets is a commutative diagram in the dg derived category.

Acknowledgments. Parts of this article were written during our visits to the Fields Institute in Toronto in spring 2012. The second author worked on this article also at BICMR, Beijing, and at the University of Padova. We would like to thank these institutions for their support and hospitality.

This article was inspired by the work of Beilinson on p-adic comparison theorems. We would like to thank him for discussions related to his work. Luc Illusie and Weizhe Zheng helped us understand the ∞-category theory involved in Beilinson’s work and made their notes [41], [64] available to us – we would like to thank them for that. We have also profited from conversations with Laurent Berger, Amnon Besser, Bharghav Bhatt, Bruno Chiarellotto, Pierre Colmez, Frédéric Déglise, Luc Illusie, Tony Scholl, and Weizhe Zheng - we are grateful for these exchanges. Moreover, we would like to thank Pierre Colmez for reading and correcting parts of this article. Special thanks go to Laurent Berger and Frédéric Déglise for writing the Appendices.

2. Preliminaries

In this section we will do some preparation. In the first part, we will collect some relevant facts from the literature concerning period rings, derived log de Rham complex, and h-topology. In the second part, we will prove vanishing results in Galois cohomology and a criterium comparing two spectral sequences that we will need to compare the syntomic descent spectral sequence with the étale Hochschild-Serre spectral sequence.

2.1. The rings of periods. Let us recall briefly the definitions of the rings of periods B_{cr}, B_{dR}, B_{st} of Fontaine [28]. Let A_{cr} denote Fontaine’s ring of crystalline periods [28, 2.2,2.3]. This is a p-adically complete ring such that A_{cr,n} := A_{cr}/p^{n} is a universal PD-thickening of \overline{V}_{n} over W_{n}(k). Let J_{cr,n} denote its PD-ideal, A_{cr,n}/J_{cr,n} = \overline{V}_{n}. We have

A_{cr,n} = H_{cr}^{0}(Spec(\overline{V}_{n})/W_{n}(k)), B_{cr}^{+} := A_{cr}[1/p], B_{cr} := B_{cr}^{+}[t^{−1}],

where t is a certain element of B_{cr}^{+} (see [28] for a precise definition of t). The ring B_{cr}^{+} is a topological K_{0}- module equipped with a Frobenius φ coming from the crystalline cohomology and a natural G_{K}-action. We have that φ(t) = pt and that G_{K} acts on t via the cyclotomic character.

Let

B_{dR}^{+} := \underleftarrow{lim}_{r}(Q ⊗ \underleftarrow{lim}_{n} A_{cr,n}/J_{cr,n}^{[r]}), B_{dR} := B_{dR}^{+}[t^{−1}].

The ring B_{dR}^{+} has a discrete valuation given by the powers of t. Its quotient field is B_{dR}. We set F^{n}B_{dR} = t^{n}B_{dR}^{+}. This defines a descending filtration on B_{dR}.

The period ring B_{st} lies between B_{cr} and B_{dR} [28, 3.1]. To define it, choose a sequence of elements s = (s_{n})_{n≥0} of \overline{V} such that s_{0} = p and s_{n+1}^{p} = s_{n}. Fontaine associates to it an element u_{s} of B_{dR}^{+} that is transcendental over B_{cr}^{+}. Let B_{st}^{+} denote the subring of B_{dR} generated by B_{cr}^{+} and u_{s}. It is a polynomial algebra in one variable over B_{cr}^{+}. The ring B_{st}^{+} does not depend on the choice of s (because for another sequence s′ = (s′_{n})_{n≥0} we have u_{s} − u_{s′} ∈ Z_{p}t ⊂ B_{cr}^{+}). The action of G_{K} on B_{dR}^{+} restricts well to B_{st}^{+}. The Frobenius φ extends to B_{st}^{+} by φ(u_{s}) = pu_{s} and one defines the monodromy operator N : B_{st}^{+} → B_{st}^{+} as the unique B_{cr}^{+}-derivation such that Nu_{s} = −1. We have Nφ = pφN and the short exact sequence

(11)  0 → B_{cr}^{+} → B_{st}^{+} →^{N} B_{st}^{+} → 0

Let B_{st} = B_{cr}[u_{s}]. We denote by ι the injection ι : B_{st}^{+} ↪ B_{dR}^{+}. The topology on B_{st} is the one induced by B_{cr} and the inductive topology; the map ι is continuous (though the topology on B_{st} is not the one induced from B_{dR}).

2.2. Derived log de Rham complex. In this subsection we collect a few facts about the relationship between crystalline cohomology and de Rham cohomology.

Let S be a log-PD-scheme on which p is nilpotent. For a log-scheme Z over S, let LΩZ/S^{•} denote the derived log de Rham complex (see [3, 3.1] for a review). This is a commutative dg 𝒪_{S}-algebra on Z_{ét} equipped with a Hodge filtration F^{m}. There is a natural morphism of filtered commutative dg 𝒪_{S}-algebras [4, 1.9.1]

(12)  κ : LΩZ/S^{•} → Ru_{Z/S∗}(𝒪_{Z/S}),

where u_{Z/S} : Z_{cr} → Z_{ét} is the projection from the log-crystalline to the étale topos. The following theorem was proved by Beilinson [4, 1.9.2] by direct computations of both sides.

Theorem 2.1. Suppose that Z, S are fine and f : Z → S is an integral, locally complete intersection morphism. Then (12) yields quasi-isomorphisms

κ_{m} : LΩZ/S^{•}/F^{m} →^{∼} Ru_{Z/S∗}(𝒪_{Z/S}/𝒥_{Z/S}^{[m]}).

Recall [11, Def. 7.20] that a log-scheme is called G-log-syntomic if it is log-syntomic and the local log-smooth models can be chosen to be of Cartier type. The next theorem, finer than Theorem 2.1, was proved by Bhatt [11, Theorem 7.22] by looking at the conjugate filtration of the l.h.s.

Theorem 2.2. Suppose that f : Z → S is G-log-syntomic. Then we have a quasi-isomorphism

κ : LΩZ/S^{•} →^{∼} Ru_{Z/S∗}(𝒪_{Z/S}).

Combining the two theorems above, we get a filtered version:

Corollary 2.3. Suppose that f : Z → S is G-log-syntomic. Then we have a quasi-isomorphism

F^{m}LΩZ/S^{•} →^{∼} Ru_{Z/S∗}(𝒥_{Z/S}^{[m]}).

Proof. Consider the following commutative diagram with exact rows

F^{m}LΩZ/S^{•} ⟶  LΩZ/S^{•}  ⟶  LΩZ/S^{•}/F^{m}
↓  ↓^{≀}  ↓^{≀}
Ru_{Z/S∗}(𝒥_{Z/S}^{[m]}) ⟶ Ru_{Z/S∗}(𝒪_{Z/S}) ⟶ Ru_{Z/S∗}(𝒪_{Z/S}/𝒥_{Z/S}^{[m]}).

and use the above theorems of Bhatt and Beilinson. □

Let X be a fine, proper, log-smooth scheme over V^{×}. Set

RΓ(X_{ét}, LΩX/W (k)^{•,∧})ˆ⊗Q_{p} := (holim_{n} RΓ(X_{ét}, LΩX_{n}/W_{n}(k)^{•,∧})) ⊗ Q

and similarly for complexes over V^{×}. Here the hat over derived log de Rham complex refers to the completion with respect to the Hodge filtration (in the sense of prosystems). For r ≥ 0, consider the following sequence of maps

(13)

RΓ_{dR}(X_{K})/F^{r} ←^{∼} RΓ(X, LΩX/V^{×}^{•}/F^{r})_{Q} →^{∼} RΓ(X_{ét}, LΩX/V^{×}^{•}/F^{r})ˆ⊗Q_{p}

→^{∼} RΓ_{cr}(X, 𝒪_{X/V^{×}}/𝒥_{X/V^{×}}^{[r]})_{Q} ← RΓ_{cr}(X, 𝒪_{X/W (k)}/𝒥_{X/W (k)}^{[r]})_{Q}

The first quasi-isomorphism follows from the fact that the natural map LΩX_{K}/K_{0}^{•}/F^{r} →^{∼} ΩX_{K}/K_{0}^{•}/F^{r} is a quasi-isomorphism since X_{K} is log-smooth over K_{0}. The second quasi-isomorphism follows from X being proper and log-smooth over V^{×}, the third one from Theorem 2.1. Define the map

γ_{r}^{−1} : RΓ_{cr}(X, 𝒪_{X/W (k)}/𝒥_{X/W (k)}^{[r]})_{Q} → RΓ_{dR}(X_{K})/F^{r}

as the composition (13).

Corollary 2.4. Let X be a fine, proper, log-smooth scheme over V^{×}. Let r ≥ 0. There exists a canonical quasi-isomorphism

γ_{r} : RΓ_{dR}(X_{K})/F^{r} →^{∼} RΓ_{cr}(X, 𝒪_{X/W (k)}/𝒥_{X/W (k)}^{[r]})_{Q}

Proof. It suffices to show that the last map in the composition (13) is also a quasi-isomorphism. By Theorem 2.1, this map is quasi-isomorphic to the map

(RΓ(X_{ét}, LΩX/W (k)^{•,∧})ˆ⊗Q_{p})/F^{r} → (RΓ(X_{ét}, LΩX/V^{×}^{•,∧})ˆ⊗Q_{p})/F^{r}

Hence it suffices to show that the natural map

gr_{F}^{i} RΓ(X_{ét}, LΩX/W (k)^{•,∧})ˆ⊗Q_{p} → gr_{F}^{i} RΓ(X_{ét}, LΩX/V^{×}^{•,∧})ˆ⊗Q_{p}

is a quasi-isomorphism for all i ≥ 0.

Fix n ≥ 1 and i ≥ 0 and recall [3, 1.2] that we have a natural identification

gr_{F}^{i} LΩX_{n}/W_{n}(k)^{•} →^{∼} LΛ_{X}^{i}(L_{X_{n}/W_{n}(k)})[−i], gr_{F}^{i} LΩX_{n}/V_{n}^{×}^{•} →^{∼} LΛ_{X}^{i}(L_{X_{n}/V_{n}^{×}})[−i],

where L_{Y/S} denotes the relative log cotangent complex [3, 3.1] and LΛ_{X}(•) is the nonabelian left derived functor of the exterior power functor. The distinguished triangle

𝒪_{X} ⊗_{V} L_{V_{n}^{×}/W_{n}(k)} → L_{X_{n}/W_{n}(k)} → L_{X_{n}/V_{n}^{×}}

yields a distinguished triangle

LΛ_{X}^{i}(𝒪_{X} ⊗_{V} L_{V_{n}^{×}/W_{n}(k)})[−i] → gr_{F}^{i} LΩX_{n}/W_{n}(k)^{•} → gr_{F}^{i} LΩX_{n}/V_{n}^{×}^{•}

Hence we have a distinguished triangle

holim_{n} RΓ(X_{ét}, LΛ_{X}^{i}(𝒪_{X}⊗_{V} L_{V_{n}^{×}/W_{n}(k)}))⊗Q[−i] → gr_{F}^{i} RΓ(X_{ét}, LΩX/W (k)^{•,∧})ˆ⊗Q_{p} → gr_{F}^{i} RΓ(X_{ét}, LΩX/V^{×}^{•,∧})ˆ⊗Q_{p}

It suffices to show that the term on the left is zero. But this will follow as soon as we show that the cohomology groups of L_{V_{n}^{×}/W_{n}(k)} are annihilated by p^{c}, where c is a constant independent of n. To show this recall that V is a log complete intersection over W (k). If π is a generator of V/W (k), f(t) its minimal polynomial then (cf. [56, 6.9]) L_{V^{×}/W (k)} is quasi-isomorphic to the cone of the multiplication by f′(π) map on V. Hence L_{V^{×}/W (k)} is acyclic in non-zero degrees, H^{0}L_{V^{×}/W (k)} = Ω_{V^{×}/W (k)} is a cyclic V -module and we have a short exact sequence

0 → Ω_{V/W (k)} → Ω_{V^{×}/W (k)} → V/𝔪_{K} → 0

Since Ω_{V/W (k)} ≃ V/𝒟_{K/K_{0}}, where 𝒟_{K/K_{0}} is the different, we get that p^{c}H^{0}L_{V^{×}/W (k)} = 0 for a constant c independent of n. Since L_{V^{×}/W (k)} ≃ L_{V^{×}/W (k)} ⊗_{V}^{L} V_{n}, we are done. □

Remark 2.5. Versions of the above corollary appear in various degrees of generality in the proofs of the p-adic comparison theorems (cf. [43, Lemma 4.5], [44, Lemma 2.7]). They are proved using computations in crystalline cohomology. We find the above argument based on Beilinson comparison Theorem 2.1 particularly conceptual and pleasing.

2.3. h-topology. In this subsection we review terminology connected with h-topology from Beilinson papers [3], [4], [11]; we will use it freely. Let 𝒱 ar_{K} be the category of varieties (i.e., reduced and separated schemes of finite type) over a field K. An arithmetic pair over K is an open embedding j : U ↪ \overline{U} with dense image of a K-variety U into a reduced proper flat V -scheme \overline{U}. A morphism (U, \overline{U}) → (T, \overline{T}) of pairs is a map \overline{U} → \overline{T} which sends U to T. In the case that the pairs represent log-regular schemes this is the same as a map of log-schemes. For a pair (U, \overline{U}), we set V_{U} := Γ(\overline{U}, 𝒪\overline{U}) and K_{U} := Γ(\overline{U}_{K}, 𝒪\overline{U}). K_{U} is a product of several finite extensions of K (labeled by the connected components of \overline{U}) and, if \overline{U} is normal, V_{U} is the product of the corresponding rings of integers. We will denote by 𝒫_{K}^{ar} the category of arithmetic pairs over K. A semistable pair (ss-pair) over K [3, 2.2] is a pair of schemes (U, \overline{U}) over (K, V) such that (i) \overline{U} is regular and proper over V, (ii) \overline{U} \ U is a divisor with normal crossings on \overline{U}, and (iii) the closed fiber \overline{U}_{0} of \overline{U} is reduced. Closed fiber is taken over the closed points of V_{U}. We will think of ss-pairs as log-schemes equipped with log-structure given by the divisor \overline{U} \ U. The closed fiber \overline{U}_{0} has the induced log-structure. We will say that the log-scheme (U, \overline{U}) is split over V_{U}. We will denote by 𝒫_{K}^{ss} the category of ss-pairs over K. A semistable pair is called strict if the irreducible components of the closed fiber are regular. We will often work with the larger category 𝒫_{K}^{log} of log-schemes (U, \overline{U}) ∈ 𝒫_{K}^{ar} log-smooth over V_{U}^{×}.

A semistable pair (ss-pair) over \overline{K} [3, 2.2] is a pair of connected schemes (T, \overline{T}) over (\overline{K}, \overline{V}) such that there exists an ss-pair (U, \overline{U}) over K and a \overline{K}-point α : K_{U} → \overline{K} such that (T, \overline{T}) is isomorphic to the base change (U\overline{K}, \overline{U}\overline{V}). We will denote by 𝒫ss ; \overline{K} the category of ss-pairs over \overline{K}.

A geometric pair over K is a pair (U, \overline{U}) of varieties over K such that \overline{U} is proper and U ⊂ \overline{U} is open and dense. We say that the pair (U, \overline{U}) is a nc-pair if \overline{U} is regular and \overline{U} \ U is a divisor with normal crossings in \overline{U}; it is strict nc-pair if the irreducible components of U \ \overline{U} are regular. A morphism of pairs f : (U_{1}, \overline{U}_{1}) → (U, \overline{U}) is a map \overline{U}_{1} → \overline{U} that sends U_{1} to U. We denote the category of nc-pairs over K by 𝒫_{K}^{nc}.

For a field K, the h-topology (cf. [61],[3, 2.3]) on 𝒱 ar_{K} is the coarsest topology finer than the Zariski and proper topologies. ^{5} It is stronger than the étale and proper topologies. It is generated by the pretopology whose coverings are finite families of maps {Y_{i} → X} such that Y := ∐ Y_{i} → X is a universal topological epimorphism (i.e., a subset of X is Zariski open if and only if its preimage in Y is open). We denote by 𝒱 ar_{K,h}, X_{h} the corresponding h-sites. For any of the categories 𝒫 mentioned above let γ : (U, \overline{U}) → U denote the forgetful functor. Beilinson proved [3, 2.5] that the categories 𝒫^{nc}, (𝒫_{K}^{ar}, γ) and (𝒫_{K}^{ss}, γ) form a base for 𝒱 ar_{K,h}. One can easily modify his argument to conclude the same about the categories (𝒫_{K}^{log}, γ).

2.4. Galois cohomology. In this subsection we review the definition of (higher) semistable Selmer groups and prove that in stable ranges they are the same as Galois cohomology groups. Our main references are [29], [30], [20], [13], [32], [47]. Recall [29], [30] that a p-adic representation V of G_{K} (i.e., a finite dimensional continuous Q_{p}-vector space representation) is called semistable (over K) if dim_{K_{0}}(B_{st} ⊗_{Q_{p}} V)^{G_{K}} = dim_{Q_{p}}(V). It is called potentially semistable if there exists a finite extension K′ of K such that V |G_{K′} is semistable over K′. We denote by Rep_{st}(G_{K}) and Rep_{pst}(G_{K}) the categories of semistable and potentially semistable representations of G_{K}, respectively.

5The latter is generated by a pretopology whose coverings are proper surjective maps.

As in [29, 4.2] a φ-module over K_{0} is a pair (D, φ), where D is a finite dimensional K_{0}-vector space, φ = φ_{D} is a φ-semilinear automorphism of D; a (φ, N)-module is a triple (D, φ, N), where (D, φ) is a φmodule and N = N_{V} is a K_{0}-linear endomorphism of D such that Nφ = pφN (hence N is nilpotent). A filtered (φ, N)-module is a (D, φ, N, F^{•}), where (D, φ, N) is a (φ, N)-module and F^{•} is a decreasing finite filtration of D_{K} by K-vector spaces. There is a notion of a (weakly) admissible filtered (φ, N)-module [20]. Denote by MF_{K}^{ad}(φ, N) ⊂ MF_{K}(φ, N) the categories of admissible filtered (φ, N)-modules and filtered (φ, N)-modules, respectively. We know [20] that the pair of functors D_{st}(V) = (B_{st} ⊗_{Q_{p}} V)^{G_{K}}, V_{st}(D) = (B_{st} ⊗_{K_{0}} D)^{φ=Id,N=0} ∩ F^{0}(B_{dR} ⊗_{K} D_{K}) defines an equivalence of categories MF_{K}^{ad}(φ, N) ≃ Rep_{st}(G_{K}).

For D ∈ MF_{K}(φ, N), set

C_{st}(D) := [ \xymatrix{D \ar[d]^{N} \ar[r]^{(1−φ,can)} & D ⊕ D_{K}/F^{0} \ar[d]^{(N,0)} \\ D \ar[r]^{1−pφ} & D} ]

Here the brackets denote the total complex of the double complex inside the brackets. Consider also the following complex

C^{+}(D) := [ \xymatrix{D ⊗_{K_{0}} B_{st}^{+} \ar[d]^{N} \ar[r]^{(1−φ,can ⊗ι)} & D ⊗_{K_{0}} B_{st}^{+} ⊕ (D_{K} ⊗_{K} B_{dR}^{+})/F^{0} \ar[d]^{(N,0)} \\ D ⊗_{K_{0}} B_{st}^{+} \ar[r]^{1−pφ} & D ⊗_{K_{0}} B_{st}^{+}} ]

Define C(D) by omitting the superscript + in the above diagram. We have C_{st}(D) = C(D)^{G_{K}}.

Remark 2.6. Recall [47, 1.19], [32, 3.3] that to every p-adic representation V of G_{K} we can associate a complex

C_{st}(V) : D_{st}(V)⟶^{(N,1−φ,ι)}D_{st}(V) ⊕ D_{st}(V) ⊕ t_{V} ⟶^{(1−pφ)−N}D_{st}(V) → 0 ⋯

where t_{V} := (V ⊗_{Q_{p}} (B_{dR}/B_{dR}^{+}))^{G_{K}} [32, I.2.2.1]. The cohomology of this complex is called H_{st}^{∗}(G_{K}, V). If V is semistable then C_{st}(V) = C_{st}(D_{st}(V)) hence H^{∗}(C_{st}(D_{st}(V))) = H_{st}^{∗}(G_{K}, V). If V is potentially semistable the groups H_{st}^{∗}(G_{K}, V) compute Yoneda extensions of Q_{p} by V in the category of potentially semistable representations [32, I.3.3.8]. In general [32, I.3.3.7], H_{st}^{0}(G_{K}, V) →^{∼} H^{0}(G_{K}, V) and H_{st}^{1}(G_{K}, V) ↪ H^{1}(G_{K}, V) computes st-extensions^{6} of Q_{p} by V.

Remark 2.7. Let D ∈ MF_{K}(φ, N). Note that (1) H^{0}(C(D)) = V_{st}(D);

(2) for i ≥ 2, H^{i}(C^{+}(D)) = H^{i}(C(D)) = 0 (because N is surjective on B_{st}^{+} and B_{st});

(3) if F^{1}D_{K} = 0 then F^{0}(D_{K} ⊗_{K} B_{dR}^{+}) = F^{0}(D_{K} ⊗_{K} B_{dR}) (in particular, the map of complexes C^{+}(D) → C(D) is an injection);

(4) if D = D_{st}(V) is admissible then we have quasi-isomorphisms

C(D) ←^{∼} V ⊗_{Q_{p}} [B_{cr} ⟶^{(1−φ,can)}B_{cr} ⊕ B_{dR}/F^{0}] ←^{∼} V ⊗_{Q_{p}} (B_{cr}^{φ=1} ∩ F^{0}) = V

and the map of complexes C_{st}(D) → C(D) represents the canonical map H_{st}^{i}(G_{K}, V) → H^{i}(G_{K}, V).

Lemma 2.8. ([28, Theorem II.5.3]) If X ⊂ B_{cr} ∩ B_{dR}^{+} and φ(X) ⊂ X then φ^{2}(X) ⊂ B_{cr}^{+}.

Proposition 2.9. If D ∈ MF_{K}(φ, N) and F^{1}D_{K} = 0 then H^{0}(C(D)/C^{+}(D)) = 0.

6Extension 0 → V_{1} → V_{2} → V_{3} → 0 is called st if the sequence 0 → D_{st}(V_{1}) → D_{st}(V_{2}) → D_{st}(V_{3}) → 0 is exact.

Proof. We will argue by induction on m such that N^{m} = 0. Assume first that m = 1 (hence N = 0). We have

C(D)/C^{+}(D) = [ \xymatrix{D ⊗_{K_{0}} (B_{st}/B_{st}^{+}) \ar[r]^{(1−φ,can ⊗ι)} \ar[d]^{1⊗N} & D ⊗_{K_{0}} (B_{st}/B_{st}^{+}) ⊕ D_{K} ⊗_{K} (B_{dR}/B_{dR}^{+}) \ar[d]^{(1⊗N,0)} \\ D ⊗_{K_{0}} (B_{st}/B_{st}^{+}) \ar[r]^{1−pφ} & D ⊗_{K_{0}} (B_{st}/B_{st}^{+})} ]

←^{∼} [D ⊗_{K_{0}} (B_{cr}/B_{cr}^{+})⟶^{(1−φ,can)}D ⊗_{K_{0}} (B_{cr}/B_{cr}^{+}) ⊕ D_{K} ⊗_{K} (B_{dR}/B_{dR}^{+})]

Write D = ⊕_{i=1}^{r}K_{0}d_{i} and, for 1 ≤ i ≤ r, consider the following maps

p_{i} : H^{0}(C(D)/C^{+}(D)) = (D⊗_{K_{0}} ((B_{cr} ∩B_{dR}^{+})/B_{cr}^{+}))^{φ=1} ⊂ ⊕_{i=1}^{r}d_{i} ⊗((B_{cr} ∩B_{dR}^{+})/B_{cr}^{+}) →^{pr_{i}} (B_{cr} ∩B_{dR}^{+})/B_{cr}^{+}

Let Y_{a}, a ∈ H^{0}(C(D)/C^{+}(D)), denote the K_{0}-subspace of (B_{cr} ∩ B_{dR}^{+})/B_{cr}^{+} spanned by p_{1}(a), …, p_{r}(a). We have (p_{1}(a), …, p_{r}(a))^{T} = Mφ(p_{1}(a), …, p_{r}(a))^{T}, for M ∈ GL_{r}(K_{0}). Hence φ(Y_{a}) ⊂ Y_{a}. Let X_{a} ⊂ B_{cr} ∩ B_{dR}^{+} be the inverse image of Y_{a} under the projection B_{cr} ∩ B_{dR}^{+} → (B_{cr} ∩ B_{dR}^{+})/B_{cr}^{+} (naturally B_{cr}^{+} ⊂ X_{a}). Then φ(X_{a}) ⊂ X_{a} + B_{cr}^{+} = X_{a}. By the above lemma φ^{2}(X_{a}) ⊂ B_{cr}^{+}. Hence φ^{2}(Y_{a}) = 0 and (applying M^{−2}) Y_{a} = 0. This implies that a = 0 and H^{0}(C(D)/C^{+}(D)) = 0, as wanted.

For general m > 0, consider the filtration D_{1} ⊂ D, where D_{1} := ker(N) with induced structures. Set D_{2} := D/D_{1} with induced structures. Then D_{1}, D_{2} ∈ MF_{K}(φ, N); N^{i} is trivial on D_{1} for i = 1 and on D_{2} for i = m − 1. Clearly F^{1}D_{1,K} = F^{1}D_{2,K} = 0. Hence, by Remark 2.7.3, we have a short exact sequence

0 → C(D_{1})/C^{+}(D_{1}) → C(D)/C^{+}(D) → C(D_{2})/C^{+}(D_{2}) → 0

By the inductive assumption H^{0}(C(D_{1})/C^{+}(D_{1})) = H^{0}(C(D_{2})/C^{+}(D_{2})) = 0. Hence H^{0}(C(D)/C^{+}(D)) = 0, as wanted. □

Corollary 2.10. If D ∈ MF_{K}(φ, N) and F^{1}D_{K} = 0 then H^{0}(C^{+}(D)) = H^{0}(C(D)) = V_{st}(D) (⊂ D ⊗_{K_{0}} B_{st}^{+}) and H^{1}(C^{+}(D)) ↪ H^{1}(C(D)).

Corollary 2.11. If D ∈ MF_{K}^{ad}(φ, N) and F^{1}D_{K} = 0 then

H^{i}(C^{+}(D)) = H^{i}(C(D)) = \begin{cases}V_{st}(D) & i = 0 \\ 0 & i ≠ 0 \end{cases}

(i.e., C^{+}(D) →^{∼} C(D)).

A filtered (φ, N, G_{K})-module is a tuple (D, φ, N, ρ, F^{•}), where

(1) D is a finite dimensional K_{0}^{nr}-vector space;

(2) φ : D → D is a Frobenius map;

(3) N : D → D is a K_{0}^{nr}-linear monodromy map such that Nφ = pφN;

(4) ρ is a K_{0}^{nr}-semilinear G_{K}-action on D (hence ρ|I_{K} is linear) that is smooth, i.e., all vectors have open stabilizers, and that commutes with φ and N;

(5) F^{•} is a decreasing finite filtration of D_{K} := (D ⊗_{K_{0}^{nr}} \overline{K})^{G_{K}} by K-vector spaces.

Morphisms between filtered (φ, N, G_{K})-modules are K_{0}^{nr}-linear maps preserving all structures. There is a notion of a (weakly) admissible filtered (φ, N, G_{K})-module [20], [30]. Denote by MF_{K}^{ad}(φ, N, G_{K}) ⊂ MF_{K}(φ, N, G_{K}) the categories of admissible filtered (φ, N, G_{K})-modules and filtered (φ, N, G_{K})-modules, respectively. We know [20] that the pair of functors D_{pst}(V) = inj lim_{H}(B_{st} ⊗_{Q_{p}} V)^{H}, H ⊂ G_{K} - an open subgroup, V_{pst}(D) = (B_{st} ⊗_{K_{0}^{nr}} D)^{φ=Id,N=0} ∩ F^{0}(B_{dR} ⊗_{K} D_{K}) define an equivalence of categories MF_{K}^{ad}(φ, N, G_{K}) ≃ Rep_{pst}(G_{K}).

For D ∈ MF_{K}(φ, N, G_{K}), set ^{7}

C_{pst}(D) := [ \xymatrix{D_{st} \ar[d]^{N} \ar[r]^{(1−φ,can)} & D_{st} ⊕ D_{K}/F^{0} \ar[d]^{(N,0)} \\ D_{st} \ar[r]^{1−pφ} & D_{st}} ]

Here D_{st} := D^{G_{K}}. Consider also the following complex (we set D\overline{K} := D ⊗_{K_{0}^{nr}} \overline{K})

C^{+}(D) := [ \xymatrix{D ⊗_{K_{0}^{nr}} B_{st}^{+} \ar[d]^{N} \ar[r]^{(1−φ,can ⊗ι)} & (D ⊗_{K_{0}^{nr}} B_{st}^{+}) ⊕ (D\overline{K} ⊗\overline{K} B_{dR}^{+})/F^{0} \ar[d]^{(N,0)} \\ D ⊗_{K_{0}^{nr}} B_{st}^{+} \ar[r]^{1−pφ} & D ⊗_{K_{0}^{nr}} B_{st}^{+}} ]

Define C(D) by omitting the superscript + in the above diagram. We have C_{pst}(D) = C(D)^{G_{K}}.

Remark 2.12. If V is potentially semistable then C_{st}(V) = C_{pst}(D_{pst}(V)) hence H^{∗}(C_{pst}(D_{pst}(V))) = H_{st}^{∗}(G_{K}, V).

Remark 2.13. If D = D_{pst}(V) is admissible then we have quasi-isomorphisms

C(D) ←^{∼} V ⊗_{Q_{p}} [B_{cr} ⟶^{(1−φ,can)}B_{cr} ⊕ B_{dR}/F^{0}] ←^{∼} V ⊗_{Q_{p}} (B_{cr}^{φ=1} ∩ F^{0}) = V

and the map of complexes C_{pst}(D) → C(D) represents the canonical map H_{st}^{i}(G_{K}, V) → H^{i}(G_{K}, V).

Remark 2.14. Let D = D_{pst}(V) be admissible. The Bloch-Kato exponential

(Z^{1}C(D))^{G_{K}} → H^{1}(G_{K}, V)

is given by the coboundary map arising from the exact sequence

0 → V → C^{0}(D) → Z^{1}C(D) → 0.

Its restriction to the de Rham part of Z^{1}C(D) is the Bloch-Kato exponential

exp_{BK} : D_{K}/F^{0} → H^{1}(G_{K}, V).

It is also obtained by applying Rf, where f(−) = (−)^{G_{K}}, to the coboundary map ∂ : Z^{1}C(D) → V [1] arising from the above exact sequence (see the proof of Theorem 4.8 for an appropriate formalism of continuous cohomology). Note that the composition of the canonical maps

Z^{1}C(D) → (σ_{≥1}C(D))[1] → C(D)[1] ←^{∼} V [1]

is not equal to ∂, but to −∂, by (18).

Corollary 2.15. If D ∈ MF_{K}^{ad}(φ, N, G_{K}) and F^{1}D_{K} = 0 then

H^{i}(C^{+}(D)) →^{∼} H^{i}(C(D)) = \begin{cases}V_{pst}(D) & i = 0 \\ 0 & i ≠ 0 \end{cases}

(i.e., C^{+}(D) →^{∼} C(D)).

Proof. By Remark 2.13 we have C(D) ≃ V_{pst}(D)[0]. To prove the isomorphism H^{i}(C^{+}(D)) →^{∼} H^{i}(C(D)), i ≥ 0, take a finite Galois extension K′/K such that D becomes semistable over K′, i.e., I_{K′} acts trivially on D. We have (D′, φ, N) ∈ MF_{K′}^{ad}(φ, N), where D′ := D^{G_{K′}} and (compatibly) D ≃ D′ ⊗_{K′_{0}} K_{0}^{nr}, F^{•}D′_{K′} ≃ F^{•}D_{K} ⊗_{K} K′. It easily follows that C^{+}(D) = C^{+}(K′, D′) and C(D) = C(K′, D′). Since F^{1}D′_{K′} = 0, our corollary is now a consequence of Corollary 2.11 □ Proposition 2.16. If D ∈ MF_{K}^{ad}(φ, N, G_{K}) and F^{1}D_{K} = 0 then, for i ≥ 0, the natural map

7We hope that the notation below will not lead to confusion with the semistable case in general but if in doubt we will add the data of the field K in the latter case.

H_{st}^{i}(G_{K}, V_{pst}(D)) →^{∼} H^{i}(G_{K}, V_{pst}(D))

is an isomorphism.

Proof. Both sides satisfy Galois descent for finite Galois extensions. We can assume, therefore, that D = D_{st}(V) for a semistable representation V of G_{K}. For i = 0 we have (even without assuming F^{1}D_{K} = 0) H^{0}(C_{st}(D)) = H^{0}(C(D)^{G_{K}}) = H^{0}(C(D))^{G_{K}} = V^{G_{K}}. For i = 1 the statement is proved in [6, Thm. 6.2, Lemme 6.5]. For i = 2 it follows from the assumption F^{1}D_{K} = 0 (by weak admissibility of D) that there is a W (k)-lattice M ⊂ D such that φ^{−1}(M) ⊂ p^{2}M, which implies that 1 − pφ = −pφ(1 − p^{−1}φ^{−1}) : D → D is surjective, hence H^{2}(C_{st}(D)) = 0 (cf. the proof of [6, Lemme 6.7]). The proof of the fact that H^{2}(G_{K}, V) = 0 if F^{1}D_{K} = 0 was kindly communicated to us by L. Berger; it is reproduced in Appendix A (cf. Theorem A.1). For i > 2 both terms vanish. □

2.5. Comparison of spectral sequences. The purpose of this subsection is to prove a derived category theorem (Theorem 2.18) that will be used later to relate the syntomic descent spectral sequence with the étale Hochschild-Serre spectral sequence (cf. Theorem 4.8). Let D be a triangulated category and H : D → A a cohomological functor to an abelian category A. A finite collection of adjacent exact triangles (a “Postnikov system” in the language of [37, IV.2, Ex. 2])

(14)  [figure: Y^{0}; Y^{1}; Y^{n}; X = X^{0}; X^{1}; [1]; X^{2}; [1]; X^{n}; X^{n+1} = 0; [1]]

gives rise to an exact couple

D_{1}^{p,q} = H^{q}(X^{p}) = H(X^{p}[q]), E_{1}^{p,q} = H^{q}(Y^{p}) ⟹ H^{p+q}(X).

The induced filtration on the abutment is given by

F^{p}H^{p+q}(X) = Im (D_{1}^{p,q} = H^{q}(X^{p}) → H^{p+q}(X)) .

Remark 2.17. In the special case when A is the heart of a non-degenerate t-structure (D^{≤n}, D^{≥n}) on D and H = τ_{≤0}τ_{≥0}, the following conditions are equivalent:

(1) E_{2}^{p,q} = 0 for p ≠ 0;

(2) D_{2}^{p,q} = 0 for all p, q;

(3) D_{r}^{p,q} = 0 for all p, q and r > 1;

(4) the sequence 0 → H^{q}(X^{p}) → H^{q}(Y^{p}) → H^{q}(X^{p+1}) → 0 is exact for all p, q;

(5) the sequence 0 → H^{q}(X) → H^{q}(Y^{0}) → H^{q}(Y^{1}) → ⋯ is exact for all q;

(6) the canonical map H^{q}(X) → E_{1}^{•,q} is a quasi-isomorphism, for all q;

(7) the triangle τ_{≤q}X^{p} → τ_{≤q}Y^{p} → τ_{≤q}X^{p+1} is exact for all p, q.

From now on until the end of 2.5 assume that D = D(A) is the derived category of A with the standard t-structure and that X^{i}, Y^{i} ∈ D^{+}(A), for all i. Furthermore, assume that f : A → A′ is a left exact functor to an abelian category A′ and that A admits a class of f-adapted objects (hence the derived functor Rf : D^{+}(A) → D^{+}(A′) exists).

Applying Rf to (14) we obtain another Postnikov system, this time in D^{+}(A′). The corresponding exact couple

(15)  ^{I}D_{1}^{p,q} = (R^{q}f)(X^{p}),  ^{I}E_{1}^{p,q} = (R^{q}f)(Y^{p}) ⟹ (R^{p+q}f)(X)

induces filtration

IF^{p}(R^{p+q}f)(X) = Im (^{I}D_{1}^{p,q} = (R^{q}f)(X^{p}) → (R^{p+q}f)(X)) .

Our goal is to compare (15), under the equivalent assumptions (2.17), to the hypercohomology exact couple

(16)  ^{II}D_{2}^{p,q} = (R^{p+q}f)(τ_{≤q−1}X),  ^{II}E_{2}^{p,q} = (R^{p}f)(H^{q}(X)) ⟹ (R^{p+q}f)(X)

for which

IIF^{p}(R^{p+q}f)(X) = Im (^{II}D_{2}^{p−1,q+1} = (R^{p+q}f)(τ_{≤q}X) → (R^{p+q}f)(X)) .

Theorem 2.18. Under the assumptions (2.17) there is a natural morphism of exact couples (u, v) : (^{I}D_{2},^{I}E_{2}) → (^{II}D_{2},^{II}E_{2}). Consequently, we have ^{I}F^{p} ⊆ ^{II}F^{p} for all p and there is a natural morphism of spectral sequences ^{I}E_{r}^{∗,∗} → ^{II}E_{r}^{∗,∗} (r > 1) compatible with the identity map on the common abutment.

Proof. Step 1: we begin by constructing a natural map u : ^{I}D_{2} → ^{II}D_{2}.

For each p > 0 there is a commutative diagram in D^{+}(A′)

\xymatrix{(R^{p+q}f)((τ_{≤q}Y^{p−1})[−p]) \ar[r] \ar[d]^{≀} & (R^{p+q}f)((τ_{≤q}X^{p})[−p]) \ar[r] \ar[d]^{≀} & (R^{p+q}f)(τ_{≤q}X) \ar[d]^{α_{II}} \\ ^{I}E_{1}^{p−1,q} = (R^{p+q}f)(Y^{p−1}[−p]) \ar[r]^{k_{1}} & ^{I}D_{1}^{p,q} = (R^{p+q}f)(X^{p}[−p]) \ar@{-->}[ur]^{u′} \ar[r]^{α_{I}} & (R^{p+q}f)(X)}

whose both rows are complexes. This defines a map u′ : ^{I}D_{1}^{p,q} → ^{II}D_{2}^{p−1,q+1} such that u′k_{1} = 0 and α_{II}u′ = α_{I} (hence ^{I}F^{p} = Im(α_{I}) ⊆ Im(α_{II}) = ^{II}F^{p}). By construction, the diagram (with exact top row)

\xymatrix{^{I}E_{1}^{p,q−1} \ar[r]^{k_{1}} \ar[dr]^{0} & ^{I}D_{1}^{p+1,q−1} \ar[r]^{i_{1}} \ar[d]^{u′} & ^{I}D_{1}^{p,q} \ar[d]^{u′} \\  & ^{II}D_{2}^{p,q} \ar[r]^{i_{2}} & ^{II}D_{2}^{p−1,q+1}}

is commutative for each p ≥ 0, which implies that the map

u = u′i_{1}^{−1} : ^{I}D_{2}^{p,q} = i_{1}(^{I}D_{1}^{p+1,q−1}) → ^{II}D_{2}^{p,q}

is well-defined and satisfies ui_{2} = i_{2}u. Step 2: for all q, the canonical quasi-isomorphism H^{q}(X) → E_{1}^{•,q} induces natural morphisms

v′ : ^{I}E_{2}^{p,q} = H^{p}(i ↦ (R^{q}f)(Y^{i})) → H^{p}(i ↦ f(H^{q}(Y^{i}))) → (R^{p}f)(i ↦ H^{q}(Y^{i}))

= (R^{p}f)(E_{1}^{•,q}) ⟵^{∼} (R^{p}f)(H^{q}(X)) = ^{II}E_{2}^{p,q};

set v = (−1)^{p}v′ : ^{I}E_{2}^{p,q} → ^{II}E_{2}^{p,q}.

It remains to show that u and v are compatible with the maps

?D_{2}^{p−1,q+1} ⟶^{j_{2}} ^{?}E_{2}^{p,q} ⟶^{k_{2}} ^{?}D_{2}^{p+1,q} (? = I, II).

Step 3: for any complex M^{•} over A denote by Z^{i}(M^{•}) = Ker(δ^{i} : M^{i} → M^{i+1}) the subobject of cycles in degree i.

If M^{•} is a resolution of an object M of A, then each exact sequence

(17)  0 ⟶ Z^{p}(M^{•}) ⟶ M^{p} ⟶^{δ^{p}} Z^{p+1}(M^{•}) ⟶ 0  (p ≥ 0)

can be completed to an exact sequence of resolutions

\xymatrix{0 \ar[r] & Z^{p}(M^{•}) \ar[r] \ar[d]^{can} & M^{p} \ar[r] \ar[d]^{can} & Z^{p+1}(M^{•}) \ar[r] \ar[d]^{− can} & 0 \\ 0 \ar[r] & (σ_{≥p}(M^{•}))[p] \ar[r] & (σ_{≥p}Cone(M^{•} →^{id} M^{•}))[p] \ar[r] & (σ_{≥p+1}(M^{•}))[p + 1] \ar[r] & 0.}

By induction, we obtain that the following diagram, whose top arrow is the composition of the natural maps Z^{i} → Z^{i−1}[1] induced by (17), commutes in D^{+}(A).

(18)  \xymatrix{Z^{p}(M^{•}) \ar[r] \ar[d]^{can} & Z^{0}(M^{•})[p] = M[p] \ar[d]^{(−1)^{p} can} \\ (σ_{≥p}(M^{•}))[p] \ar[r]^{can} & M^{•}[p]}

We are going to apply this statement to M = H^{q}(X) and M^{•} = E1^{•,q}, when Z^{p}(M^{•}) = D_{1}^{p,q} = H^{q}(X^{p}) and Z^{0}(M^{•}) = H^{q}(X). Step 4: we are going to investigate ^{I}E_{2}^{p,q}.

Complete the morphism Y^{p} → Y^{p+1} to an exact triangle U^{p} → Y^{p} → Y^{p+1} in D^{+}(A) and fix a lift X^{p} → U^{p} of the morphism X^{p} → Y^{p}.

There are canonical epimorphisms

(19)  (R^{q}f)(U^{p}) ↠ Ker((R^{q}f)(Y^{p}) ⟶^{j_{1}k_{1}} (R^{q}f)(Y^{p+1})) = Z^{p}(^{I}E1^{•,q}) ↠ ^{I}E_{2}^{p,q}

and the map

k_{2} : ^{I}E_{2}^{p,q} → ^{I}D_{2}^{p+1,q} = Ker(^{I}D_{1}^{p+1,q} ⟶^{j_{1}} ^{I}E_{1}^{p+1,q})

is induced by the restriction of k_{1} : ^{I}E_{1}^{p,q} → ^{I}D_{1}^{p+1,q} to Z^{p}(^{I}E1^{•,q}).

The following octahedron (in which we have drawn only the four exact faces)

\xymatrix{X^{p+2} \ar[dr]_{[1]} &  & Y^{p+1} \ar[ll] & X^{p+2} \ar[dd]_{[1]} &  & Y^{p+1} \ar[dl] \\  & X^{p+1} \ar@{-->}[ur] \ar[dl] &  &  & U^{p}[1] \ar@{-->}[ul] \ar@{-->}[dr]^{[1]} \\ X^{p}[1] \ar[rr]^{[1]} &  & Y^{p} \ar[ul] & X^{p}[1] \ar@{-->}[ur] &  & Y^{p} \ar[uu]}

shows that the triangle X^{p} → U^{p} → X^{p+2}[−1] is exact and the diagrams

\xymatrix{U^{p}[1] \ar[r] \ar[d] & Y^{p}[1] \ar[d] & (R^{q}f)(U^{p}) \ar[r] \ar[d] & Z^{p}(^{I}E1^{•,q}) \ar[d]_{k_{1}} \\ X^{p+2} \ar[r] & X^{p+1}[1] & (R^{q}f)(X^{p+2}[−1]) = ^{I}D_{2}^{p+2,q−1} \ar[r]^{i_{1}} & ^{I}D_{2}^{p+1,q}}

commute. The previous discussion implies that the composite map

(R^{q}f)(U^{p}) ↠ Z^{p}(^{I}E1^{•,q}) ↠ ^{I}E_{2}^{p,q} ⟶^{k_{2}} ^{I}D_{2}^{p+1,q} →^{u} ^{II}D_{2}^{p+1,q} = (R^{q}f)((τ_{≤q−1}X)[p + 1])

is obtained by applying R^{q}f to

(20)  τ_{≤q} U^{p} → τ_{≤q}(X^{p+2}[−1]) = (τ_{≤q−1}X^{p+2})[−1] → (τ_{≤q−1} X)[p + 1].

Step 5: all boundary maps H^{q}(X^{p+2}[−1]) → H^{q}(X^{p}) vanish by (2.17), which means that the following triangles are exact.

τ_{≤q} X^{p} → τ_{≤q} U^{p} → τ_{≤q}(X^{p+2}[−1]) = (τ_{≤q−1} X^{p+2})[−1]

The commutative diagram

[figure: τ_{≤q} U^{p}; H^{q}(U^{p})[−q]; Ker (H^{q}(Y^{p}) → H^{q}(Y^{p+1})) [−q]; τ_{≤q} X^{p}; H^{q}(X^{p})[−q]]

gives rise to an octahedron

\xymatrix{V^{p} \ar[dr]_{[1]} &  & H^{q}(X^{p})[−q] \ar[ll] & V^{p} \ar[dd]_{[1]} &  & H^{q}(X^{p})[−q] \ar@{-->}[dl] \\  & τ_{≤q} U^{p} \ar@{-->}[ur] \ar@{-->}[dl] &  &  & (τ_{≤q−1} X^{p})[1] \ar@{-->}[ul] \ar@{-->}[dr]^{[1]} \\ τ_{≤q}(X^{p+2}[−1]) \ar[rr]^{[1]} &  & τ_{≤q} X^{p} \ar@{-->}[ul] & X^{p}[1] \ar@{-->}[ur] &  & Y^{p} \ar[uu]}

In particular, the following diagram commutes.

(21)  \xymatrix{τ_{≤q} U^{p} \ar[r] \ar[d] & H^{q}(X^{p})[−q] \ar[d] \\ τ_{≤q}(X^{p+2}[−1]) \ar[r] & (τ_{≤q−1} X^{p})[1]}

Step 6: the diagram (18) implies that the composition of v : ^{I}E_{2}^{p,q} → ^{II}E_{2}^{p,q} with the second epimorphism in (19) is equal to the composite map

Z^{p}(^{I}E1^{•,q}) = Ker ((R^{q}f)(τ_{≤q} Y^{p}) → (R^{q}f)(τ_{≤q} Y^{p+1})) →

→ Ker ((R^{q}f)(H^{q}(Y^{p})[−q]) → (R^{q}f)(H^{q}(Y^{p+1})[−q])) =

= (R^{q}f)(Z^{p}(E1^{•,q})[−q]) → (R^{q}f)(Z^{0}(E1^{•,q})[−q + p]) = (R^{p}f)(H^{q}(X)) = ^{II}E_{2}^{p,q}.

As a result, the composition of v with (19) is obtained by applying R^{q}f to

(22)  τ_{≤q} U^{p} → H^{q}(X^{p})[q] → H^{q}(X)[−q + p].

Consequently, the composite map

ID_{1}^{p,q} = (R^{q}f)(τ_{≤q} X^{p}) ⟶^{j_{1}} Z^{p}(^{I}E1^{•,q}) ↠ ^{I}E_{2}^{p,q} →^{v} ^{II}E_{2}^{p,q}

is given by applying R^{q}f to

τ_{≤q} X^{p} → H^{q}(X^{p})[q] → H^{q}(X)[−q + p],

hence is equal to j_{2}u′. It follows that vj_{2} = vj_{1}i_{1}^{−1} = j_{2}u′i_{1}^{−1} = j_{2}u. Step 7: the diagram (21) implies that the map (20) coincides with the composition of (22) with the canonical map H^{q}(X)[−q + p] → (τ_{≤q−1} X)[p + 1], hence uk_{2} = k_{2}v. Theorem is proved. □

Example 2.19. If K^{•} is a bounded below filtered complex over A (with a finite filtration)

K^{•} = F^{0}K^{•} ⊃ F^{1}K^{•} ⊃ ⋯ ⊃ F^{n}K^{•} ⊃ F^{n+1}K^{•} = 0,

then the objects

X^{p} = F^{p}K^{•}[p], Y^{p} = (F^{p}K^{•}/F^{p+1}K^{•})[p] = gr_{F}^{p} (K^{•})[p] ∈ D^{+}(A)

form a Postnikov system of the kind considered in (14). The corresponding spectral sequences are equal to

E_{1}^{p,q} = H^{p+q}(gr_{F}^{p} (K^{•})) ⟹ H^{p+q}(K^{•}), ^{I}E_{1}^{p,q} = (R^{p+q}f)(gr_{F}^{p} (K^{•})) ⟹ (R^{p+q}f)(K^{•}).

In the special case when K^{•} is the total complex associated to a first quadrant bicomplex C^{•,•} and the filtration F^{p} is induced by the column filtration on C^{•,•}, then the complex f(K^{•}) over A′ is equipped with a canonical filtration (fF^{p})(f(K^{•})) = f(F^{p}K^{•}) satisfying

gr_{f(F)}^{p}(f(K^{•})) = f(gr_{F}^{p} (K^{•})).

Under the assumptions (2.17), the corresponding exact couple

fD_{1}^{p,q} = H^{p+q}(f(F^{p}K^{•})), ^{f}E_{1}^{p,q} = H^{p+q}(gr_{f(F)}^{p}(f(K^{•}))) = H^{p+q}(f(gr_{F}^{p} (K^{•}))) ⟹ H^{p+q}(f(K^{•}))

then naturally maps to the exact couple (15), hence (beginning from (D_{2}, E_{2})) to the exact couple (16), by Theorem 2.18.

3. Syntomic cohomology

In this section we will define the arithmetic and geometric syntomic cohomologies of varieties over K and \overline{K}, respectively, and study their basic properties.

3.1. Hyodo-Kato morphism revisited. We will need to use the Hyodo-Kato morphism on the level of derived categories and vary it in h-topology. Recall that the original morphism depends on the choice of a uniformizer and a change of such is encoded in a transition function involving exponential of the monodromy. Since the fields of definition of semistable models in the bases for h-topology change we will need to use these transitions functions. The problem though is that in the most obvious (i.e., crystalline) definition of the Hyodo-Kato complexes the monodromy is (at best) homotopically nilpotent - making the exponential in the transition functions impossible to define. Beilinson [4] solves this problem by representing Hyodo-Kato complexes using modules with nilpotent monodromy. In this subsection we will summarize what we need from his approach.

At first a quick reminder. Let (U, \overline{U}) be a log-scheme, log-smooth over V^{×}. For any r ≥ 0, consider its absolute (meaning over W (k)) log-crystalline cohomology complexes

RΓ_{cr}(U, \overline{U}, 𝒥^{[r]})_{n} : = RΓ(\overline{U}_{ét}, Ru_{U_{n}^{×}/W_{n}(k)∗}𝒥_{U_{n}^{×}/W_{n}(k)}^{[r]}), RΓ_{cr}(U, \overline{U}, 𝒥^{[r]}) := holim_{n} RΓ_{cr}(U, \overline{U}, 𝒥^{[r]})_{n},
RΓ_{cr}(U, \overline{U}, 𝒥^{[r]})_{Q} : = RΓ_{cr}(U, \overline{U}, 𝒥^{[r]}) ⊗ Q_{p},

where U^{×} denotes the log-scheme (U, \overline{U}) and u_{U_{n}^{×}/W_{n}(k)} : (U_{n}^{×}/W_{n}(k))_{cr} → \overline{U}_{ét} is the projection from

the log-crystalline to the étale topos. For r ≥ 0, we write 𝒥_{U_{n}^{×}/W_{n}(k)}^{[r]} for the r’th divided power of

the canonical PD-ideal 𝒥_{U_{n}^{×}/W_{n}(k)}; for r ≤ 0, we set 𝒥_{U_{n}^{×}/W_{n}(k)}^{[r]} := 𝒪_{U_{n}^{×}/W_{n}(k)} and we will often omit it from the notation. The absolute log-crystalline cohomology complexes are filtered E_{∞} algebras over W_{n}(k), W (k), or K_{0}, respectively. Moreover, the rational ones are filtered commutative dg algebras.

Remark 3.1. The canonical pullback map

RΓ(\overline{U}_{ét}, Ru_{U_{n}^{×}/W_{n}(k)∗}𝒥_{U_{n}^{×}/W_{n}(k)}^{[r]}) →^{∼} Ru_{U_{n}^{×}/Z/p^{n}∗}𝒥_{U_{n}^{×}/Z/p^{n}}^{[r]})

is a quasi-isomorphism. In what follows we will often call both the ”absolute crystalline cohomology”.

Let W (k) < t_{l} > be the divided powers polynomial algebra generated by elements t_{l}, l ∈ 𝔪_{K}/𝔪_{K}^{2} \{0}, subject to the relations t_{al} = [\overline{a}]t_{l}, for a ∈ V^{∗}, where [\overline{a}] ∈ W (k) is the Teichmüller lift of \overline{a} - the reduction mod 𝔪_{K} of a. Let R_{V} (or simply R) be the p-adic completion of the subalgebra of W (k) < t_{l} > generated by t_{l} and t_{l}^{ie_{K}} /i!, i ≥ 1. For a fixed l, the ring R is the following W (k)-subalgebra of K_{0}[[t_{l}]]:

R = \{∑_{i=0}^{∞} a_{i} \frac{t_{l}^{i}}{⌊i/e_{K}⌋!} | a_{i} ∈ W (k), lim_{i→∞} a_{i} = 0\}.

One extends the Frobenius φ_{R} (semi-linearly) to R by setting φ_{R}(t_{l}) = t_{l}^{p} and defines a monodromy operator N_{R} as a W (k)-derivation by setting N_{R}(t_{l}) = −t_{l}. Let E := Spec(R) equipped with the logstructure generated by the t_{l}’s.

We have two exact closed embeddings

i_{0} : W (k)^{0} ↪ E, i_{π} : V^{×} ↪ E.

The first one is canonical and induced by t_{l} ↦ 0. The second one depends on the choice of the class of the uniformizing parameter π ∈ 𝔪_{K}/p𝔪_{K} up to multiplication by Teichmüller elements. It is induced by t_{l} ↦ [\overline{l/π}]π.

Assume that (U, \overline{U}) is of Cartier type (i.e., the special fiber \overline{U}_{0} is of Cartier type). Consider the log-crystalline and the Hyodo-Kato complexes (cf. [4, 1.16])

RΓ_{cr}((U, \overline{U})/R, 𝒥^{[r]})_{n} := RΓ_{cr}((U, \overline{U})_{n}/R_{n}, 𝒥_{\overline{U}_{n}/R_{n}}^{[r]}), RΓ_{HK}(U, \overline{U})_{n} := RΓ_{cr}((U, \overline{U})_{0}/W_{n}(k)^{0}).

Let RΓ_{cr}((U, \overline{U})/R, 𝒥^{[r]}) and RΓ_{HK}(U, \overline{U}) be their homotopy inverse limits. The last complex is called the Hyodo-Kato complex. The complex RΓ_{cr}((U, \overline{U})/R) is R-perfect and

RΓ_{cr}((U, \overline{U})/R)_{n} ≃ RΓ_{cr}((U, \overline{U})/R) ⊗_{R}^{L} R_{n} ≃ RΓ_{cr}((U, \overline{U})/R) ⊗^{L} Z/p^{n}.

In general, we have RΓ_{cr}((U, \overline{U})/R, 𝒥^{[r]})_{n} ≃ RΓ_{cr}((U, \overline{U})/R, 𝒥^{[r]}) ⊗^{L} Z/p^{n}. The complex RΓ_{HK}(U, \overline{U}) is W (k)-perfect and

RΓ_{HK}(U, \overline{U})_{n} ≃ RΓ_{HK}(U, \overline{U}) ⊗_{W (k)}^{L} W_{n}(k) ≃ RΓ_{HK}(U, \overline{U}) ⊗^{L} Z/p^{n}.

We normalize the monodromy operators N on the rational complexes RΓ_{cr}((U, \overline{U})/R)_{Q} and RΓ_{HK}(U, \overline{U})_{Q} by replacing the standard N [40, 3.6] by N_{R} := e_{K}^{−1}N. This makes them compatible with base change. The embedding i_{0} : (U, \overline{U})_{0} ↪ (U, \overline{U}) over i_{0} : W_{n}(k)^{0} ↪ E_{n} yields compatible morphisms i_{0,n}^{∗} : RΓ_{cr}((U, \overline{U})/R)_{n} → RΓ_{HK}(U, \overline{U})_{n}. Completing, we get a morphism

i_{0}^{∗} : RΓ_{cr}((U, \overline{U})/R) → RΓ_{HK}(U, \overline{U}),

which induces a quasi-isomorphism i_{0}^{∗} : RΓ_{cr}((U, \overline{U})/R) ⊗_{R}^{L} W (k) →^{∼} RΓ_{HK}(U, \overline{U}). All the above objects have an action of Frobenius and these morphisms are compatible with Frobenius. The Frobenius action is invertible on RΓ_{HK}(U, \overline{U})_{Q}.

The map i_{0}^{∗} : RΓ_{cr}((U, \overline{U})/R)_{Q} → RΓ_{HK}(U, \overline{U})_{Q} admits a unique (in the classical derived category) W (k)-linear section ι_{π} [4, 1.16], [62, 4.4.6] that commutes with φ and N. The map ι_{π} is functorial and its R-linear extension is a quasi-isomorphism

ι_{π} : R ⊗_{W (k)} RΓ_{HK}(U, \overline{U})_{Q} →^{∼} RΓ_{cr}((U, \overline{U})/R)_{Q}.

The composition (the Hyodo-Kato map)

ι_{dR,π} := γ_{r}^{−1}i_{π}^{∗} · ι_{π} : RΓ_{HK}(U, \overline{U})_{Q} → RΓ_{dR}(U, \overline{U}_{K}),

where

γ_{r}^{−1} : RΓ_{cr}(U, \overline{U}, 𝒪/𝒥^{[r]})_{Q} →^{∼} RΓ_{dR}(U, \overline{U}_{K})/F^{r}

is the quasi-isomorphism from Corollary 2.4, induces a K-linear functorial quasi-isomorphism (the Hyodo-Kato quasi-isomorphism) [62, 4.4.8, 4.4.13]

(23)  ι_{dR,π} : RΓ_{HK}(U, \overline{U}) ⊗_{W (k)} K →^{∼} RΓ_{dR}(U, \overline{U}_{K})

We are going now to describe the Beilinson-Hyodo-Kato morphism and to study it on a few examples. Let S_{n} = Spec(Z/p^{n}) equipped with the trivial log-structure and let S = Spf(Z_{p}) be the induced formal log-scheme. For any log-scheme Y → S_{1} let D_{φ}((Y/S)_{cr}, 𝒪_{Y/S}) denote the derived category of Frobenius 𝒪_{Y/S}-modules and D_{φ}^{pcr}(Y/S) its thick subcategory of perfect F-crystals, i.e., those Frobenius modules that are perfect crystals [4, 1.11]. We call a perfect F-crystal (ℱ , φ) non-degenerate if the map Lφ^{∗}(ℱ) → ℱ is an isogeny. The corresponding derived category is denoted by D_{φ}^{pcr}(Y/S)^{nd}. It has a dg category structure [4, 1.14] that we denote by 𝒟_{φ}^{pcr}(Y/S)^{nd}. We will omit S if understood.

Suppose now that Y is a fine log-scheme that is affine. Assume also that there is a PD-thickening P = Spf R of Y that is formally smooth over S and such that R is a p-adically complete ring with no p-torsion. Let f : Z → Y be a log-smooth map of Cartier type with Z fine and proper over Y. Beilinson [4, 1.11,1.14] proves the following theorem.

Theorem 3.2. The complex ℱ := Rf_{cr ∗}(𝒪_{Z/S}) is a non-degenerate perfect F-crystal.

Let D_{φ,N}(K_{0}) denote the bounded derived category of (φ, N)-modules. By [4, 1.15], it has a dg category structure that we will denote by 𝒟_{φ,N}(K_{0}). We call (φ, N)-module effective if it contains a W (k)-lattice preserved by φ and N. Denote by 𝒟_{φ,N}(K_{0})^{eff} ⊂ 𝒟_{φ,N}(K_{0}) the bounded derived category of the abelian category of effective modules.

Let f : Y → k^{0} be a log-scheme. We think of k^{0} as W (k)_{1}^{×}. Then the map f is given by a k-structure on Y plus a section l = f^{∗}(\overline{p}) ∈ Γ(Y, M_{Y}) such that its image in Γ(Y, 𝒪_{Y}) equals 0. We will often write f = f_{l}, l = l_{f}.

Beilinson proves the following theorem [4, 1.15].

Theorem 3.3. (1) There is a natural functor

(24)  ε_{f} = ε_{l} : 𝒟_{φ,N}(K_{0})^{eff} → 𝒟_{φ}^{pcr}(Y)^{nd} ⊗ Q.

(2) ε_{f} is compatible with base change, i.e., for any θ : Y′ → Y one has a canonical identification ε_{fθ} →^{∼} Lθ_{cr}^{∗}ε_{f}. For any a ∈ k^{∗}, m ∈ Z_{>0}, there is a canonical identification ε_{al^{m}}(V, φ, N) →^{∼} ε_{l}(V, φ, mN).

(3) Suppose that Y is a local scheme with residue field k and nilpotent maximal ideal, M_{Y} /𝒪_{Y}^{∗} = Z_{>0}, and the map f^{∗} : M_{k^{0}}/k^{∗} → M_{Y} /𝒪_{Y}^{∗} is injective. Then (24) is an equivalence of dg categories.

In particular, we have an equivalence of dg categories

ε := ε\overline{p} : 𝒟_{φ,N}(K_{0})^{eff} →^{∼} 𝒟_{φ}^{pcr}(k^{0})^{nd} ⊗ Q

and a canonical identification ε_{f} = Lf_{cr}^{∗}ε.

On the level of sections the functor (24) has a simple description [4, 1.15.3]. Assume that Y = Spec(A/J), where A is a p-adic algebra and J is a PD-ideal in A, and that we have a PD-thickening i : Y ↪ T = Spf(A). Let λ_{l,n} be the preimage of l under the map Γ(T_{n}, M_{T_{n}}) → i_{∗}Γ(Y, M_{Y}). It is a trivial (1 + J_{n})^{×}-torsor. Set λ_{A} := \underleftarrow{lim}_{n}Γ(T_{n}, λ_{l,n}). It is a (1 + J)^{×}-torsor. Let τ_{A_{Q}} be the Fontaine-

Hyodo-Kato torsor: A_{Q}-torsor obtained from λ_{A} by the pushout by (1 + J)^{×} →^{log} J → A_{Q}. We call the 𝔾_{a}-torsor Spec A_{Q}^{τ} over Spec A_{Q} with sections τ_{A_{Q}} the same name. Denote by N_{τ} the A_{Q}-derivation of A_{Q}^{τ} given by the action of the generator of Lie_{𝔾_{a}}.

Let M be an (φ, N)-module. Integrating the action of the monodromy N_{M} we get an action of the group 𝔾_{a} on M. Denote by M_{A_{Q}}^{τ} the τ_{A_{Q}}-twist of M_{A_{Q}} := M ⊗_{K_{0}} A_{Q}. It can be represented as the module of maps v : τ_{A_{Q}} → M_{A_{Q}} that are A_{Q}-equivariant, i.e., such that v(τ + a) = exp(aN)(v(τ)), τ ∈ τ_{A_{Q}}, a ∈ A_{Q}. We can also write

M_{A_{Q}}^{τ} = (M ⊗_{K_{0}} A_{Q}^{τ})^{𝔾_{a}} = (M ⊗_{K_{0}} A_{Q}^{τ})^{N=0},

where N := N_{M} ⊗ 1 + 1 ⊗ N_{τ}. Now, by definition,

(25)  ε_{f}(M)(Y, T) = M_{A_{Q}}^{τ}

The algebra A_{Q}^{τ} has a concrete description. Take the natural map a : τ_{A_{Q}} → A_{Q}^{τ} of A_{Q}-torsors which maps τ ∈ τ_{A_{Q}} to a function a(τ) ∈ A_{Q}^{τ} whose value on any τ′ ∈ τ_{A_{Q}} is τ − τ′ ∈ A_{Q}. This map is compatible with the logarithm log : (1 + J)^{×} → A. The algebra A_{A_{Q}}^{τ} is freely generated over A_{Q} by a(τ) for any τ ∈ τ_{A_{Q}}; the A_{Q}-derivation N_{τ} is defined by N_{τ} (a(τ)) = −1. That is, for chosen τ ∈ τ_{A_{Q}}, we can write

A_{Q}^{τ} = A_{Q}[a(τ)], N_{τ} (a(τ)) = −1

For every lifting φ_{T} of Frobenius to T we have φ_{T}^{∗} λ_{A} = λ_{A}^{p}. Hence Frobenius φ_{T} extends canonically to a Frobenius φ_{τ} on A_{Q}^{τ} in such a way that N_{τ} φ_{τ} = pφ_{τ} N_{τ}. The isomorphism (25) is compatible with Frobenius.

Example 3.4. As an example, consider the case when the pullback map f^{∗} : Q = (M_{k^{0}}/k^{∗})^{gp} ⊗ Q →^{∼} (Γ(Y, M_{Y})/k^{∗})^{gp} ⊗ Q is an isomorphism. We have a surjection v : (Γ(T, M_{T})/k^{∗})^{gp} ⊗ Q → Q with the kernel log : (1 + J)_{Q}^{×} →^{∼} J_{Q} = A_{Q}. We obtain an identification of A_{Q}-torsors τ_{A_{Q}} ≃ v^{−1}(1). Hence every non-invertible t ∈ Γ(T, M_{T}) yields an element t^{1/v(t)} ∈ v^{−1}(1) and a trivialization of τ_{A_{Q}}.

For a fixed element t^{1/v(t)} ∈ v^{−1}(1), we can write

A_{A_{Q}}^{τ} = A_{Q}[a(t^{1/v(t))}], N_{τ} (a(t^{1/v(t))}) = −1

For an (φ, N)-module M, the twist M_{A_{Q}}^{τ} can be trivialized

β_{t} : M ⊗_{K_{0}} A_{Q} →^{∼} M_{A_{Q}}^{τ} = (M ⊗_{K_{0}} A_{Q}[a(t^{1/v(t)})])^{N=0}
m ↦ exp(N_{M}(m)a(t^{1/v(t))})

For a different choice t_{1}^{1/v(t_{1})} ∈ v^{−1}(1), the two trivializations β_{t}, β_{t_{1}} are related by the formula

β_{t_{1}} = β_{t} exp(N_{M}(m)a(t_{1}, t)), a(t_{1}, t) = a(t_{1})/v(t_{1}) − a(t)/v(t).

Consider the map f : V_{1}^{×} → k^{0}. By Theorem 3.3, we have the equivalences of dg categories

ε : 𝒟_{φ,N}(K_{0})^{eff} →^{∼} 𝒟_{φ}^{pcr}(k^{0})^{nd} ⊗ Q,
ε_{f} = Lf_{cr}^{∗}ε : 𝒟_{φ,N}(K_{0})^{eff} →^{∼} 𝒟_{φ}^{pcr}(V_{1}^{×})^{nd} ⊗ Q

Let Z_{1} → V_{1}^{×} be a log-smooth map of Cartier type with Z_{1} fine and proper over V_{1}. By Theorem 3.2 Rf_{cr ∗}(𝒪_{Z_{1}/Z_{p}}) is a non-degenerate perfect F-crystal on V_{1,cr}. Set

RΓ_{HK}^{B}(Z_{1}) := ε_{f}^{−1}Rf_{cr ∗}(𝒪_{Z_{1}/Z_{p}})_{Q} ∈ 𝒟_{φ,N}(K_{0}).

We will call it the Beilinson-Hyodo-Kato complex [4, 1.16.1].

Example 3.5. To get familiar with the Beilinson-Hyodo-Kato complexes we will work out some examples.

(1) Let g : X → V^{×} be a log-smooth log-scheme, proper, and of Cartier type. Adjunction yields a quasi-isomorphism

(26) ε_{f}RΓ_{HK}^{B}(X_{1}) = ε_{f}ε_{f}^{−1}Rg_{cr ∗}(𝒪_{X_{1}/Z_{p}})_{Q} →^{∼} Rg_{cr ∗}(𝒪_{X_{1}/Z_{p}})_{Q}

Evaluating it on the PD-thickening V_{1}^{×} ↪ V^{×} (here A = V, J = pV, l = \overline{p}, λ_{V} = p(1 + J)^{×}, τ_{K} = p(1 + J)^{×} ×_{(1+J)^{×}} K), we get a map

RΓ_{HK}^{B}(X_{1})_{K}^{τ} = ε_{f}RΓ_{HK}^{B}(X_{1})(V_{1}^{×} ↪ V^{×}) →^{∼} Rg_{cr ∗}(𝒪_{X_{1}/Z_{p}})(V_{1}^{×} ↪ V^{×})_{Q} = RΓ_{cr}(X_{1}/V^{×})_{Q}
≃ RΓ_{cr}(X/V^{×})_{Q} ≃ RΓ_{dR}(X_{K})

We will call it the Beilinson-Hyodo-Kato map [4, 1.16.3]

(27)  ι_{dR}^{B} : RΓ_{HK}^{B}(X_{1})_{K}^{τ} →^{∼} RΓ_{dR}(X_{K})

Recall that

RΓ_{HK}^{B}(X_{1})_{K}^{τ} = (RΓ_{HK}^{B}(X_{1}) ⊗_{K_{0}} K[a(τ)])^{N=0}, τ ∈ τ_{K}

This makes it clear that the Beilinson-Hyodo-Kato map is not only functorial for log-schemes over V^{×} but, by Theorem 3.3, it is also compatible with base change of V^{×}. Moreover, if we use the canonical trivialization by p

β = β_{p} : RΓ_{HK}^{B}(X_{1})_{K} →^{∼} RΓ_{HK}^{B}(X)_{K}^{τ} = (RΓ_{HK}^{B}(X_{1}) ⊗_{K_{0}} K[a(p)])^{N=0}
x ↦ exp(N(x)a(p))

we get that the composition (which we we also call the Beilinson-Hyodo-Kato map and denote by ι_{dR}^{B})

ι_{dR}^{B} = ι_{dR}^{B}β : RΓ_{HK}^{B}(X_{1}) → RΓ_{dR}(X_{K})

is functorial and compatible with base change.

(2) Evaluating the map (26) on the PD-thickening V_{1}^{×} ↪ E associated to a uniformizer π (here A = R, l = \overline{p}), we get a map

(28)  κ_{R} : RΓ_{HK}^{B}(X_{1})_{R_{Q}}^{τ} →^{∼} RΓ_{cr}(X/R)_{Q}

as the composition

RΓ_{HK}^{B}(X_{1})_{R_{Q}}^{τ} = ε_{f}RΓ_{HK}^{B}(X_{1})(V_{1}^{×} ↪ E) →^{∼} Rg_{cr ∗}(𝒪_{X_{1}/Z_{p}})(V_{1}^{×} ↪ E)_{Q} = RΓ_{cr}(X_{1}/R)_{Q}
≃ RΓ_{cr}(X/R)_{Q}

We have

RΓ_{HK}^{B}(X_{1})_{R_{Q}}^{τ} = (RΓ_{HK}^{B}(X_{1}) ⊗_{K_{0}} R_{Q}[a(τ)])^{N=0}, τ ∈ τ_{R_{Q}}

Since the map κ_{R} is compatible with the log-connection on R it is also compatible with the normalized monodromy operators. Specifically, if we define the monodromy on the left hand side of (28) as

N : RΓ_{HK}^{B}(X_{1})_{R_{Q}}^{τ} → RΓ_{HK}^{B}(X_{1})_{R_{Q}}^{τ},
∑_{I} m_{τ_{I}} ⊗ r_{τ_{I}} a^{k_{I}} (τ_{I}) ↦ ∑_{I}(N_{M}(m_{τ_{I}}) ⊗ r_{τ_{I}} a^{k_{I}} (τ_{I}) + m_{τ_{I}} ⊗ N_{R}(r_{τ_{I}})a^{k_{I}} (τ_{I}))

the two operators will correspond under the map κ_{R}.

The exact immersion i_{π} : V^{×} ↪ E, yields a commutative diagram

\xymatrix{RΓ_{HK}^{B}(X_{1})_{R_{Q}}^{τ} \ar[r]^{∼} \ar[d]^{i_{π}^{∗}} & RΓ_{cr}(X/R)_{Q} \ar[d]^{i_{π}^{∗}} \\ RΓ_{HK}^{B}(X_{1})_{K}^{τ} \ar[r]^{∼} & RΓ_{cr}(X/V^{×})_{Q}}

If p = uπ^{e_{K}}, u ∈ V^{×}, we have λ_{R} = ũt_{π}^{e_{K}} (1+J)^{×}, where ũ ∈ R is such that ũ lifts u. Alternatively, λ_{R} = [\overline{u}]t_{π}^{e_{K}} (1 + J)^{×}. We have the associated trivialization

β_{π} : RΓ_{HK}^{B}(X_{1}) ⊗_{K_{0}} R_{Q} →^{∼} RΓ_{HK}^{B}(X_{1})_{R_{Q}}^{τ} = (RΓ_{HK}^{B}(X_{1}) ⊗_{K_{0}} R_{Q}[a(τ_{π})])^{N=0}, τ_{π} := [\overline{u}]t_{π}^{e_{K}} ,
x ↦ exp(N(x)a(τ_{π}))

(3) Consider the log-scheme k_{1}^{0}: the scheme Spec(k) with the log-structure induced by the exact closed immersion i : k_{1}^{0} ↪ V_{1}^{×}. We have the commutative diagram

\xymatrix{X_{0} \ar@{^{(}->}[r]^{i} \ar[d]^{g_{0}} & X_{1} \ar[d]^{g} \\ k_{1}^{0} \ar[dr]_{f_{0}} \ar@{^{(}->}[r]^{i} & V_{1}^{×} \ar[d]^{f} \\  & k^{0}}

The morphisms f, f_{0} map \overline{p} to \overline{p}. By log-smooth base change we have a canonical quasiisomorphism Li^{∗}Rg_{cr ∗}(𝒪_{X_{1}/Z_{p}}) ≃ Rg_{0 cr ∗}(𝒪_{X_{1}/Z_{p}}). By Theorem 3.3 we have the equivalence of dg categories

ε_{f_{0}} : 𝒟_{φ,N}(K_{0})^{eff} →^{∼} 𝒟_{φ}^{pcr}(k_{1}^{0})^{nd} ⊗ Q, ε_{f_{0}} = Li^{∗}ε_{f}.

This implies the natural quasi-isomorphisms

RΓ_{HK}^{B}(X_{1}) = ε_{f}^{−1}Rg_{cr ∗}(𝒪_{X_{1}/Z_{p}})_{Q} ≃ ε_{f_{0}}^{−1}Li^{∗}Rg_{cr ∗}(𝒪_{X_{1}/Z_{p}})_{Q}
≃ ε_{f_{0}}^{−1}Rg_{0 cr ∗}(𝒪_{X_{0}/Z_{p}})_{Q}

Hence, by adjunction,

ε_{f_{0}}RΓ_{HK}^{B}(X_{1}) = ε_{f_{0}}ε_{f_{0}}^{−1}Rg_{0 cr ∗}(𝒪_{X_{0}/Z_{p}})_{Q} ≃ Rg_{0 cr ∗}(𝒪_{X_{0}/Z_{p}})_{Q}

We will evaluate both sides on the PD-thickening k_{1}^{0} ↪ W (k)^{0}. Here we write the log-structure on W (k)^{0} as associated to the map Γ(V^{×}, M_{V^{×}}) → k → W (k), a ↦ \overline{a}. We take A = W (k), l = p, J = pW (k), λ_{W (k)} = \overline{p}(1 + pW (k))^{×}, τ_{K_{0}} = \overline{p}(1 + pW (k))^{×} ×_{(1+pW (k))^{×}} K_{0}. We get a quasi-isomorphism

κ : RΓ_{HK}^{B}(X_{1})_{K_{0}}^{τ} →^{∼} RΓ_{HK}(X)_{Q}

as the composition

RΓ_{HK}^{B}(X_{1})_{K_{0}}^{τ} = ε_{f_{0}}RΓ_{HK}^{B}(X_{1})(k_{1}^{0} ↪ W (k)^{0}) ≃ Rg_{0 cr ∗}(𝒪_{X_{0}/Z_{p}})(k_{1}^{0} ↪ W (k)^{0})_{Q}
= RΓ_{cr}(X_{0}/W (k)^{0})_{Q} = RΓ_{HK}(X)_{Q}

To compare the monodromy operators on both sides of the map κ, note that by Theorem 3.3, we have the canonical identification

Rg_{0 cr ∗}(𝒪_{X_{0}/Z_{p}})_{Q} ≃ ε_{f_{0}}(RΓ_{HK}^{B}(X_{1}), N) ≃ ε\overline{p}(RΓ_{HK}^{B}(X_{1}), e_{K}N)

Hence, from the description of the Hyodo-Kato monodromy in [40, 3.6], it follows easily that the map κ pairs the operator N on RΓ_{HK}^{B}(X_{1})_{K_{0}}^{τ} defined by

N(∑_{I} m_{τ_{I}} ⊗ r_{τ_{I}} a^{k_{I}} (τ_{I})) = ∑_{I}(N_{M}(m_{τ_{I}}) ⊗ r_{τ_{I}} a^{k_{I}} (τ_{I}) + m_{τ_{I}} ⊗ N_{R}(r_{τ_{I}})a^{k_{I}} (τ_{I})),

with the normalized Hyodo-Kato monodromy on RΓ_{HK}(X)_{Q}.

Composing the map κ with the trivialization

β = β_{p} : RΓ_{HK}^{B}(X_{1}) →^{∼} RΓ_{HK}^{B}(X_{1})_{K_{0}}^{τ} = (RΓ_{HK}^{B}(X_{1})[a(\overline{p})])^{N=0}
x ↦ exp(N(x)a(\overline{p}))

we get a quasi-isomorphism between Beilinson-Hyodo-Kato complexes and the (classical) Hyodo-Kato complexes.

(29) κ = βκ : RΓ_{HK}^{B}(X_{1}) →^{∼} RΓ_{HK}(X)_{Q}

The trivialization above is compatible with Frobenius and the normalized monodromy hence so is the quasi-isomorphism (29). It is clearly functorial and, by Theorem 3.3, compatible with base change.

By functoriality (Theorem 3.3), the morphism of PD-thickenings (exact closed immersion) i_{0} : (k_{1}^{0} ↪ W (k)^{0}) ↪ (V_{1}^{×} ↪ R) yields the right square in the following diagram

(30)  \xymatrix{RΓ_{HK}(X)_{Q} \ar[r]^{ι_{π}} & RΓ_{cr}(X_{1}/R)_{Q} \ar[r]^{i_{0}^{∗}} & RΓ_{HK}(X)_{Q} \\ RΓ_{HK}^{B}(X_{1})_{K_{0}}^{τ} \ar[u]^{≀}_{κ} \ar[r]^{ι_{π}} & RΓ_{HK}^{B}(X_{1})_{R_{Q}}^{τ} \ar[u]^{≀}_{κ_{R}} \ar[r]^{i_{0}^{∗}} & RΓ_{HK}^{B}(X_{1})_{K_{0}}^{τ} \ar[u]^{≀}_{κ}}

In the left square the bottom map ι_{π} is induced by the natural map K_{0} → R and by sending a(\overline{p}) ↦ a(τ_{π}). It is a (right) section to i_{0}^{∗} and it (together with the vertical maps) commutes with Frobenius. By uniqueness of the top map ι_{π} this makes the left square commute in the classical derived category (of abelian groups).

It is easy to check that we have the following commutative diagram

\xymatrix{RΓ_{HK}^{B}(X_{1})_{K_{0}}^{τ} \ar[r]^{ι_{π}} & RΓ_{HK}^{B}(X_{1})_{R_{Q}}^{τ} \ar[r]^{i_{π}^{∗}} & RΓ_{HK}^{B}(X_{1})_{K}^{τ} \\ RΓ_{HK}^{B}(X_{1}) \ar[u]^{β_{p}}_{≀} \ar[rr]^{can} &  & RΓ_{HK}^{B}(X_{1})_{K} \ar[u]^{β_{p}}_{≀}}

and that the composition of maps on the top of it is equal to the map induced by the canonical map K_{0} → K and the map λ_{W (k)^{0}} → λ_{V}^{×}, \overline{p} → p.

Combining the commutative diagrams in parts (2) and (3) of this example we get the following commutative diagram.

\xymatrix{RΓ_{HK}(X) \ar[r]^{ι_{π}} & RΓ_{cr}(X_{1}/R)_{Q} \ar[r]^{i_{π}^{∗}} & RΓ_{cr}(X_{1}/V^{×})_{Q} \\ RΓ_{HK}^{B}(X_{1})_{K_{0}}^{τ} \ar[u]^{≀}_{κ} \ar[r]^{ι_{π}} & RΓ_{HK}^{B}(X_{1})_{R_{Q}}^{τ} \ar[r]^{i_{π}^{∗}} \ar[u]^{≀}_{κ_{R}} & RΓ_{HK}^{B}(X_{1})_{K}^{τ} \ar[u]^{≀}_{ι_{dR}^{B}} \\ RΓ_{HK}^{B}(X_{1}) \ar[u]^{β_{p}}_{≀} \ar[rr]^{can} &  & RΓ_{HK}^{B}(X_{1})_{K} \ar[u]^{β_{p}}_{≀}}

Since the composition of the top maps is equal to the Hyodo-Kato map ι_{dR} and the bottom maps is just the canonical map RΓ_{HK}(X_{1}) → RΓ_{HK}(X_{1})_{K} we obtain that the Hyodo-Kato and the Beilinson-Hyodo-Kato maps are related by a natural quasi-isomorphism, i.e., that the following diagram commutes.

(31)  \xymatrix{RΓ_{HK}(X) \ar[r]^{ι_{dR,π}} & RΓ_{dR}(X_{K}) \\ RΓ_{HK}^{B}(X_{1}) \ar[ur]_{ι_{dR}^{B}} \ar[u]^{≀}_{κ}}

The above examples can be generalized [4, 1.16]. It turns out that the relative crystalline cohomology of all the base changes of the map f can be described using the Beilinson-Hyodo-Kato complexes [4, 1.16.2]. Namely, let θ : Y → V_{1}^{×} be an affine log-scheme and let T be a p-adic PD-thickening of Y, T = Spf(A), Y = Spec(A/J). Denote by f_{Y} : Z_{1Y} → Y the θ-pullback of f. Beilinson proves the following theorem [4, 1.16.2].

Theorem 3.6. (1) The A-complex RΓ_{cr}(Z_{1Y} /T, 𝒪_{Z_{1Y /T}}) is perfect, and one has

RΓ_{cr}(Z_{1Y} /T_{n}, 𝒪_{Z_{1Y /Tn}}) = RΓ_{cr}(Z_{1Y} /T, 𝒪_{Z_{1Y /T}}) ⊗^{L} Z/p^{n}.

(2) There is a canonical Beilinson-Hyodo-Kato quasi-isomorphism of A_{Q}-complexes

κ_{A_{Q}}^{B} : RΓ_{HK}^{B}(Z_{1})_{A_{Q}}^{τ} →^{∼} RΓ_{cr}(Z_{1Y} /T, 𝒪_{Z_{1Y /T}})_{Q}

If there is a Frobenius lifting φ_{T}, then κ_{A_{Q}}^{B} commutes with its action.

3.2. Log-syntomic cohomology. We will study now (rational) log-syntomic cohomology. Let (U, \overline{U}) be log-smooth over V^{×}. For r ≥ 0, define the mod p^{n}, completed, and rational log-syntomic complexes

(32)  RΓ_{syn}(U, \overline{U}, r)_{n} := Cone(RΓ_{cr}(U, \overline{U}, 𝒥^{[r]})_{n}⟶^{p^{r}−φ}RΓ_{cr}(U, \overline{U})_{n})[−1],
RΓ_{syn}(U, \overline{U}, r) := holim_{n} RΓ_{syn}(U, \overline{U}, r)_{n},
RΓ_{syn}(U, \overline{U}, r)_{Q} := Cone(RΓ_{cr}(U, \overline{U}, 𝒥^{[r]})_{Q}⟶^{1−φ_{r}}RΓ_{cr}(U, \overline{U})_{Q})[−1].

Here the Frobenius φ is defined by the composition

φ : RΓ_{cr}(U, \overline{U}, 𝒥^{[r]})_{n} → RΓ_{cr}(U, \overline{U})_{n} →^{∼} RΓ_{cr}((U, \overline{U})_{1}/W (k))_{n} →^{φ} RΓ_{cr}((U, \overline{U})_{1}/W (k))_{n}

←^{∼} RΓ_{cr}(U, \overline{U})_{n}

and φ_{r} := φ/p^{r}. The mapping fibers are taken in the ∞-derived category of abelian groups. The direct sums

⨁_{r≥0} RΓ_{syn}(U, \overline{U}, r)_{n}, ⨁_{r≥0} RΓ_{syn}(U, \overline{U}, r), ⨁_{r≥0} RΓ_{syn}(U, \overline{U}, r)_{Q}

are graded E_{∞} algebras over Z/p^{n}, Z_{p}, and Q_{p}, respectively [39, 1.6]. The rational log-syntomic complexes are moreover graded commutative dg algebras over Q_{p} [39, 4.1], [38, 3.22], [46]. Explicit definition of syntomic product structure can be found in [62, 2.2].

We have RΓ_{syn}(U, \overline{U}, r)_{n} ≃ RΓ_{syn}(U, \overline{U}, r) ⊗^{L} Z/p^{n}. There is a canonical quasi-isomorphism of graded E_{∞} algebras

RΓ_{syn}(U, \overline{U}, r)_{n} →^{∼} Cone(RΓ_{cr}(U, \overline{U})_{n} ⟶^{(p^{r}−φ,can)}RΓ_{cr}(U, \overline{U})_{n} ⊕ RΓ_{cr}(U, \overline{U}, 𝒪/𝒥^{[r]})_{n})[−1].

Similarly in the completed and rational cases.

Since, by Corollary 2.4, there is a quasi-isomorphism

γ_{r}^{−1} : RΓ_{cr}(U, \overline{U}, 𝒪/𝒥^{[r]})_{Q} →^{∼} RΓ_{dR}(U, \overline{U}_{K})/F^{r},

we have a particularly nice canonical description of rational log-syntomic cohomology

RΓ_{syn}(U, \overline{U}, r)_{Q} →^{∼} [RΓ_{cr}(U, \overline{U})_{Q} ⟶^{(1−φ_{r},γ_{r}^{−1})}RΓ_{cr}(U, \overline{U})_{Q} ⊕ RΓ_{dR}(U, \overline{U}_{K})/F^{r})],

where square brackets stand for mapping fiber.

Remark 3.7. In the above definition one can replace the map 1 − φ_{r} with any polynomial map P ∈ 1 + XK[X] to obtain the analog of Besser’s finite polynomial cohomology. This was studied in [10].

For arithmetic pairs (U, \overline{U}) that are log-smooth over V^{×} and of Cartier type this can be simplified further by using Hyodo-Kato complexes (cf. Proposition 3.8 below). To do that, consider the following sequence of maps of homotopy limits. Homotopy limits are taken in the ∞-derived category (to do that we define the maps ι_{π} by the zigzag from diagram (30)). We will describe the coherence data only if they are nonobvious.

RΓ_{syn}(U, \overline{U}, r)_{Q} →^{∼} \xymatrix{[RΓ_{cr}(U, \overline{U})_{Q} \ar[r]^{(1−φ_{r},γ_{r}^{−1})} & RΓ_{cr}(U, \overline{U})_{Q} ⊕ RΓ_{dR}(U, \overline{U}_{K})/F^{r}]}

→^{∼} [ \xymatrix{RΓ_{cr}((U, \overline{U})/R)_{Q} \ar[r]^{(1−φ_{r},i_{π}^{∗}γ_{r}^{−1})} \ar[d]^{N} & RΓ_{cr}((U, \overline{U})/R)_{Q} ⊕ RΓ_{dR}(U, \overline{U}_{K})/F^{r} \ar[d]^{(N,0)} \\ RΓ_{cr}((U, \overline{U})/R)_{Q} \ar[r]^{1−φ_{r−1}} & RΓ_{cr}((U, \overline{U})/R)_{Q}} ]

←^{ι_{π}} [ \xymatrix{RΓ_{HK}(U, \overline{U})_{Q} \ar[r]^{(1−φ_{r},ι_{dR,π})} \ar[d]^{N} & RΓ_{HK}(U, \overline{U})_{Q} ⊕ RΓ_{dR}(U, \overline{U}_{K})/F^{r} \ar[d]^{(N,0)} \\ RΓ_{HK}(U, \overline{U})_{Q} \ar[r]^{1−φ_{r−1}} & RΓ_{HK}(U, \overline{U})_{Q}} ]

The first map was described above. The second one is induced by the distinguished triangle

RΓ_{cr}(U, \overline{U}) → RΓ_{cr}((U, \overline{U})/R) →^{N} RΓ_{cr}((U, \overline{U})/R)

The third one - by the section ι_{π} : RΓ_{HK}(U, \overline{U})_{Q} → RΓ_{cr}((U, \overline{U})/R)_{Q} (notice that ι_{dR,π} = γ_{r}^{−1}i_{π}^{∗}ι_{π}). We will show below that the third map is a quasi-isomorphism.

Set C_{st}(RΓ_{HK}(U, \overline{U})\{r\}) equal to the last homotopy limit in the above diagram.

Proposition 3.8. Let (U, \overline{U}) be an arithmetic pair that is log-smooth over V^{×} and of Cartier type. Let r ≥ 0. Then the above diagram defines a canonical quasi-isomorphism.

α_{syn,π} : RΓ_{syn}(U, \overline{U}, r)_{Q} →^{∼} C_{st}(RΓ_{HK}(U, \overline{U})\{r\}).

Proof. We need to show that the map ι_{π} in the above diagram is a quasi-isomorphism. Define complexes (r ≥ −1)

RΓ_{cr}((U, \overline{U})/R, r) := Cone(RΓ_{cr}((U, \overline{U})/R)_{Q} ⟶^{1−φ_{r}} RΓ_{cr}((U, \overline{U})/R)_{Q})[−1],
RΓ_{HK}(U, \overline{U}, r) := Cone(RΓ_{HK}(U, \overline{U})_{Q} ⟶^{1−φ_{r}} RΓ_{HK}(U, \overline{U})_{Q})[−1]

It suffices to prove that the following maps

(33)  i_{0}^{∗} : RΓ_{cr}((U, \overline{U})/R, r) →^{∼} RΓ_{HK}(U, \overline{U}, r), ι_{π} : RΓ_{HK}(U, \overline{U}, r) →^{∼} RΓ_{cr}((U, \overline{U})/R, r)

are quasi-isomorphisms. Since i_{0}^{∗}ι_{π} = Id, it suffices to show that the map i_{0}^{∗} is a quasi-isomorphism. Base-changing to W (\overline{k}), we may assume that the residue field of V is algebraically closed. It suffices to show that, for i ≥ 0, t ≥ −1, in the commutative diagram

H_{HK}^{i}(U, \overline{U})_{Q} ⟶^{p^{t}−φ} H_{HK}^{i}(U, \overline{U})_{Q}
↑^{i_{0}^{∗}}  ↑^{i_{0}^{∗}}
H_{cr}^{i}((U, \overline{U})/R)_{Q} ⟶^{p^{t}−φ} H_{cr}^{i}((U, \overline{U})/R)_{Q}

the vertical maps induce isomorphisms between the kernels and cokernels of the horizontal maps. Since the W (k)-linear map ι_{π} commutes with φ and its R-linear extension is a quasi-isomorphism

ι_{π} : R ⊗_{W (k)} RΓ_{HK}(U, \overline{U})_{Q} →^{∼} RΓ_{cr}((U, \overline{U})/R)_{Q}

it suffices to show that in the following commutative diagram

H_{HK}^{i}(U, \overline{U})_{Q}  ⟶^{p^{t}−φ}  H_{HK}^{i}(U, \overline{U})_{Q}
↑^{i_{0}⊗Id}  ↑^{i_{0}⊗Id}
R ⊗_{W (k)} H_{HK}^{i}(U, \overline{U})_{Q} ⟶^{p^{t}−φ} R ⊗_{W (k)} H_{HK}^{i}(U, \overline{U})_{Q}

the vertical maps induce isomorphisms between the kernels and cokernels of the horizontal maps. This will follow if we show that the following map

I ⊗_{W (k)} H_{HK}^{i}(U, \overline{U})_{Q} ⟶^{p^{t}−φ} I ⊗_{W (k)} H_{HK}^{i}(U, \overline{U})_{Q},

for I ⊂ R - the kernel of the projection i_{0} : R_{Q} → K_{0}, t_{l} ↦ 0, is an isomorphism. We argue as Langer in [44, p. 210]. Let M := H_{HK}^{i}(U, \overline{U})/tor. It is a lattice in H_{HK}^{i}(U, \overline{U})_{Q} that is stable under Frobenius. Consider the formal inverse ψ := ∑_{n≥0}(p^{−t}φ)^{n} of 1 − p^{−t}φ. It suffices to show that, for y ∈ I ⊗_{W (k)} M, ψ(y) ∈ I ⊗_{W (k)} M. Fix l and let T^{{k}} := t_{l}^{k}/⌊k/e_{K}⌋!. We will show that, for any m ∈ M, ψ(T^{{k}} ⊗ m) ∈ I ⊗_{W (k)} M and the infinite series converges uniformly in k. We have

(p^{−t}φ)^{n}(T^{\{k\}} ⊗ m) = \frac{⌊kp^{n}/e_{K}⌋!}{⌊k/e_{K}⌋!p^{tn}} T^{\{kp^{n}\}} ⊗ m′

and ord_{p}(⌊kp^{n}/e_{K}⌋!/⌊k/e_{K}⌋!) ≥ p^{n−1}. Hence \frac{⌊kp^{n}/e_{K}⌋!}{⌊k/e_{K}⌋!p^{tn}} converges p-adically to zero, uniformly in k, as wanted. □

Remark 3.9. It was Langer [44, p.193] (cf. [48, Lemma 2.13] in the good reduction case) who observed the fact that while, in general, the crystalline cohomology RΓ_{cr}(U, \overline{U}) behaves badly (it is ”huge”), after taking ”filtered Frobenius eigenspaces” we obtain syntomic cohomology RΓ_{syn}(U, \overline{U}, r)_{Q} that behaves well (it is ”small”). In [49, 3.5] this phenomenon is explained by relating syntomic cohomology to the complex C_{st}(RΓ_{HK}(U, \overline{U})\{r\}).

Remark 3.10. The construction of the map α_{syn,π} depends on the choice of the uniformizer π, which makes the h-sheafification impossible. We will show now that there is a functorial and compatible with base change quasi-isomorphism α′_{syn} between rational syntomic cohomology and certain complexes built from Hyodo-Kato cohomology and de Rham cohomology that h-sheafify well.

Set

α′_{syn} : RΓ_{syn}(U, \overline{U}, r)_{Q} →^{∼} [RΓ_{cr}(U, \overline{U}, r)⟶^{γ_{r}^{−1}}RΓ_{dR}(U, \overline{U}_{K})/F^{r}]

→^{β} [RΓ_{HK}(U, \overline{U}, r)^{N=0}⟶^{ι′_{dR}}RΓ_{dR}(U, \overline{U}_{K})/F^{r}]

Here the two morphisms β and ι′_{dR} are defined as the following compositions

β : RΓ_{cr}(U, \overline{U}, r) →^{∼} RΓ_{cr}(U_{0}, \overline{U}_{0}, r) →^{∼} RΓ_{HK}(U, \overline{U}, r)^{N=0}
ι′_{dR} : RΓ_{HK}(U, \overline{U}, r)^{N=0} ←^{β} RΓ_{cr}(U, \overline{U}, r) →^{γ_{r}^{−1}} RΓ_{dR}(U, \overline{U}_{K}),

where (⋯)^{N=0} denotes the mapping fiber of the monodromy. The map β is a quasi-isomorphism because so is each of the intermediate maps. To see this for the map i_{0}^{∗} : RΓ_{cr}(U, \overline{U}, r) → RΓ_{cr}(U_{0}, \overline{U}_{0}, r), consider the following factorization

F^{m} : RΓ_{cr}(U, \overline{U}, r) →^{i_{0}^{∗}} RΓ_{cr}(U_{0}, \overline{U}_{0}, r) →^{ψ_{m}} RΓ_{cr}(U, \overline{U}, r)

of the m’th power of the Frobenius, where m is large enough. We also have i_{0}^{∗}ψ_{m} = F^{m}. Since Frobenius is a quasi-isomorphism on RΓ_{cr}(U, \overline{U}, r) and RΓ_{cr}(U_{0}, \overline{U}_{0}, r) both i_{0}^{∗} and ψ_{m} are quasi-isomorphisms as well. The second morphism in the sequence defining β is a quasi-isomorphism by an argument similar to the one we used in the proof of Proposition 3.8.

Define the complex

C′_{st}(RΓ_{HK}(U, \overline{U})\{r\}) := [RΓ_{HK}(U, \overline{U}, r)^{N=0}⟶^{ι′_{dR}}RΓ_{dR}(U, \overline{U}_{K})/F^{r}].

We have obtained a quasi-isomorphism

α′_{syn} : RΓ_{syn}(U, \overline{U}, r)_{Q} →^{∼} C′_{st}(RΓ_{HK}(U, \overline{U})\{r\})

It is clearly functorial but it is also easy to check that it is compatible with base change (of the base V).

Define the complex

C_{st}(RΓ_{HK}^{B}(U, \overline{U})\{r\}) := [RΓ_{HK}^{B}(U_{1}, \overline{U}_{1}, r)^{N=0}⟶^{ι_{dR}^{B}}RΓ_{dR}(U, \overline{U}_{K})/F^{r}].

From the commutative diagram (31) we obtain the natural quasi-isomorphisms

γ : C_{st}(RΓ_{HK}^{B}(U, \overline{U})\{r\}) →^{∼} C_{st}(RΓ_{HK}(U, \overline{U})\{r\})
α_{syn,π}^{B} := γ^{−1}α_{syn,π} : RΓ_{syn}(U, \overline{U}, r)_{Q} →^{∼} C_{st}(RΓ_{HK}^{B}(U, \overline{U})\{r\})

We will show now that log-syntomic cohomology satisfies finite Galois descent. Let (U, \overline{U}) be a fine log-scheme, log-smooth over V^{×}, and of Cartier type. Let r ≥ 0. Let K′ be a finite Galois extension of K and let G = Gal(K′/K). Let (T, \overline{T}) = (U ×_{V} V′, \overline{U} ×_{V} V′), V′ - the ring of integers in K′, be the base change of (U, \overline{U}) to (K′, V′), and let f : (T, \overline{T}) → (U, \overline{U}) be the canonical projection. Take R = R_{V}, N, e, π associated to V. Similarly, we define R′ := R_{V′}, N′, e′, π′. Write the map α_{syn,π}^{B} as

[figure: RΓ_{syn}(U, \overline{U}, r)_{Q}; α_{syn,π}^{B}; ≀; ∼; h; [RΓ_{HK}^{B,τ} ((U, \overline{U})_{R}, r)^{N=0}; i_{π}^{∗}; RΓ_{dR}(U, \overline{U}_{K})/F^{r}]; C_{st}(RΓ_{HK}^{B}(U, \overline{U})\{r\}); ∼; [RΓ_{HK}^{B}(U, \overline{U}, r)^{N=0}; ι_{dR}^{B}; ≀; ι_{π}β; RΓ_{dR}(U, \overline{U}_{K})/F^{r}]]

Here we defined the map h as the composition

(34)  RΓ_{syn}(U, \overline{U}, r)_{Q} → RΓ_{cr}((U, \overline{U})/R)_{Q} ←^{∼} RΓ_{HK}^{B}(U_{1}, \overline{U}_{1})_{R_{Q}}^{τ}

From the construction of the Beilinson-Hyodo-Kato map ι_{dR}^{B} : RΓ_{HK}^{B}(T_{1}, \overline{T}_{1}) → RΓ_{dR}(T, \overline{T}_{K′}) it follows that it is G-equivariant; hence the complex C_{st}(RΓ_{HK}^{B}(T, \overline{T})\{r\}) is equipped with a natural Gaction. We claim that the map α_{syn,π′}^{B} induces a natural map

α̃_{syn,π′}^{B} : RΓ(G, RΓ_{syn}(T, \overline{T} , r)_{Q}) → RΓ(G, C_{st}(RΓ_{HK}^{B}(T, \overline{T})\{r\})),
α̃_{syn,π′}^{B} := (1/|G|) ∑_{g∈G} α_{syn,g(π′)}^{B}

To see this it suffices to show that, for every g ∈ G, we have a commutative diagram

RΓ_{syn}(T, \overline{T} , r)_{Q} ⟶^{α_{syn,π′}^{B}} C_{st}(RΓ_{HK}^{B}(T, \overline{T})\{r\})
↓^{g^{∗}}  ↓^{g^{∗}}
RΓ_{syn}(T, \overline{T} , r)_{Q} ⟶^{α_{syn,g(π′)}^{B}} C_{st}(RΓ_{HK}^{B}(T, \overline{T})\{r\})

We accomplish this by constructing natural morphisms

g^{∗} : RΓ_{cr}((T, \overline{T})/R′_{π′}) → RΓ_{cr}((T, \overline{T})/R′_{g(π′)}),
g^{∗} : RΓ_{HK}^{B}(T_{1}, \overline{T}_{1})_{R′_{π′}}^{τ} → RΓ_{HK}^{B}(T_{1}, \overline{T}_{1})_{R′_{g(π′)}}^{τ}

that are compatible with the maps in (34) that define h, the maps ι_{?} and i_{?}^{∗}, and the trivialization β. We define the pullbacks g^{∗} from a map g : R′_{π′} → R′_{g(π′)} constructed by lifting the action of g from V′_{1} to R′ by setting g(t′_{π′}) = t′_{g(π′)} and taking the induced action of g on W (k′). This map is compatible with Frobenius and monodromy. The induced pullbacks g^{∗} are clearly compatible with the map i_{0}^{∗} and the maps ι_{?}, the maps i_{π′}^{∗}, i_{g(π′)}^{∗}, and the trivialization β. From the construction of the Beilinson-Hyodo-Kato map, the pullbacks g^{∗} are also compatible with the maps κ_{R′_{?}}; hence with the map h, as wanted.

Proposition 3.11. (1) The following diagram commutes in the (classical) derived category.

RΓ_{syn}(U, \overline{U}, r)_{Q}  ⟶^{f^{∗}} RΓ(G, RΓ_{syn}(T, \overline{T} , r)_{Q})
↓^{α_{syn,π}^{B}}  ↓^{}^{α̃_{syn,π′}^{B}}
C_{st}(RΓ_{HK}^{B}(U, \overline{U})\{r\}) ⟶^{f^{∗}} RΓ(G, C_{st}(RΓ_{HK}^{B}(T, \overline{T})\{r\}))

(2) The natural map

f^{∗} : RΓ_{syn}(U, \overline{U}, r)_{Q} →^{∼} RΓ(G, RΓ_{syn}(T, \overline{T} , r)_{Q})

is a quasi-isomorphism.

Proof. The second claim of the proposition follows from the first one and the fact that the Hyodo-Kato and de Rham cohomologies satisfy finite Galois decent.

Since everything in sight is functorial and satisfies finite unramified Galois descent we may assume that the extension K′/K is totally ramified. First, we will construct a G-equivariant (for the trivial action of G on R) map

f^{∗} : RΓ_{cr}((U, \overline{U})/R, r)^{N=0} → RΓ_{cr}((T, \overline{T})/R′, r)^{N′=0}

such that the following diagram commutes

(35)

RΓ_{cr}(U, \overline{U}, r)  ⟶^{f^{∗}}  RΓ_{cr}(T, \overline{T} , r)
↓^{≀}  ↓^{≀}
RΓ_{cr}((U, \overline{U})/R, r)^{N=0} ⟶^{f^{∗}} RΓ_{cr}((T, \overline{T})/R′, r)^{N′=0}
≀↑^{ι_{π}}  ^{≀}↑^{ι_{π′}}
RΓ_{HK}(U, \overline{U}, r)^{N=0} ⟶_{∼}^{f^{∗}} RΓ_{HK}(T, \overline{T} , r)^{N′=0}

Remark 3.12. Note that the bottom map is an isomorphism because f^{∗} acts trivially on the Hyodo-Kato complexes. The commutativity of the above diagram and the quasi-isomorphisms (33) will imply that a totally ramified Galois extension does not change the log-crystalline complexes RΓ_{cr}(U, \overline{U}, r) and RΓ_{cr}((U, \overline{U})/R, r)^{N=0}.

Let e_{1} be the ramification index of V′/V. Set v = (π′)^{e_{1}}π^{−1}, and choose an integer s such that (π′)^{p^{s}} ∈ pV′. Set T := t_{π}, T′ := t_{π′} and define the morphism a : R → R′ by T ↦ (T′)^{e_{1}}[\overline{v}]^{−1}. Since V′_{1} and V_{1} are defined by pR + T^{e}R and by pR′ + (T′)^{e′}R′, respectively, a induces a morphism a_{1} : V_{1} → V′_{1}. We have F^{s}a_{1} = F^{s}f_{1}, where F is the absolute Frobenius on Spec(V_{1}). Notice that in general f_{1} ≠ a_{1} if v[\overline{v}]^{−1} ≇ 1 mod pV′. The morphism φ_{R}^{s}a : Spec(R′) → Spec(R) is compatible with F^{s}f_{1} : Spec(V′_{1}) → Spec(V_{1}) and it commutes with the operators N and p^{s}N′. We have the following commutative diagram

\xymatrix{(T, \overline{T})_{1} \ar[r]^{F^{s}f_{1}} \ar[d] & (U, \overline{U})_{1} \ar[d] \\ Spec(V′_{1}) \ar[r]^{F^{s}a_{1}=F^{s}f_{1}} \ar[d] & Spec(V_{1}) \ar[d] \\ Spec(R′) \ar[r]^{φ_{R}^{s}a} & Spec(R)}

Hence the commutative diagram of distinguished triangles

(36)

RΓ_{cr}(U, \overline{U})_{Q} ⟶ RΓ_{cr}((U, \overline{U})/R)_{Q} ⟶^{eN} RΓ_{cr}((U, \overline{U})/R)_{Q}
↓^{f^{∗}F^{s}}  ↓^{f^{∗}F^{s}}  ↓^{p^{s}e_{1}f^{∗}F^{s}}
RΓ_{cr}(T, \overline{T})_{Q} ⟶ RΓ_{cr}((T, \overline{T})/R′)_{Q} ⟶^{e′N′} RΓ_{cr}((T, \overline{T})/R′)_{Q}.

To see how this diagram arises we may assume (by the usual Čech argument) that we have a fine affine log-scheme X_{n}/V_{n}^{×} that is log-smooth over V_{n}^{×}. We can also assume that we have a lifting of X_{n} ↪ Z_{n} over Spec(W_{n}(k)[T]) (with the log-structure coming from T) and a lifting of Frobenius φ_{Z} on Z_{n} that is compatible with the Frobenius φ_{R}. Recall [42, Lemma 4.2] that the horizontal distinguished triangles in the above diagram arise from an exact sequence of complexes of sheaves on X_{n,ét}

(37)  0 → C′_{V} [−1]⟶^{∧ dlog T}C_{V} → C′_{V} → 0

where C_{V} := R_{n} ⊗_{W_{n}(k)[T]} Ω_{Z_{n}/W_{n}(k)}^{·} and C′_{V} := R_{n} ⊗_{W_{n}(k)[T]} Ω_{Z_{n}/W_{n}(k)[T]}^{·}. Now consider the base change of Z_{n}/W_{n}(k)[T] by the map F^{s}a : Spec(W_{n}(k)[T′]) → Spec(W_{n}(k)[T]) and the related complexes (37). We get a commutative diagram of complexes of sheaves on X_{n,ét} (note that X_{V′,n,ét} = X_{n,ét})

0 ⟶ C′_{V′}[−1] ⟶^{∧ dlog T′} C_{V′} ⟶ C′_{V′} ⟶ 0
↑^{p^{s}e_{1}a^{∗}φ_{Z}^{s}}  ↑^{a^{∗}φ_{Z}^{s}}  ↑^{a^{∗}φ_{Z}^{s}}
0 ⟶ C′_{V} [−1] ⟶^{∧ dlog T} C_{V} ⟶ C′_{V} ⟶ 0

Hence diagram (36).

Combining diagram (36) with Frobenius we obtain the following commutative diagram

RΓ_{cr}(U, \overline{U}, r)  ⟵^{F^{s}}  RΓ_{cr}(U, \overline{U}, r)  ⟶^{f^{∗}F^{s}}  RΓ_{cr}(T, \overline{T} , r)
↓^{≀}  ↓^{≀}  ↓^{≀}
RΓ_{cr}((U, \overline{U})/R, r)^{N=0} ⟵^{(F^{s},p^{s}F^{s})} RΓ_{cr}((U, \overline{U})/R, r)^{N=0} ⟶^{(a^{∗}F^{s},p^{s}a^{∗}F^{s})} RΓ_{cr}((T, \overline{T})/R′, r)^{N′=0}
≀↓^{i_{0}^{∗}}  ^{≀}↓^{i_{0}^{∗}}  ^{≀}↓^{i_{0}^{∗}}
RΓ_{HK}(U, \overline{U}, r)^{N=0} ⟵_{∼}^{(F^{s},p^{s}F^{s})} RΓ_{HK}(U, \overline{U}, r)^{N=0}  ⟶_{∼}^{(F^{s},p^{s}F^{s})}  RΓ_{HK}(T, \overline{T} , r)^{N′=0}

It follows that all the maps in the above diagram are quasi-isomorphisms. We define the map

f^{∗} : RΓ_{cr}((U, \overline{U})/R, r)^{N=0} → RΓ_{cr}((T, \overline{T})/R′, r)^{N′=0}

by the middle row. Since, for any g ∈ G, we have v_{g(π′)} = g(v_{π′}), the map f^{∗} is G-equivariant. In the (classical) derived category, this definition is independent of the constant s we have chosen. Since i_{0}^{∗} is a quasi-isomorphism and i_{0}^{∗}ι_{?∗} = Id, the diagram (35) commutes as well, as wanted.

We define the map

(38)  f^{∗} : RΓ_{HK}^{B,τ} ((U, \overline{U})/R, r)^{N=0} → RΓ_{HK}^{B,τ} ((T, \overline{T})/R′, r)^{N′=0}

in an analogous way. By the above diagram and by compatibility of the Beilinson-Hyodo-Kato constructions with base change and with Frobenius, the two pullback maps f^{∗} are compatible via the morphism h, i.e., the following diagram commutes

\xymatrix{RΓ_{cr}(U, \overline{U}, r) \ar[r] \ar[d]^{f^{∗}} & RΓ_{cr}((U, \overline{U})/R, r)^{N=0} \ar[d]^{f^{∗}} & RΓ_{HK}^{B,τ} ((U, \overline{U})/R, r)^{N=0} \ar[l]_{κ_{R}} \ar[d]^{f^{∗}} \\ RΓ_{cr}(T, \overline{T} , r) \ar[r] & RΓ_{cr}((T, \overline{T})/R′, r)^{N′=0} & RΓ_{HK}^{B,τ} ((T, \overline{T})/R′, r)^{N′=0} \ar[l]_{κ_{R′}}}

From the analog of diagram (35) for the Beilinson-Hyodo-Kato complexes and by the universal nature of the trivialization at \overline{p} we obtain that the pullback map f^{∗} is compatible with the maps βι_{?}. It remains to show that we have a commutative diagram

\xymatrix{RΓ_{HK}^{B}(U, \overline{U}, r)^{N=0} \ar[r]^{f^{∗}}_{∼} \ar[d]^{ι_{dR}^{B}} & RΓ_{HK}^{B}(T, \overline{T} , r)^{N′=0} \ar[d]^{ι_{dR}^{B}} \\ RΓ_{dR}(U, \overline{U}_{K})/F^{r} \ar[r]^{f^{∗}} & RΓ_{dR}(T, \overline{T}_{K′})/F^{r}}

But this follows since the Beilinson-Hyodo-Kato map is compatible with base change. □

3.3. Arithmetic syntomic cohomology. We are now ready to introduce and study arithmetic syntomic cohomology, i.e., syntomic cohomology over K. Let 𝒥_{cr}^{[r]}, 𝒜_{cr}, and 𝒮 (r) for r ≥ 0 be the h-sheafifications on 𝒱 ar_{K} of the presheaves sending (U, \overline{U}) ∈ 𝒫_{K}^{ss} to RΓ_{cr}(U, \overline{U}, J^{[r]}), RΓ_{cr}(U, \overline{U}), and RΓ_{syn}(U, \overline{U}, r), respectively. Let 𝒥_{cr,n}^{[r]}, 𝒜_{cr,n}, and 𝒮_{n}(r) denote the h-sheafifications of the mod-p^{n} versions of the respective presheaves. We have

𝒮_{n}(r) ≃ Cone(𝒥_{cr,n}^{[r]} ⟶^{p^{r}−φ} 𝒜_{cr,n})[−1], 𝒮 (r) ≃ Cone(𝒥_{cr}^{[r]} ⟶^{p^{r}−φ} 𝒜_{cr})[−1].

For r ≥ 0, define 𝒮 (r)_{Q} as the h-sheafification of the preasheaf sending ss-pairs (U, \overline{U}) to RΓ_{syn}(U, \overline{U}, r)_{Q}. We have

𝒮 (r)_{Q} ≃ Cone(𝒥_{cr,Q}^{[r]}⟶^{1−φ_{r}}𝒜_{cr,Q})[−1]

For X ∈ 𝒱 ar_{K}, set RΓ_{syn}(X_{h}, r)_{n} = RΓ(X_{h}, 𝒮_{n}(r)), RΓ_{syn}(X_{h}, r) := RΓ(X_{h}, 𝒮 (r)_{Q}). We have

RΓ_{syn}(X_{h}, r)_{n} ≃ Cone(RΓ(X_{h}, 𝒥_{cr,n}^{[r]}) ⟶^{p^{r}−φ} RΓ(X_{h}, 𝒜_{cr,n}))[−1],
RΓ_{syn}(X_{h}, r) ≃ Cone(RΓ(X_{h}, 𝒥_{cr,Q}^{[r]}) ⟶^{1−φ_{r}} RΓ(X_{h}, 𝒜_{cr,Q}))[−1].

We will often write RΓ_{cr}(X_{h}) for RΓ(X_{h}, 𝒜_{cr}) if this does not cause confusion.

Let 𝒜_{HK} be the h-sheafification of the presheaf (U, \overline{U}) ↦ RΓ_{HK}(U, \overline{U})_{Q} on 𝒫_{K}^{ss}; this is an h-sheaf of E_{∞} K_{0}-algebras on 𝒱 ar_{K} equipped with a φ-action and a derivation N such that Nφ = pφN. For X ∈ 𝒱 ar_{K}, set RΓ_{HK}(X_{h}) := RΓ(X_{h}, 𝒜_{HK}). Similarly, we define h-sheaves 𝒜_{HK}^{B} and the complexes RΓ_{HK}^{B}(X_{h}) := RΓ(X_{h}, 𝒜_{HK}^{B}). The maps κ : RΓ_{HK}^{B}(U_{1}, \overline{U}_{1}) → RΓ_{HK}(U, \overline{U})_{Q} h-sheafify and we obtain functorial quasi-isomorphisms

κ : 𝒜_{HK}^{B} →^{∼} 𝒜_{HK}, κ : RΓ_{HK}^{B}(X_{h}) →^{∼} RΓ_{HK}(X_{h}).

Remark 3.13. The complexes 𝒥_{cr,n}^{[r]} and 𝒮_{n}(r) (and their completions) have a concrete description. For the complexes 𝒥_{cr,n}^{[r]}: we can represent the presheaves (U, \overline{U}) ↦ RΓ_{cr}(U, \overline{U}, 𝒥_{n}^{[r]}) by Godement resolutions (on the crystalline site), sheafify them for the h-topology on 𝒫_{K}^{ss}, and then move them to 𝒱 ar_{K}. For the complexes 𝒮_{n}(r): the maps p^{r} − φ can be lifted to the Godement resolutions and their mapping fiber (defining 𝒮_{n}(r)(U, \overline{U})) can be computed in the abelian category of complexes of abelian groups. To get 𝒮_{n}(r) we h-sheafify on 𝒫_{K}^{ss} and pass to 𝒱 ar.

Let, for a moment, K be any field of characteristic zero. Consider the presheaf (U, \overline{U}) ↦ RΓ_{dR}(U, \overline{U}) := RΓ(\overline{U}, Ω(U,\overline{U})^{•}) of filtered dg K-algebras on 𝒫_{K}^{nc}. Let 𝒜_{dR} be its h-sheafification. It is a sheaf of filtered K-algebras on 𝒱 ar_{K}. For X ∈ 𝒱 ar_{K}, we have Deligne’s de Rham complex of X equipped with Deligne’s Hodge filtration: RΓ_{dR}(X_{h}) := RΓ(X_{h}, 𝒜_{dR}). Beilinson proves the following comparison statement.

Proposition 3.14. ([3, 2.4])

(1) For (U, \overline{U}) ∈ 𝒫_{K}^{nc}, the canonical map RΓ_{dR}(U, \overline{U}) →^{∼} RΓ_{dR}(U_{h}) is a filtered quasi-isomorphism.

(2) The cohomology groups H_{dR}^{i}(X_{h}) := H^{i}RΓ_{dR}(X_{h}) are K-vector spaces of dimension equal to the

rank of H^{i}(X_{\overline{K},ét}, Q_{p}).

Corollary 3.15. For a geometric pair (U, \overline{U}) over K that is saturated and log-smooth, the canonical map

RΓ_{dR}(U, \overline{U}) →^{∼} RΓ_{dR}(U_{h})

is a filtered quasi-isomorphism.

Proof. Recall [52, Theorem 5.10] that there is a log-blow-up (U, \overline{T}) → (U, \overline{U}) that resolves singularities of (U, \overline{U}), i.e., such that (U, \overline{T}) ∈ 𝒫_{K}^{nc}. We have a commutative diagram

\xymatrix{RΓ_{dR}(U, \overline{T}) \ar[r]^{∼} & RΓ_{dR}(U_{h}) \\ RΓ_{dR}(U, \overline{U}) \ar[u]^{≀} \ar[ur]}

The vertical map is a filtered quasi-isomorphism; the horizontal map is a filtered quasi-isomorphism by the above proposition. Our corollary follows. □

Remark 3.16. Another proof of the above result (and a mild generalization) that does not use resolution of singularities can be found in [4, 1.19] (where it is attributed to A.Ogus).

Return now to our p-adic field K.

Remark 3.17. By construction the complexes RΓ(X_{h}, 𝒥_{cr,Q}^{[r]}), RΓ_{syn}(X_{h}, r), RΓ_{HK}(X_{h}), RΓ_{HK}^{B}(X_{h}), and RΓ_{dR}(X_{h}) satisfy h-descent. In particular, since h-topology is finer than the étale topology, they satisfy Galois descent for finite extensions. Hence, for any finite Galois extension K_{1}/K, the natural maps

RΓ_{?}^{∗}(X_{h}) →^{∼} RΓ(G, RΓ_{?}^{∗}(X_{K_{1},h})), ? = cr, syn, HK, dR; ∗ = B, ∅

where G = Gal(K_{1}/K), are (filtered) quasi-isomorphisms. Since G is finite, it follows that the natural maps

RΓ_{HK}^{∗}(X_{h}) ⊗_{K_{0}} K_{1,0} →^{∼} RΓ_{HK}^{∗}(X_{K_{1},h}), RΓ_{dR}(X_{h}) ⊗_{K} K_{1} →^{∼} RΓ_{dR}(X_{K_{1},h})

are (filtered) quasi-isomorphisms as well.

Recall [4, 2.5], Proposition 41, that for a fine, log-scheme X, log-smooth over V^{×}, and of Cartier type we have a quasi-isomorphism RΓ_{cr}(X\overline{V} , 𝒥_{X\overline{V} /W (k)}^{[r]})_{Q} ≃ RΓ(X_{\overline{K},h}, 𝒥_{cr}^{[r]})_{Q}. We can descend this result to K but on the level of rational log-syntomic cohomology; the key observation being that the field extensions introduced by the alterations are harmless since, by Proposition 3.11, log-syntomic cohomology satisfies finite Galois descent. Along the way we will get an analogous comparison quasi-isomorphism for the Hyodo-Kato cohomology.

Proposition 3.18. For any arithmetic pair (U, \overline{U}) that is fine, log-smooth over V^{×}, and of Cartier type, and r ≥ 0, the canonical maps

RΓ_{HK}^{∗}(U, \overline{U})_{Q} →^{∼} RΓ_{HK}^{∗}(U_{h}), RΓ_{syn}(U, \overline{U}, r)_{Q} →^{∼} RΓ_{syn}(U_{h}, r)

are quasi-isomorphisms.

Proof. It suffices to show that for any h-hypercovering (U_{•}, \overline{U}_{•}) → (U, \overline{U}) by pairs from 𝒫_{K}^{log} the natural maps

RΓ_{HK}(U, \overline{U})_{Q} → RΓ_{HK}(U_{•}, \overline{U}_{•})_{Q}, RΓ_{syn}(U, \overline{U}, r)_{Q} → RΓ_{syn}(U_{•}, \overline{U}_{•}, r)_{Q}

are (modulo taking a refinement of (U_{•}, \overline{U}_{•})) quasi-isomorphisms. For the second map, since we have a canonical quasi-isomorphism

RΓ_{syn}(U, \overline{U}, r)_{Q} →^{∼} Cone(RΓ_{cr}(U, \overline{U}, r)_{Q} → RΓ_{cr}(U, \overline{U}, 𝒪/𝒥^{[r]})_{Q})[−1]

it suffices to show that, up to a refinement of the hypercovering, we have quasi-isomorphisms

RΓ_{cr}(U, \overline{U}, 𝒪/𝒥^{[r]})_{Q} →^{∼} RΓ_{cr}(U_{•}, \overline{U}_{•}, 𝒪/𝒥^{[r]})_{Q}, RΓ_{cr}(U, \overline{U}, r)_{Q} →^{∼} RΓ_{cr}(U_{•}, \overline{U}_{•}, r)_{Q}.

For the first of these maps, by Corollary 2.4 this amounts to showing that the following map is a quasiisomorphism

RΓ(\overline{U}_{K}, Ω(U,\overline{U}_{K})^{•})/F^{r} →^{∼} RΓ(\overline{U}_{•,K}, Ω(U_{•},\overline{U}_{•,K})^{•})/F^{r}.

But, by Corollary 3.15 this map is quasi-isomorphic to the map

RΓ_{dR}(U_{h})/F^{r} → RΓ_{dR}(U_{•,h})/F^{r},

which is clearly a quasi-isomorphism.

Hence it suffices to show that, up to a refinement of the hypercovering, we have quasi-isomorphisms

RΓ_{HK}(U, \overline{U})_{Q} →^{∼} RΓ_{HK}(U_{•}, \overline{U}_{•})_{Q}, RΓ_{cr}(U, \overline{U}, r)_{Q} →^{∼} RΓ_{cr}(U_{•}, \overline{U}_{•}, r)_{Q}.

Fix t ≥ 0. To show that H^{t}RΓ_{cr}(U, \overline{U}, r)_{Q} →^{∼} H^{t}RΓ_{cr}(U_{•}, \overline{U}_{•}, r)_{Q} is a quasi-isomorphism we will often work with (t+1)-truncated h-hypercovers. This is because τ_{≤t}RΓ_{cr}(U_{•}, \overline{U}_{•}, r) ≃ τ_{≤t}RΓ_{cr}((U_{•}, \overline{U}_{•})_{≤t+1}, r), where (U_{•}, \overline{U}_{•})_{≤t+1} denotes the t+1-truncation. Assume first that we have an h-hypercovering (U_{•}, \overline{U}_{•}) → (U, \overline{U}) of arithmetic pairs over K, where each pair (U_{i}, \overline{U}_{i}), i ≤ t+1, is log-smooth over V^{×} and of Cartier type. We claim that then already the maps

(39) τ_{≤t}RΓ_{HK}(U, \overline{U})_{Q} →^{∼} τ_{≤t}RΓ_{HK}((U_{•}, \overline{U}_{•})_{≤t+1})_{Q}; τ_{≤t}RΓ_{cr}(U, \overline{U})_{Q} →^{∼} τ_{≤t}RΓ_{cr}((U_{•}, \overline{U}_{•})_{≤t+1})_{Q}

are quasi-isomorphisms. To see the second quasi-isomorphism consider the following commutative diagram of distinguished triangles (R = R_{V})

RΓ_{cr}(U, \overline{U})  ⟶ RΓ_{cr}((U, \overline{U})/R) ⟶^{N} RΓ_{cr}((U, \overline{U})/R)
↓  ↓  ↓
RΓ_{cr}((U_{•}, \overline{U}_{•})_{≤t+1}) ⟶ RΓ_{cr}((U_{•}, \overline{U}_{•})_{≤t+1}/R) ⟶^{N} RΓ_{cr}((U_{•}, \overline{U}_{•})_{≤t+1}/R)

It suffices to show that the two right vertical arrows are rational quasi-isomorphisms in degrees less or equal to t. But we have the R-linear quasi-isomorphisms

ι : R⊗_{W (k)}RΓ_{HK}(U, \overline{U})_{Q} →^{∼} RΓ((U, \overline{U})/R)_{Q}, ι : R⊗_{W (k)}RΓ_{HK}((U_{•}, \overline{U}_{•})_{≤t+1})_{Q} →^{∼} RΓ((U_{•}, \overline{U}_{•})_{≤t+1}/R)_{Q}.

Hence to show both quasi-isomorphisms (39), it suffices to show that the map

τ_{≤t}RΓ_{HK}(U, \overline{U})_{Q} → τ_{≤t}RΓ_{HK}((U_{•}, \overline{U}_{•})_{≤t+1})_{Q}

is a quasi-isomorphism.

Tensoring over K_{0} with K and using the Hyodo-Kato quasi-isomorphism (23) we reduce to showing that the map

τ_{≤t}RΓ(\overline{U}_{K}, Ω_{(U,\overline{U}_{K})}^{·}) → τ_{≤t}RΓ(\overline{U}_{•K,≤t+1}, Ω_{(U_{•},\overline{U}_{•,K})_{≤t+1}}^{·})

is a quasi-isomorphism. And this we have done above.

To treat the general case, set X = (U, \overline{U}), Y_{•} = (U_{•}, \overline{U}_{•}). We will do a base change to reduce to the case discussed above. We may assume that all the fields K_{n,i}, K_{U_{n}} ≃ ∏ K_{n,i} are Galois over K. Choose a finite Galois extension (V′, K′)/(V, K) for K′ Galois over all the fields K_{n,i}, n ≤ t+1. Write N_{X}(X_{V′}) for the ”Čech nerve” of X_{V′}/X. The term N_{X}(X_{V′})_{n} is defined as the (n + 1)-fold fiber product of X_{V′} over X: N_{X}(X_{V′})_{n} = (U ×_{K} K^{′,n+1}, (\overline{U} ×_{V} V^{′,n+1})^{norm}), where V^{′,n+1}, K^{′,n+1} are defined as the (n + 1)-fold product of V′ over V and of K′ over K, respectively. Normalization is taken with respect to the open regular subscheme U ×_{K} K^{′,n+1}. Note that N_{X}(X_{V′})_{n} ≃ (U ×_{K} K′×G^{n}, \overline{U} ×_{V} V′×G^{n}), G = Gal(K′/K). Hence it is a log-smooth scheme over V^{′,×}, of Cartier type. The augmentation N_{X}(X_{V′}) → X is an hhypercovering.

Consider the bi-simplicial scheme Y_{•} ×_{X} N_{X}(X_{V′})_{•},

(Y_{•} ×_{X} N_{X}(X_{V′})_{•})_{n,m} := Y_{n} ×_{X} N_{X}(X_{V′})_{m} ≃ (U_{n} ×_{U} U ×_{K} K^{′,m+1}, (\overline{U}_{n} ×\overline{U} (\overline{U} ×_{V} V^{′,m+1})^{norm})^{norm})

≃ ∐_{i}(U_{n} ×_{K_{n,i}} K_{n,i} ×_{K} K^{′,m+1}, \overline{U}_{n} ×_{V_{n,i}} (V_{n,i} ×_{V} V^{′,m+1})^{norm}).

Hence (Y_{•} ×_{X} N_{X}(X_{V′})_{•})_{n,m} ∈ 𝒫_{K}^{log}. For n, m ≤ t + 1, we have

(Y_{•} ×_{X} N_{X}(X_{V′})_{•})_{n,m} ≃ ∐_{i}(U_{n} ×_{K_{n,i}} K′ × G_{n,i} × G^{m}, \overline{U}_{n} ×_{V_{n,i}} V′ × G_{n,i} × G^{m}),

where G_{n,i} = Gal(K_{n,i}/K). It is a log-scheme log-smooth over V^{′,×}, of Cartier type.

Consider now its diagonal Y_{•} ×_{X} N_{X}(X_{V′}) := Δ(Y_{•} ×_{X} N_{X}(X_{V′})_{•}). It is an h-hypercovering of X refining Y_{•} such that, for n ≤ t+1, (Y_{•} ×_{X} N_{X}(X_{V′}))_{n} is log-smooth over V^{′,×}, of Cartier type. It suffices to show that the compositions

(40)  RΓ_{HK}(X)_{Q} → RΓ_{HK}(Y_{•})_{Q} →^{pr_{1}^{∗}} RΓ_{HK}(Y_{•} ×_{X} N_{X}(X_{V′}))_{Q};
RΓ_{cr}(X, r)_{Q} → RΓ_{cr}(Y_{•}, r)_{Q} →^{pr_{1}^{∗}} RΓ_{cr}(Y_{•} ×_{X} N_{X}(X_{V′}), r)_{Q}

are quasi-isomorphisms in degrees less or equal to t. Using the commutative diagram of bi-simplicial schemes

Y_{•} ×_{X} N_{X}(X_{V′}) ⟶^{Δ} Y_{•} ×_{X} N_{X}(X_{V′})_{•} ⟶^{pr_{1}} Y_{•}
↓^{pr}^{2}  ↓
N_{X}(X_{V′})  ⟶^{f} X

we can write the second composition as

RΓ_{cr}(X, r)_{Q} →^{f^{∗}} RΓ_{cr}(N_{X}(X_{V′}), r)_{Q} →^{pr_{2}^{∗}} RΓ_{cr}(Y_{•} ×_{X} N_{X}(X_{V′})_{•}, r)_{Q} →^{Δ^{∗}} RΓ_{cr}(Y_{•} ×_{X} N_{X}(X_{V′}), r)_{Q}

We claim that all of these maps are quasi-isomorphisms in degrees less or equal to t. The map Δ^{∗} is a quasi-isomorphism (in all degrees) by [34, Prop. 2.5]. For the second map, fix n ≤ t + 1 and consider the induced map pr_{2} : (Y_{•} ×_{X} N_{X}(X_{V′})_{•})_{•,n} → N_{X}(X_{V′})_{n}. It is an h-hypercovering whose (t + 1)-truncation is built from log-schemes, log-smooth over (V′, K′), of Cartier type. It suffices to show that the induced map τ_{≤t}RΓ_{cr}(N_{X}(X_{V′})_{n}, r)_{Q} →^{pr_{2}^{∗}} τ_{≤t}RΓ_{cr}((Y_{•} ×_{X} N_{X}(X_{V′}))_{•,n}, r)_{Q} is a quasi-isomorphism. Since all maps are defined over K′, this follows from the case considered at the beginning of the proof.

To prove that the map f^{∗} : RΓ_{cr}(X, r)_{Q} → RΓ_{cr}(N_{X}(X_{V′}), r)_{Q} is a quasi-isomorphism consider first the case when the extension V′/V is unramified. Then RΓ_{cr}(X_{V′}) ≃ RΓ_{cr}(X) ⊗_{W (k)} W (k′) and the map f^{∗} is a quasi-isomorphism by finite étale descent for crystalline cohomology.

Assume now that the extension V′/V is totally ramified and let π and π′ be uniformizers of V and V′, respectively. Consider the target of f^{∗} as a double complex. To show that f^{∗} is a quasi-isomorphism it suffices to show that, for each s ≥ 0, the sequence

0 → H^{s}RΓ_{cr}(X, r)_{Q} →^{f^{∗}} H^{s}RΓ_{cr}(N_{X}(X_{V′})_{0}, r)_{Q} →^{d_{0}^{∗}} H^{s}RΓ_{cr}(N_{X}(X_{V′})_{1}, r)_{Q} →^{d_{1}^{∗}} H^{s}RΓ_{cr}(N_{X}(X_{V′})_{2}, r)_{Q} …

is exact. Embed it into the following diagram

[figure: 0; H^{s}RΓ_{cr}(X, r)_{Q}; ≀; α_{syn,π}^{B}; f^{∗}; H^{s}RΓ_{cr}(N_{X}(X_{V′})_{0}, r)_{Q}; ≀; α̃_{syn,π′}^{B}; d_{0}^{∗}; H^{s}RΓ_{cr}(N_{X}(X_{V′})_{1}, r)_{Q}; ≀; α̃_{syn,π′}^{B}; 0; H^{s}RΓ_{HK}^{B}(X, r)_{Q}^{N=0}; f^{∗}; H^{s}RΓ_{HK}^{B}(N_{X}(X_{V′})_{0}, r)_{Q}^{N′=0}; d_{0}^{∗}; H^{s}RΓ_{HK}^{B}(N_{X}(X_{V′})_{1}, r)_{Q}^{N′=0}]

Note that, since all the maps d_{i}^{∗} are induced from automorphisms of V′/V, by the proof of Proposition 3.11 (take the map f used there to be a given automorphism g ∈ G = Gal(K′/K) and π′, g(π′) for the uniformizers of V′) and the proof of Proposition 3.8, we get the vertical maps above that make all the squares commute.

Hence it suffices to show that the following sequence of Hyodo-Kato cohomology groups is exact:

0 → H^{s}RΓ_{HK}(X)_{Q} →^{f^{∗}} H^{s}RΓ_{HK}(N_{X}(X_{V′})_{0})_{Q} →^{d_{0}^{∗}} H^{s}RΓ_{HK}(N_{X}(X_{V′})_{1})_{Q} →^{d_{1}^{∗}} H^{s}RΓ_{HK}(N_{X}(X_{V′})_{2})_{Q} →

But this sequence is isomorphic to the following sequence

0 → H^{s}RΓ_{HK}(X)_{Q} →^{f^{∗}} H^{s}RΓ_{HK}(X_{V′})_{Q} →^{d_{0}^{∗}} H^{s}RΓ_{HK}(X_{V′})_{Q} × G →^{d_{1}^{∗}} H^{s}RΓ_{HK}(X_{V′})_{Q} × G^{2} →

representing the (augmented) G-cohomology of H^{s}RΓ_{HK}(X)_{Q}. Since G is finite, this complex is exact in degrees at least 1. It remains to show that H^{0}(G, H^{s}RΓ_{HK}(X_{V′})_{Q}) ≃ H^{s}RΓ_{HK}(X)_{Q}. Since K′/K is totally ramified, we have H^{s}RΓ_{HK}(X_{V′}) ≃ H^{s}RΓ_{HK}(X). Hence the action of G on H^{s}RΓ_{HK}(X_{V′}) is trivial and we get the right H^{0} as well. We have proved the second quasi-isomorphism from (40). Notice that along the way we have actually proved the first quasi-isomorphism. □

For X ∈ 𝒱 ar_{K}, we define a canonical K_{0}-linear map (the Beilinson-Hyodo-Kato morphism)

ι_{dR}^{B} : RΓ_{HK}^{B}(X_{h})→RΓ_{dR}(X_{h})

as sheafification of the map ι_{dR}^{B} : RΓ_{HK}^{B}(U_{1}, \overline{U}_{1})→RΓ_{dR}(U, \overline{U}_{K}). It follows from Proposition 3.22 that we prove in the next section that the cohomology groups H_{HK}^{i}(X_{h}) := H^{i}RΓ_{HK}^{B}(X_{h}) are finite rank K_{0}-vector spaces and that they vanish for i > 2 dim X. This implies the following lemma.

Lemma 3.19. The syntomic cohomology groups H_{syn}^{i}(X_{h}, r) := H^{i}RΓ_{syn}(X_{h}, r) vanish for i > 2 dim X+ 2.

Proof. The map ι′_{dR} : RΓ_{HK}(U, \overline{U}, r)^{N=0} → RΓ_{dR}(U, \overline{U}_{K})/F^{r} from Remark 3.10 sheafifies and so does the quasi-isomorphism α′_{syn} : RΓ_{syn}(U, \overline{U}, r)_{Q} →^{∼} C′_{st}(RΓ_{HK}(U, \overline{U})\{r\}). Hence RΓ_{syn}(X_{h}, r) is quasiisomorpic via α′_{syn} to the mapping fiber

C′_{st}(RΓ_{HK}(X_{h}){r}) := [RΓ_{HK}(X_{h}, r)^{N=0}⟶^{ι′_{dR}}RΓ_{dR}(X_{h})/F^{r}]

The statement of the lemma follows. □

For X ∈ 𝒱 ar_{K} and r ≥ 0, define the complex

C_{st}(RΓ_{HK}^{B}(X_{h})\{r\}) := [ \xymatrix{RΓ_{HK}^{B}(X_{h}) \ar[r]^{(1−φ_{r},ι_{dR}^{B})} \ar[d]^{N} & RΓ_{HK}^{B}(X_{h}) ⊕ RΓ_{dR}(X_{h})/F^{r} \ar[d]^{(N,0)} \\ RΓ_{HK}^{B}(X_{h}) \ar[r]^{1−φ_{r−1}} & RΓ_{HK}^{B}(X_{h})} ]

Proposition 3.20. For X ∈ 𝒱 ar_{K} and r ≥ 0, there exists a canonical (in the classical derived category) quasi-isomorphism

α_{syn} : RΓ_{syn}(X_{h}, r) →^{∼} C_{st}(RΓ_{HK}^{B}(X_{h}){r}).

Moreover, this morphism is compatible with finite base change (of the field K).

Proof. To construct the map α_{syn}, take a number t ≥ 2 dim X + 2 and let Y_{•} → X, Y_{•} = (U_{•}, \overline{U}_{•}), be an h-hypercovering of X by ss-pairs over K. Choose a finite Galois extension (V′, K′)/(V, K) and a uniformizer π′ of V′ as in the proof of Proposition 3.18. Keeping the notation from that proof, refine our hypercovering to the h-hypercovering Y_{•} ×_{V} V′ → X_{K′}. Then the truncation (Y_{•} ×_{V} V′)_{≤t+1} is built from log-schemes log-smooth over V^{′,×} and of Cartier type. We have the following sequence of quasi-isomorphisms

γ_{π′} : RΓ_{syn}(X_{K′,h}) ←^{∼} τ_{≤t}RΓ_{syn}(X_{K′,h}) →^{∼} τ_{≤t}RΓ_{syn}((U_{•} ×_{K} K′)_{≤t+1,h}) ←^{∼} τ_{≤t}RΓ_{syn}((Y_{•} ×_{V} V′)_{≤t+1})_{Q}

→^{∼} C_{st}(τ_{≤t}RΓ_{HK}^{B}((Y_{•} ×_{V} V′)_{≤t+1}){r}) →^{∼} C_{st}(τ_{≤t}RΓ_{HK}^{B}((U_{•} ×_{K} K′)_{≤t+1,h}){r})

←^{∼} C_{st}(τ_{≤t}RΓ_{HK}^{B}(X_{K′,h}){r}) →^{∼} C_{st}(RΓ_{HK}^{B}(X_{K′,h}){r})

The first quasi-isomorphism follows from Lemma 3.19. The third and fifth quasi-isomorphisms follow from Proposition 3.18. The fourth quasi-isomorphism (the map α̃_{syn,π′}^{B}), since all the log-schemes involved are log-smooth over V^{′,×} and of Cartier type, follows from Proposition 3.8.

Now, set G := Gal(K′/K). Passing from γ_{π′} to its G-fixed points we obtain the map

α_{syn} := α_{syn,π′} : RΓ_{syn}(X_{h}) → C_{st}(RΓ_{HK}^{B}(X_{h}){r})

as the composition

RΓ_{syn}(X_{h}) → RΓ_{syn}(X_{K′,h})^{G} →^{γ_{π′}} C_{st}(RΓ_{HK}^{B}(X_{K′,h}){r})^{G} ←^{∼} C_{st}(RΓ_{HK}^{B}(X_{K,h}){r})

It remains to check that so defined map is independent of all choices. For that, it suffices to check that, in the above construction, for a finite Galois extension (V_{1}, K_{1}) of (V′, K′), H = Gal(K_{1}/K′), the corresponding maps α_{syn,?} : RΓ_{syn}(X_{h}) → C_{st}(RΓ_{HK}^{B}(X_{h}){r}) are the same in the classical derived category (note that this includes trivial extensions). Easy diagram chase shows that this amounts to checking that the following diagram commutes

\xymatrix{RΓ_{syn}((Y_{•} ×_{V} V′)_{≤t+1})_{Q} \ar[r]^{∼}_{α_{syn,π′}} \ar[d] & C_{st}(RΓ_{HK}^{B}((Y_{•} ×_{V} V′)_{≤t+1})\{r\}) \ar[d] \\ RΓ_{syn}((Y_{•} ×_{V} V_{1})_{≤t+1})_{Q}^{H} \ar[r]^{∼}_{α_{syn,π_{1}}} & C_{st}(RΓ_{HK}^{B}((Y_{•} ×_{V} V_{1})_{≤t+1})\{r\})^{H}}

But this we have shown in Proposition 3.11.

For the compatibility with finite base change, consider a finite field extension L/K. We can choose in the above a Galois extension K′/K that works for both fields. We get the same maps γ_{π′} for both L and K. Consider now the following commutative diagram. The top and bottom rows define the maps α_{syn,π′}^{L} and α_{syn,π′}^{K}, respectively.

\xymatrix{RΓ_{syn}(X_{L,h}) \ar[r] & RΓ_{syn}(X_{K′,h})^{G_{L}} \ar[r]^{γ_{π′}} & C_{st}(RΓ_{HK}^{B}(X_{K′,h})\{r\})^{G_{L}} & C_{st}(RΓ_{HK}^{B}(X_{L,h})\{r\}) \ar[l]_{∼} \\ RΓ_{syn}(X_{h}) \ar[u] \ar[r] & RΓ_{syn}(X_{K′,h})^{G} \ar[u] \ar[r]^{γ_{π′}} & C_{st}(RΓ_{HK}^{B}(X_{K′,h})\{r\})^{G} \ar[u] & C_{st}(RΓ_{HK}^{B}(X_{K,h})\{r\}) \ar[l]_{∼} \ar[u]}

This proves the last claim of our proposition. □

3.4. Geometric syntomic cohomology. We will now study geometric syntomic cohomology, i.e., syntomic cohomology over \overline{K}. Most of the constructions related to syntomic cohomology over K have their analogs over \overline{K}. We will summarize them briefly. For details the reader should consult [62], [4].

For (U, \overline{U}) ∈ 𝒫ss ; \overline{K}, r ≥ 0, we have the absolute crystalline cohomology complexes and their completions

RΓ_{cr}(U, \overline{U}, 𝒥^{[r]})_{n} : = RΓ_{cr}(\overline{U}_{ét}, Ru_{\overline{U}_{n}/W_{n}(k)∗}𝒥_{\overline{U}_{n}/W_{n}(k)}^{[r]}), RΓ_{cr}(U, \overline{U}, 𝒥^{[r]}) := holim_{n} RΓ_{cr}(U, \overline{U}, 𝒥^{[r]})_{n},
RΓ_{cr}(U, \overline{U}, 𝒥^{[r]})_{Q} : = RΓ_{cr}(U, \overline{U}, 𝒥^{[r]}) ⊗ Q_{p}

By [4, Theorem 1.18], the complex RΓ_{cr}(U, \overline{U}) is a perfect A_{cr}-complex and RΓ_{cr}(U, \overline{U})_{n} ≃ RΓ_{cr}(U, \overline{U})⊗_{A_{cr}}^{L} A_{cr}/p^{n} ≃ RΓ_{cr}(U, \overline{U}) ⊗^{L} Z/p^{n}. In general, we have RΓ_{cr}(U, \overline{U}, 𝒥^{[r]})_{n} ≃ RΓ_{cr}(U, \overline{U}, 𝒥^{[r]}) ⊗^{L} Z/p^{n}. Moreover, J_{cr}^{[r]} = RΓ_{cr}(Spec(\overline{K}), Spec(\overline{V}), 𝒥^{[r]}) [62, 1.6.3,1.6.4]. The absolute log-crystalline cohomology complexes are filtered E_{∞} algebras over A_{cr,n}, A_{cr}, or A_{cr,Q}, respectively. Moreover, the rational ones are filtered commutative dg algebras.

For r ≥ 0, the mod-p^{n}, completed, and rational log-syntomic complexes RΓ_{syn}(U, \overline{U}, r)_{n}, RΓ_{syn}(U, \overline{U}, r), and RΓ_{syn}(U, \overline{U}, r)_{Q} are defined by analogs of formulas (32). We have RΓ_{syn}(U, \overline{U}, r)_{n} ≃ RΓ_{syn}(U, \overline{U}, r)⊗^{L} Z/p^{n}. Let 𝒥_{cr}^{[r]}, 𝒜_{cr}, and 𝒮 (r) be the h-sheafifications on 𝒱 ar\overline{K} of the presheaves sending (U, \overline{U}) ∈ 𝒫ss ; \overline{K}

to RΓ_{cr}(U, \overline{U}, 𝒥^{[r]}), RΓ_{cr}(U, \overline{U}), and RΓ_{syn}(U, \overline{U}, r), respectively. Let 𝒥_{cr,n}^{[r]}, 𝒜_{cr,n}, and 𝒮_{n}(r) denote the h-sheafifications of the mod-p^{n} versions of the respective presheaves; and let 𝒥_{cr,Q}^{[r]}, 𝒜_{cr,Q}, 𝒮 (r)_{Q} be the h-sheafification of the rational versions of the same presheaves.

For X ∈ 𝒱 ar\overline{K}, set RΓ_{cr}(X_{h}) := RΓ(X_{h}, 𝒜_{cr}). It is a filtered (by RΓ(X_{h}, 𝒥_{cr}^{[r]}), r ≥ 0, ) E_{∞} A_{cr}-algebra equipped with the Frobenius action φ. The Galois group G_{K} acts on 𝒱 ar\overline{K} and it acts on X ↦ RΓ_{cr}(X_{h}) by transport of structure. If X is defined over K then G_{K} acts naturally on RΓ_{cr}(X_{h}).

For r ≥ 0, set RΓ_{syn}(X_{h}, r)_{n} = RΓ(X_{h}, 𝒮_{n}(r)), RΓ_{syn}(X_{h}, r) := RΓ(X_{h}, 𝒮 (r)_{Q}). We have

RΓ_{syn}(X_{h}, r)_{n} ≃ Cone(RΓ(X_{h}, 𝒥_{cr,n}^{[r]}) ⟶^{p^{r}−φ} RΓ(X_{h}, 𝒜_{cr,n}))[−1],
RΓ_{syn}(X_{h}, r) ≃ Cone(RΓ(X_{h}, 𝒥_{cr,Q}^{[r]}) ⟶^{1−φ_{r}} RΓ(X_{h}, 𝒜_{cr,Q}))[−1].

The direct sum ⨁_{r≥0} RΓ_{syn}(X_{h}, r) is a graded E_{∞} algebra over Z_{p}.

Let \overline{f} : Z_{1} → Spec(\overline{V}_{1})^{×} be an integral, quasi-coherent log-scheme. Suppose that \overline{f} is the base change of \overline{f}_{L} : Z_{L,1} → Spec(𝒪_{L,1})^{×} by θ_{1} : Spec(\overline{𝒪_{L,1}} )^{×} → Spec(𝒪_{L,1})^{×}, for a finite extension L/K. That is, we have a map θ_{L,1} : Z_{1} → Z_{L,1} such that the square (\overline{f}, \overline{f}_{L}, θ_{1}, θ_{L,1}) is Cartesian. Assume that \overline{f}_{L} is log-smooth of Cartier type and that the underlying map of schemes is proper. Such data (L, Z_{1}, θ_{L,1}) form a directed set Σ_{1} and, for a morphism (L′, Z′_{1}, θ′_{L′,1}) → (L, Z_{1}, θ_{L,1}), we have a canonical base change identification compatible with φ-action [4, 1.18]

RΓ_{HK}^{B}(Z_{L,1}) ⊗_{L_{0}} L′_{0} →^{∼} RΓ_{HK}^{B}(Z′_{L′,1}).

These identifications can be made compatible with respect to L, so we can set

RΓ_{HK}^{B}(Z_{1}) := \underrightarrow{lim}_{Σ_{1}} RΓ_{HK}^{B}(Z_{L,1})

It is a complex of (φ, N)-modules over K_{0}^{nr}, functorial with respect to morphisms of Z_{1}.

Consider the scheme E_{cr} := Spec(A_{cr}). We have E_{cr,1} = Spec(\overline{V}_{1}) and we equip E_{cr,1} with the induced log-structure. This log-structure extends uniquely to a log-structure on E_{cr,n} and the PD-thickening Spec(\overline{V})_{1}^{×} ↪ E_{cr,n} is universal over Z/p^{n}. Set E_{cr} := Spec(A_{cr}) with the limit log-structure. Since we have [4, 1.18.1]

RΓ_{cr}(Z_{1}) →^{∼} RΓ_{cr}(Z_{1}/E_{cr}),

Theorem 3.6 yields a canonical quasi-isomorphism of B_{cr}^{+}-complexes (called the crystalline Beilinson-Hyodo-Kato quasi-isomorphism)

ι_{cr}^{B} : RΓ_{HK}^{B}(Z_{1})_{B_{cr}^{+}}^{τ} →^{∼} RΓ_{cr}(Z_{1})_{Q}

compatible with the action of Frobenius. But we have

RΓ_{HK}^{B}(Z_{1})_{B_{cr}^{+}}^{τ} = (RΓ_{HK}^{B}(Z_{1}) ⊗_{K_{0}^{nr}} A_{cr,Q}^{τ})^{N=0}

and there is a canonical isomorphism A_{cr,Q}^{τ} →^{∼} B_{st}^{+} that is compatible with Frobenius and monodromy. This implies that the above quasi-isomorphism amounts to a quasi-isomorphism of B_{cr}^{+}-complexes

ι_{cr}^{B} : RΓ_{HK}^{B}(Z_{1})_{B_{st}^{+}} →^{∼} RΓ_{cr}(Z_{1}) ⊗_{A_{cr}}^{L} B_{st}^{+}

compatible with the action of φ and N. The crystalline Beilinson-Hyodo-Kato map can be canonically trivialized at [p̃], where p̃ is a sequence of p^{n}’th roots of p,:

β = β_{[p̃]} : RΓ_{HK}^{B}(Z_{1}) ⊗_{K_{0}^{nr}} B_{cr}^{+} = (RΓ_{HK}^{B}(Z_{1}) ⊗_{K_{0}^{nr}} B_{cr}^{+}[a([p̃])])^{N=0}
x ↦ exp(N(x)a([p̃]))

This trivialization is compatible with Frobenius and monodromy.

Suppose now that \overline{f}_{1} : Z_{1} → Spec(\overline{V}_{1})^{×} is a reduction mod p of a log-scheme \overline{f} : Z → Spec(\overline{V})^{×}. Suppose that \overline{f} is the base change of \overline{f}_{L} : Z_{L} → Spec(𝒪_{L})^{×} by θ : Spec(\overline{𝒪_{L}} )^{×} → Spec(𝒪_{L})^{×}, for a finite extension L/K. That is, we have a map θ_{L} : Z → Z_{L} such that the square (\overline{f}, \overline{f}_{L}, θ, θ_{L}) is Cartesian. Assume that \overline{f}_{L} is log-smooth of Cartier type and that the underlying map of schemes is proper. Such data (L, Z, θ_{L}) form a directed set Σ and the reduction mod p map Σ → Σ_{1} is cofinal. The Beilinson-Hyodo-Kato quasi-isomorphisms (27) are compatible with morphisms in Σ and their colimit yields a natural quasi-isomorphism (called again the Beilinson-Hyodo-Kato quasi-isomorphism)

ι_{dR}^{B} : RΓ_{HK}^{B}(Z_{1})τ ; \overline{K} →^{∼} RΓ(Z\overline{K}, ΩZ/\overline{K}^{•}).

The trivializations by p are also compatible with the maps in Σ hence we obtain the Beilinson-Hyodo-Kato maps

ι_{dR}^{B} := ι_{dR}^{B}β_{p} : RΓ_{HK}^{B}(Z_{1}) → RΓ(Z\overline{K}, ΩZ/\overline{K}^{•}).

For an ss-pair (U, \overline{U}) over \overline{K}, set RΓ_{HK}^{B}(U, \overline{U}) := RΓ_{HK}^{B}((U, \overline{U})_{1}). Let 𝒜_{HK}^{B} be h-sheafification of the presheaf (U, \overline{U}) ↦ RΓ_{HK}^{B}(U, \overline{U}) on 𝒫ss ; \overline{K}. This is an h-sheaf of E_{∞} K_{0}^{nr}-algebras equipped with a φ-action and locally nilpotent derivation N such that Nφ = pφN. For X ∈ 𝒱 ar\overline{K}, set RΓ_{HK}^{B}(X_{h}) := RΓ(X_{h}, 𝒜_{HK}^{B}).

Proposition 3.21. (1) For any (U, \overline{U}) ∈ 𝒫ss ; \overline{K}, the canonical maps

(41)  RΓ_{cr}(U, \overline{U}, 𝒥^{[r]})_{Q} →^{∼} RΓ(U_{h}, 𝒥_{cr}^{[r]})_{Q}, RΓ_{HK}^{B}(U, \overline{U}) →^{∼} RΓ_{HK}^{B}(U_{h})

are quasi-isomorphisms.

(2) For every X ∈ 𝒱 ar\overline{K}, the cohomology groups H_{cr}^{n}(X_{h}) := H^{n}RΓ_{cr}(X_{h})_{Q}, resp. H_{HK}^{n}(X_{h}) := H^{n}RΓ_{HK}^{B}(X_{h}), are free B_{cr}^{+}-modules, resp. K_{0}^{nr}-modules, of rank equal to the rank of H^{n}(X_{ét}, Q_{p}).

Proof. Only the filtered statement in part (1) for r > 0 requires argument since the rest has been proven by Beilinson in [4, 2.4]. Take r > 0. To prove that we have a quasi-isomorphism RΓ_{cr}(U, \overline{U}, 𝒥^{[r]})_{Q} →^{∼} RΓ(U_{h}, 𝒥_{cr}^{[r]})_{Q} it suffices to show that the map RΓ_{cr}(U, \overline{U}, 𝒪/𝒥^{[r]})_{Q} → RΓ(U_{h}, 𝒜_{cr}/𝒥_{cr}^{[r]})_{Q} is a quasiisomorphism. Since, for an ss-pair (T, \overline{T}) over K, by Corollary 2.4, RΓ_{cr}(T, \overline{T} , 𝒪/𝒥^{[r]})_{Q} ≃ RΓ(\overline{T}_{K}, Ω(T,\overline{T}_{K})^{•}/F^{r})

this is equivalent to showing that the map RΓ(\overline{U}_{K}, Ω(U,\overline{U}_{K})^{•}/F^{r}) → RΓ(U_{h}, 𝒜_{dR}/F^{r}) is a quasi-isomophism. And this follows from Proposition 3.14. □

Proposition 3.22. Let X ∈ 𝒱 ar_{K}. The natural projection ε : X_{\overline{K},h} → X_{h} defines pullback maps

(42)  ε^{∗} : RΓ_{HK}^{B}(X_{h}) → RΓ_{HK}^{B}(X_{\overline{K},h})^{G_{K}} , ε^{∗} : RΓ_{dR}(X_{h}) → RΓ_{dR}(X_{\overline{K},h})^{G_{K}} .

These are (filtered) quasi-isomosphisms.

Proof. Notice that the action of G_{K} on RΓ_{HK}^{B}(X_{\overline{K},h})\{r\} and RΓ_{dR}(X_{\overline{K},h}) is smooth, i.e., the stabilizer of every element is an open subgroup of G_{K}. We will prove only the first quasi-isomorphism - the proof of the second one being analogous. By Proposition 3.18, it suffices to show that for any ss-pair over K the natural map

RΓ_{HK}^{B}(U_{1}, \overline{U}_{1}) → RΓ_{HK}^{B}((U, \overline{U}) ⊗_{K} \overline{K})^{G_{K}}

is a quasi-isomorphism. Passing to a finite extension of K_{U}, if necessary, we may assume that (U, \overline{U}) is log-smooth of Cartier type over a finite Galois extension K_{U} of K. Then

RΓ_{HK}^{B}((U, \overline{U}) ⊗_{K} \overline{K}) ≃ RΓ_{HK}^{B}(U_{1}, \overline{U}_{1}) ⊗_{K_{U,0}} K_{0}^{nr} × H, H = Gal(K_{U}/K).

Taking G_{K}-fixed points of this quasi-isomorphism we obtain the first quasi-isomorphism of (42), as wanted. □

Let (U, \overline{U}) be an ss-pair over \overline{K}. Set

RΓ_{dR}^{♮}(U, \overline{U}) :=RΓ(\overline{U}_{ét}, LΩ(U,\overline{U})/W (k)^{•,∧}), RΓ_{dR}^{♮}(U, \overline{U})_{n} := RΓ_{dR}^{♮}(U, \overline{U}) ⊗^{𝕃} Z/p^{n} ≃ RΓ(\overline{U}_{ét}, LΩ(U,\overline{U})_{n}/W_{n}(k)^{•,∧}),
RΓ_{dR}^{♮}(U, \overline{U})ˆ⊗Z_{p} := holim_{n} RΓ_{dR}^{♮}(U, \overline{U})_{n}, RΓ_{dR}^{♮}(U, \overline{U})ˆ⊗Q_{p} := (RΓ_{dR}^{♮}(U, \overline{U})ˆ⊗Z_{p}) ⊗ Q.

These are F -filtered E_{∞} algebras. Take the associated presheaves on 𝒫ss ; \overline{K}. Denote by 𝒜_{dR}^{♮}, 𝒜_{dR,n}^{♮}, 𝒜_{dR}^{♮} ˆ⊗Z_{p}, 𝒜_{dR}^{♮} ˆ⊗Q_{p} their sheafifications in the h-topology of 𝒱 ar\overline{K}. These are sheaves of F -filtered E_{∞} algebras (viewed as the projective system of quotients modulo F^{i}). Set A_{dR} := LΩ\overline{V} /V^{•,∧}. By [3, Lemma 3.2] we have

A_{dR} = 𝒜_{dR}^{♮}(Spec(\overline{K})) = RΓ_{dR}^{♮}(\overline{K}, \overline{V}). The corresponding F -filtered algebras A_{dR,n}, A_{dR} ˆ⊗Z_{p}, A_{dR} ˆ⊗Q_{p} are acyclic in nonzero degrees and the projections ·/F^{m+1} → ·/F^{m} are surjective. Thus (we set lim_{F} := holim_{F})

A_{dR,n}^{⋄} := lim_{F} A_{dR,n} = \underleftarrow{lim}_{m} H^{0}(A_{dR,n}/F^{m}), A_{dR}^{⋄} := lim_{F}(A_{dR} ˆ⊗Z_{p}) = \underleftarrow{lim}_{m} H^{0}(A_{dR} ˆ⊗Z_{p}/F^{m})
lim_{F} A_{dR} ˆ⊗Q_{p} = \underleftarrow{lim}_{m} H^{0}(A_{dR} ˆ⊗Q_{p}/F^{m}) = B_{dR}^{+}, A_{dR} ˆ⊗Q_{p}/F^{m} = B_{dR}^{+}/F^{m}

For any (U, \overline{U}) over \overline{K}, the complex RΓ_{dR}^{♮}(U, \overline{U}) is an F -filtered E_{∞} filtered A_{dR}-algebra hence lim_{F} RΓ_{dR}^{♮}(U, \overline{U})_{n} is an A_{dR,n}^{⋄}-algebra, lim_{F} (RΓ_{dR}^{♮}(U, \overline{U})ˆ⊗Q_{p}) is a B_{dR}^{+}-algebra, etc. We have canonical morphisms

κ′_{r,n} : RΓ_{cr}(U, \overline{U})_{n} → RΓ_{cr}(U, \overline{U})_{n}/F^{r} →^{∼} RΓ_{dR}^{♮}(U, \overline{U})_{n}/F^{r}

In the case of (\overline{K}, \overline{V}), from Theorem 2.1, we get isomorphisms κ′_{r,n} = κ_{r}^{−1} : A_{cr,n}/J^{[r]} →^{∼} A_{dR,n}/F^{r}. Hence A_{dR}^{⋄} is the completion of A_{cr} with respect to the J^{[r]}-topology.

For X ∈ 𝒱 ar\overline{K}, set RΓ_{dR}^{♮}(X_{h}) := RΓ(X_{h}, 𝒜_{dR}^{♮}). Since A_{dR,Q} = \overline{K}, for any variety X over \overline{K}, we have a filtered quasi-isomorphism of \overline{K}-algebras [3, 3.2] RΓ_{dR}^{♮}(X_{h})_{Q} →^{∼} RΓ_{dR}(X_{h}) obtained by h-sheafification of the quasi-isomorphism

(43)  RΓ_{dR}^{♮}(U, \overline{U})_{Q} →^{∼} RΓ_{dR}(U, \overline{U}_{Q}).

Concerning the p-adic coefficients, we have a quasi-isomorphism

(44)  γ_{r} : (RΓ_{dR}(X_{h}) ⊗\overline{K} B_{dR}^{+})/F^{r} →^{∼} RΓ(X_{h}, 𝒜_{dR}^{♮} ˆ⊗Q_{p})/F^{r}

To define it, consider, for any ss-pair (U, \overline{U}) over \overline{K}, the natural map RΓ_{dR}^{♮}(U, \overline{U}) → RΓ_{dR}^{♮}(U, \overline{U})ˆ⊗Z_{p}. It yields, by extension to A_{dR} ˆ⊗Q_{p} and by the quasi-isomorphism (43), a quasi-isomorphism of F -filtered \overline{K}-algebras [4, 3.5]

γ : RΓ_{dR}(U, \overline{U})_{Q} ⊗\overline{K} (A_{dR} ˆ⊗Q_{p}) →^{∼} RΓ_{dR}^{♮}(U, \overline{U})ˆ⊗Q_{p}

Its mod F^{r}-version γ_{r} after h-sheafification yields the quasi-isomorphism

γ_{r} : (𝒜_{dR} ⊗\overline{K} B_{dR}^{+})/F^{r} →^{∼} 𝒜_{dR}^{♮} ˆ⊗Q_{p}/F^{r}

Passing to RΓ(X_{h},•) we get the quasi-isomorphism (44).

For X ∈ 𝒱 ar\overline{K}, we have canonical quasi-isomorphisms

ι_{cr}^{B} : RΓ_{HK}^{B}(X_{h})_{B_{cr}^{+}}^{τ} →^{∼} RΓ_{cr}(X_{h})_{Q}, ι_{dR}^{B} : RΓ_{HK}^{B}(X_{h})τ ; \overline{K} →^{∼} RΓ_{dR}(X_{h}),

compatible with the Gal(\overline{K}/K)-action. Here _{B_{cr}^{+}}^{τ} and τ ; \overline{K} denote the h-sheafification of the crystalline and the de Rham Beilinson-Hyodo-Kato twists [4, 2.5.1]. Trivializing the first map at [p̃] and the second map at p we get the Beilinson-Hyodo-Kato maps

ι_{cr}^{B} := ι_{cr}^{B}β_{[p̃]} : RΓ_{HK}^{B}(X_{h}) ⊗_{K_{0}^{nr}} B_{cr}^{+} → RΓ_{cr}(X_{h})_{Q}, ι_{dR} := ι_{dR}β_{p} : RΓ_{HK}^{B}(X_{h}) → RΓ_{dR}(X_{h}).

Using the quasi-isomorphism κ_{r}^{−1} : 𝒜_{cr,Q}/𝒥_{cr,Q}^{[r]} →^{∼} (𝒜_{dR}^{♮} ˆ⊗Q_{p})/F^{r} from Theorem 2.1, we obtain the following quasi-isomorphisms of complexes of sheaves on X_{\overline{K},h}

𝒮 (r)_{Q} \xymatrix{→^{∼} & [𝒥_{cr,Q}^{[r]} \ar[r]^{1−φ_{r}} & 𝒜_{cr,Q}] & →^{∼} & [𝒜_{cr,Q} \ar[r]^{(1−φ_{r},can)} & 𝒜_{cr,Q} ⊕ 𝒜_{cr,Q}/𝒥_{cr,Q}^{[r]}]}
←^{∼} \xymatrix{[𝒜_{cr,Q} \ar[r]^{(1−φ_{r},κ_{r}^{−1})} & 𝒜_{cr,Q} ⊕ (𝒜_{dR}^{♮} ˆ⊗Q_{p})/F^{r}]}

Applying RΓ(X_{h},•) and the quasi-isomorphism γ_{r}^{−1} : RΓ(X_{h}, 𝒜_{dR}^{♮} ˆ⊗Q_{p})/F^{r} →^{∼} (RΓ_{dR}(X_{h}) ⊗\overline{K} B_{dR}^{+})/F^{r} from (44) we obtain the following quasi-isomorphisms

(45)  RΓ_{syn}(X_{h}, r) →^{∼} \xymatrix{[RΓ_{cr}(X_{h})_{Q} \ar[r]^{(1−φ_{r},κ_{r}^{−1})} & RΓ_{cr}(X_{h})_{Q} ⊕ RΓ(X_{h}, 𝒜_{dR}^{♮} ˆ⊗Q_{p})/F^{r}]}
→^{∼} \xymatrix{[RΓ_{cr}(X_{h})_{Q} \ar[r]^{(1−φ_{r},γ_{r}^{−1}κ_{r}^{−1})} & RΓ_{cr}(X_{h})_{Q} ⊕ (RΓ_{dR}(X_{h}) ⊗\overline{K} B_{dR}^{+})/F^{r}]}

Corollary 3.23. For any (U, \overline{U}) ∈ 𝒫ss ; \overline{K}, the canonical map

RΓ_{syn}(U, \overline{U}, r)_{Q} →^{∼} RΓ_{syn}(U_{h}, r)

is a quasi-isomorphism.

Proof. Arguing as above we find quasi-isomorphisms

RΓ_{syn}(U, \overline{U}, r)_{Q} →^{∼} \xymatrix{[RΓ_{cr}(U, \overline{U})_{Q} \ar[r]^{(1−φ_{r},κ_{r}^{−1})} & RΓ_{cr}(U, \overline{U})_{Q} ⊕ (RΓ^{♮}(U, \overline{U})ˆ⊗Q_{p})/F^{r}]}
→^{∼} \xymatrix{[RΓ_{cr}(U, \overline{U})_{Q} \ar[r]^{(1−φ_{r},γ_{r}^{−1}κ_{r}^{−1})} & RΓ_{cr}(U, \overline{U})_{Q} ⊕ (RΓ_{dR}(U, \overline{U}) ⊗\overline{K} B_{dR}^{+})/F^{r}]}

Comparing them with quasi-isomorphisms (45) we see that it suffices to check that the natural maps

RΓ_{cr}(U, \overline{U})_{Q} →^{∼} RΓ_{cr}(U_{h})_{Q}, RΓ_{dR}(U, \overline{U}) →^{∼} RΓ_{dR}(U_{h}),

are (filtered) quasi-isomorphism. But this is known by Proposition 41 and Proposition 3.14. □

Consider the following composition of morphisms

RΓ_{syn}(X_{h}, r) →^{∼} [ \xymatrix{RΓ_{cr}(X_{h})_{Q} \ar[r]^{(1−φ_{r},γ_{r}^{−1}κ_{r}^{−1})} & RΓ_{cr}(X_{h})_{Q} ⊕ (RΓ_{dR}(X_{h}) ⊗\overline{K} B_{dR}^{+})/F^{r}} ]

←^{∼} [ \xymatrix{RΓ_{HK}^{B}(X_{h}) ⊗_{K_{0}^{nr}} B_{st}^{+} \ar[r]^{(1−φ_{r},ι_{dR}^{B}⊗ι)} \ar[d]^{N} & RΓ_{HK}^{B}(X_{h}) ⊗_{K_{0}^{nr}} B_{st}^{+} ⊕ (RΓ_{dR}(X_{h}) ⊗\overline{K} B_{dR}^{+})/F^{r} \ar[d]^{(N,0)} \\ RΓ_{HK}^{B}(X_{h}) ⊗_{K_{0}^{nr}} B_{st}^{+} \ar[r]^{1−φ_{r−1}} & RΓ_{HK}^{B}(X_{h}) ⊗_{K_{0}^{nr}} B_{st}^{+}} ]

(46)

The second quasi-isomorphism uses the map

(RΓ_{HK}^{B}(X_{h}) ⊗_{K_{0}^{nr}} B_{st}^{+})^{N=0} = RΓ_{HK}^{B}(X_{h})_{B_{cr}^{+}}^{τ} →^{ι_{cr}^{B}} RΓ_{cr}(X_{h})_{Q}

(that is compatible with the action of N and φ) and the following lemma.

Lemma 3.24. The following diagrams commute

\xymatrix{RΓ_{cr}(X_{h})_{Q} ⊗_{B_{cr}^{+}} B_{st}^{+} \ar[r]^{γ_{r}^{−1}κ_{r}^{−1}⊗ι} & (RΓ_{dR}(X_{h}) ⊗\overline{K} B_{dR}^{+})/F^{r} & RΓ_{cr}(X_{h})_{Q} ⊗_{A_{cr}} B_{dR} \ar[r]^{γ_{dR}}_{∼} & RΓ_{dR}(X_{h}) ⊗\overline{K} B_{dR} \\ RΓ_{HK}^{B}(X_{h}) ⊗_{K_{0}^{nr}} B_{st}^{+} \ar[ur]_{ι_{dR}^{B}⊗ι} \ar[u]^{ι_{cr}^{B}}_{≀} &  & RΓ_{HK}^{B}(X_{h}) ⊗_{K_{0}^{nr}} B_{st} \ar[u]^{ι_{cr}^{B}⊗ι} \ar@{-->}[ur]_{ι_{dR}^{B}⊗ι}}

Here γ_{dR} is the map defined by Beilinson in [4, 3.4.1].

Proof. We will start with the left diagram. It suffices to show that it canonically commutes with X_{h} replaced by any ss-pair \overline{Y} = (U, \overline{U}) over \overline{K} – a base change of an ss-pair Y split over (V, K). Proceeding as in Example 3.5, we obtain the following diagram in which all squares but the one in the left bottom clearly commute.

\xymatrix{RΓ_{HK}^{B}(Y_{1})_{K}^{τ} \ar[d]^{Id ⊗1} \ar[r]^{ι_{K}^{B}} & RΓ_{cr}(Y_{1}/V^{×})_{Q}/F^{r} \ar[d] & RΓ(Y_{ét}, LΩY/V^{×}^{•,∧})ˆ⊗Q_{p}/F^{r} \ar[l]_{κ_{r}}^{∼} \ar[d] & RΓ_{dR}(Y_{K})/F^{r} \ar[l]_{γ_{r}}^{∼} \ar[d] \\ RΓ_{HK}^{B}(\overline{Y}_{1})τ ; \overline{K} ⊗\overline{K} B_{dR}^{+} \ar[r]^{ιB ; \overline{K}⊗κ_{r}} & RΓ_{cr}(\overline{Y}_{1}/V^{×})_{Q}/F^{r} & RΓ(\overline{Y}_{ét}, LΩ\overline{Y} /V^{×}^{•,∧})ˆ⊗Q_{p}/F^{r} \ar[l]_{κ_{r}}^{∼} & (RΓ_{dR}(\overline{Y}_{K}) ⊗\overline{K} B_{dR}^{+})/F^{r} \ar[l]_{γ_{r}}^{∼} \ar[dl]_{γ_{r}}^{∼} \\ RΓ_{HK}^{B}(\overline{Y}_{1})_{B_{cr}^{+}}^{τ} ⊗_{B_{cr}^{+}} B_{st}^{+} \ar[u]^{δ} \ar[r]^{ι_{cr}^{B}⊗κ_{r}ι} & RΓ_{cr}(\overline{Y}_{1}/A_{cr})_{Q}/F^{r} \ar[u]^{≀} & (RΓ_{dR}^{♮}(\overline{Y})ˆ⊗Q_{p})/F^{r} \ar[l]_{κ_{r}}^{∼} \ar[u]^{≀}}

Here we think B_{dR}^{+}/F^{m} = RΓ_{dR}^{♮}(\overline{K}, \overline{V})ˆ⊗Q_{p} and the map δ is defined as the composition

δ : RΓ_{HK}^{B}(\overline{Y}_{1})_{B_{cr}^{+}}^{τ}⊗_{B_{cr}^{+}}B_{st}^{+} = (RΓ_{HK}^{B}(\overline{Y}_{1})⊗_{K_{0}^{nr}}B_{st}^{+})^{N=0}⊗_{B_{cr}^{+}}B_{st}^{+} →^{∼} RΓ_{HK}^{B}(\overline{Y}_{1})⊗_{K_{0}^{nr}}B_{st}^{+}⟶^{β_{p}⊗ι}RΓ_{HK}^{B}(\overline{Y}_{1})τ ; \overline{K}⊗\overline{K}B_{dR}^{+}

Recall that for the map ι_{dR}^{B} : RΓ_{HK}^{B}(Y_{1})_{K}^{τ} → RΓ_{dR}(Y_{K})/F^{r} we have ι_{dR}^{B} = γ_{r}^{−1}κ_{r}^{−1}ι_{K}^{B}. Everything in sight being compatible with change of the ss-pairs Y - more specifically with maps in the directed system Σ - if this diagram commutes so does its Σ colimit and the left diagram in the lemma for the pair (U, \overline{U}).

It remains to show that the left bottom square in the above diagram commutes. To do that consider the ring Â_{n} defined as the PD-envelope of the closed immersion

\overline{V} _{1}^{×} ↪ A_{cr,n} ×_{W_{n}(k)} V_{n}^{×}

That is, Â_{n} is the product of the PD-thickenings (\overline{V} _{1}^{×} ↪ A_{cr,n}) and (V_{1}^{×} ↪ V_{n}^{×}) over (W_{1}(k) ↪ W_{n}(k)). By [4, Lemma 1.17], this makes \overline{V} _{1}^{×} ↪ Â_{cr,n} into the universal PD-thickening in the log-crystalline site of \overline{V} _{1}^{×} over V_{n}^{×}. Let Â:= inj lim_{n} Â_{cr,n} with the limit log-structure. Set B̂_{cr}^{+} := Â_{cr}[1/p].

Using Theorem 3.6 , we obtain a canonical quasi-isomorphism

ιB ; B̂_{cr}^{+} : RΓ_{HK}^{B}(\overline{Y}_{1})τ ; B̂_{cr}^{+} →^{∼} RΓ_{cr}(\overline{Y}_{1}/Â_{cr})_{Q}

By construction, we have the maps of PD-thickenings

\xymatrix{(V_{1}^{×} ↪ V^{×}) & (\overline{V} _{1}^{×} ↪ Â_{cr}) \ar[l]_{^{pr^{1}}} \ar[r]^{^{pr^{2}}} & (\overline{V} _{1}^{×} ↪ A_{cr})}

Consider the following diagram

\xymatrix{RΓ_{HK}^{B}(\overline{Y}_{1})τ ; B̂_{cr}^{+} \ar[ddd]_{ιB ; B̂{+ \atop cr}} &  & RΓ_{HK}^{B}(\overline{Y}_{1})τ ; \overline{K} ⊗\overline{K} B_{dR}^{+}/F^{r} \ar[ll]_{pr_{1}^{∗} ⊗ pr_{1}^{∗} κ_{r}} \ar[ddd]^{ιB ; \overline{K}⊗κ_{r}} \\  & RΓ_{HK}^{B}(\overline{Y}_{1})_{B_{cr}^{+}}^{τ} \ar[d]^{ι_{cr}^{B}} \ar[ul]^{_{pr{∗ \atop 2}}} \ar[ur]^{δ} \\  & RΓ_{cr}(\overline{Y}_{1}/A_{cr})_{Q}/F^{r} \ar[dl]_{_{pr{∗ \atop 2}}}^{∼} \ar[dr]^{∼} \\ RΓ_{cr}(\overline{Y}_{1}/Â_{cr})_{Q}/F^{r} &  & RΓ_{cr}(\overline{Y}_{1}/V^{×})_{Q}/F^{r} \ar[ll]^{_{pr{∗ \atop 1}}}_{∼}}

The bottom triangle commutes since RΓ_{cr}(\overline{Y}_{1}/A_{cr}) = RΓ_{cr}(\overline{Y}_{1}/W (k)). The pullback maps

pr_{1}^{∗} : RΓ_{cr}(\overline{Y}_{1}/V^{×}) →^{∼} RΓ_{cr}(\overline{Y} /Â_{cr}),
pr_{2}^{∗} : RΓ_{cr}(\overline{Y} /A_{cr})_{Q}/F^{r} →^{∼} RΓ_{cr}(\overline{Y} /Â_{cr})_{Q}/F^{r}

are quasi-isomorphisms. Indeed, in the case of the first pullback this follows from the universal property of Â_{cr}; in the case of the second one - it follows from the commutativity of the bottom triangle since the right slanted map is a quasi-isomorphism as shown by the first diagram in our proof.

The left trapezoid and the big square commute by the definition of the Beilinson-Bloch-Kato maps. To see that the top triangle commutes it suffices to show that for an element

x ∈ RΓ_{HK}^{B}(\overline{Y}_{1})_{B_{cr}^{+}}^{τ} = (RΓ_{HK}^{B}(\overline{Y}_{1}) ⊗_{K_{0}^{nr}} B_{st}^{+})^{N=0}, x = b ∑_{i≥0} N^{i}(m)a([p̃])^{[i]}, m ∈ RΓ_{HK}^{B}(\overline{Y}_{1}), b ∈ B_{cr}^{+},

we have pr_{2}^{∗}(x) = pr_{1}^{∗} δ(x). Since ι(a([p̃])) = log([p̃]/p) [28, 4.2.2], we calculate

δ(x) = δ(b ∑_{i≥0} N^{i}(m)a([p̃])^{[i]}) = b ∑_{i≥0}(∑_{j≥0} N^{i+j}(m)a(p)^{[j]}) log([p̃]/p)^{[i]}

= b ∑_{k≥0} N^{k}(m)(a(p) + log([p̃]/p))^{[k]}

Since in B̂_{cr}^{+} we have [p̃] = ([p̃]/p)p and [p̃]/p ∈ 1 + JB̂_{cr}^{+}, it follows that a([p̃]) = log([p̃]/p) + a(p) and

pr_{1}^{∗} δ(x) = pr_{1}^{∗}(b ∑_{k≥0} N^{k}(m)(a(p)+log([p̃]/p))^{[k]}) = b ∑_{k≥0} N^{k}(m)a([p̃])^{[k]} = pr_{2}^{∗}(b ∑_{k≥0} N^{k}(m)a([p̃])^{[k]}) = pr_{2}^{∗}(x),

as wanted. It follows now that the right trapezoid in the above diagram commutes as well and that so does the left diagram in our lemma.

To check the commutativity of the right diagram, consider the following map obtained from the maps κ′_{r,n} by passing to F -limit

κ′_{n} : RΓ_{cr}(\overline{Y})_{n} ⊗_{A_{cr,n}}^{L} A_{dR,n} →^{∼} \underleftarrow{lim}_{F} RΓ_{cr}(\overline{Y})_{n}/F^{r}

By [4, 3.6.2], this is a quasi-isomorphism. Beilinson [4, 3.4.1] defines the map

γ_{dR} : RΓ_{cr}(\overline{Y})_{Q} ⊗_{A_{cr}} B_{dR}^{+} →^{∼} RΓ_{dR}(\overline{Y}_{K}) ⊗\overline{K} B_{dR}^{+}

by B_{dR}^{+}-linearization of the composition \underleftarrow{lim}_{r}(γ_{r}^{−1}κ_{r}^{−1}) holim_{n} κ′_{n}. We have

γ_{dR} = γ_{r}^{−1}κ_{r}^{−1} : RΓ_{cr}(\overline{Y})_{Q} → (RΓ_{dR}(\overline{Y}_{K}) ⊗\overline{K} B_{dR}^{+})/F^{r}

Hence the commutativity of the right diagram follows from that of the left one. □

Let C^{+}(RΓ_{HK}^{B}(X_{h}){r}) denote the second homotopy limit in the diagram (46); denote by C(RΓ_{HK}^{B}(X_{h}){r}) the complex C^{+}(RΓ_{HK}^{B}(X_{h}){r}) with all the pluses removed. We have defined a map α_{syn} : RΓ_{syn}(X_{h}, r) → C^{+}(RΓ_{HK}^{B}(X_{h}){r}) and proved the following proposition.

Proposition 3.25. There is a functorial G_{K}-equivariant quasi-isomorphism

α_{syn} : RΓ_{syn}(X_{h}, r) = RΓ(X_{h}, 𝒮 (r)_{Q}) ≃ C^{+}(RΓ_{HK}^{B}(X_{h}){r}).

Corollary 3.26. For (U, \overline{U}) ∈ 𝒫_{K}^{ss}, we have a long exact sequence

→ H_{syn}^{i}((U, \overline{U})\overline{K}, r) → (H_{HK}^{i}(U, \overline{U})_{Q} ⊗_{K_{0}} B_{st}^{+})^{φ=p^{r},N=0} → (H_{dR}^{i}(U, \overline{U}) ⊗_{K} B_{dR}^{+})/F^{r} → H_{syn}^{i+1}((U, \overline{U})\overline{K}, r) →

Proof. By diagram (46), it suffices to show that

H^{i}[RΓ_{HK}^{B}((U, \overline{U})_{1}) ⊗_{K_{0}} B_{st}^{+}]^{φ=p^{r},N=0} ≃ (H_{HK}^{i}(U, \overline{U})_{Q} ⊗_{K_{0}} B_{st}^{+})^{φ=p^{r},N=0},
H^{i}(RΓ_{dR}(U, \overline{U}) ⊗_{K} B_{dR}^{+})/F^{r}) ≃ (H_{dR}^{i}(U, \overline{U}) ⊗_{K} B_{dR}^{+})/F^{r}

The second isomorphism is a consequence of the degeneration of the Hodge-de Rham spectral sequence. Keeping in mind that the Beilinson-Hyodo-Kato complexes RΓ_{HK}^{B}((U, \overline{U})_{1}) are built from (φ, N)-modules, the first isomorphism follows from the following short exact sequences (for a (φ, N)-module M)

0 →M ⊗_{K_{0}} B_{cr}^{+} → M ⊗_{K_{0}} B_{st}^{+} →^{N} M ⊗_{K_{0}} B_{st}^{+} → 0,
0 →(M ⊗_{K_{0}} B_{cr}^{+})^{φ=p^{r}} → M ⊗_{K_{0}} B_{cr}^{+} →^{1−φ_{r}} M ⊗_{K_{0}} B_{cr}^{+} → 0.

The first one follows, by induction on m such that N^{m} = 0 on M, from the exact sequence (11) and the fact that (M ⊗_{K_{0}} B_{st}^{+})^{N=0} ≃ M ⊗_{K_{0}} B_{cr}^{+}. The second one follows from [21, Remark ??]. □

4. Relation between syntomic cohomology and étale cohomology

In this section we will study the relationship between syntomic and étale cohomology in both the geometric and the arithmetic situation.

4.1. Geometric case. We start with the geometric case. In this subsection, we will construct the geometric syntomic period map from syntomic to étale cohomology. We will prove that in the torsion case, on the level of h-sheaves it is a quasi-isomorphism modulo a universal constant; in the rational case it induces an isomorphism on cohomology groups in a stable range. Finally, we will construct the syntomic descent spectral sequence.

We will first recall the de Rham and Crystalline Poincaré Lemmas of Beilinson and Bhatt [3], [4], [11].

Theorem 4.1. (de Rham Poincaré Lemma [3, 3.2]) The maps A_{dR} ⊗^{L} Z/p^{n} → 𝒜_{dR}^{♮} ⊗^{L} Z/p^{n} are filtered quasi-isomorphisms of h-sheaves on 𝒱 ar\overline{K}.

Theorem 4.2. (Filtered Crystalline Poincaré Lemma [4, 2.3], [11, Theorem 10.14]) The map J_{cr,n}^{[r]} → 𝒥_{cr,n}^{[r]} is a quasi-isomorphism of h-sheaves on 𝒱 ar\overline{K}.

Proof. We have the following map of distinguished triangles

J_{cr,n}^{[r]} ⟶ A_{cr,n} ⟶ A_{cr,n}/J_{cr,n}^{[r]}
↓  ↓^{≀}  ↓^{≀}
𝒥_{cr,n}^{[r]} ⟶ 𝒜_{cr,n} ⟶ 𝒜_{cr,n}/𝒥_{cr,n}^{[r]}

The middle map is a quasi-isomorphism by the Crystalline Poincaré Lemma proved in [4, 2.3]. Hence it suffices to show that so is the rightmost map. But, by [4, 1.9.2], this map is quasi-isomorphic to the map A_{dR,n}/F^{r} → 𝒜_{dR,n}^{♮}/F^{r}. Since the last map is a quasi-isomorphism by the de Rham Poincaré Lemma (4.1) we are done. □

We will now recall the definitions of the crystalline, Beilinson-Hyodo-Kato, and de Rham period maps [4, 3.1], [3, 3.5]. Let X ∈ 𝒱 ar\overline{K}. To define the crystalline period map

ρ_{cr} : RΓ_{cr}(X_{h}) → RΓ(X_{ét}, Z_{p})ˆ⊗A_{cr},

consider the natural map α_{n} : RΓ_{cr}(X_{h}) → RΓ(X_{h}, 𝒜_{cr,n}) and the composition

β_{n} : RΓ(X_{ét}, Z_{p}(r)) ⊗_{Z_{p}}^{L} A_{cr,n} →^{∼} RΓ(X_{ét}, A_{cr,n}) →^{∼} RΓ(X_{h}, A_{cr,n}) →^{∼} RΓ(X_{h}, 𝒜_{cr,n}).

Set ρ_{cr,n} := β_{n}^{−1}α_{n} and ρ_{cr} := holim_{n} ρ_{cr,n}. The Hyodo-Kato period map

ρ_{HK} : RΓ_{HK}^{B}(X_{h})_{B_{cr}^{+}}^{τ} → RΓ(X_{ét}, Q_{p}) ⊗ B_{cr}^{+}, ρ_{HK} = ρ_{cr,Q}ι_{cr}^{B},

is obtained by composing the map ρ_{cr,Q} with the quasi-isomorphism ι_{cr}^{B} : RΓ_{HK}^{B}(X_{h})_{B_{cr}^{+}}^{τ} →^{∼} RΓ_{cr}(X_{h})_{Q}. The maps ρ_{cr}, ρ_{HK} are morphisms of E_{∞} A_{cr}- and B_{cr}^{+}-algebras equipped with a Frobenius action; they are compatible with the action of the Galois group G_{K}.

To define the de Rham period map ρ_{dR} : RΓ_{dR}(X_{h}) ⊗\overline{K} B_{dR}^{+} → RΓ(X_{ét}, Q_{p}) ⊗ B_{dR}^{+} consider the compositions

α : RΓ_{dR}(X_{h}) →^{∼} RΓ_{dR}^{♮}(X_{h}) ⊗ Q → RΓ_{dR}^{♮}(X_{h})ˆ⊗Q_{p},
β : RΓ(X_{ét}, Z) ⊗^{L} A_{dR} →^{∼} RΓ(X_{ét}, A_{dR}) → RΓ(X_{h}, A_{dR}) → RΓ(X_{h}, 𝒜_{dR}^{♮}) = RΓ_{dR}^{♮}(X_{h}).

After tensoring the map β with Z/p^{n} and using the de Rham Poincaré Lemma we get a quasi-isomorphism

β_{n} : RΓ(X_{ét}, Z/p^{n}) ⊗^{L} A_{dR} →^{∼} RΓ_{dR}^{♮}(X_{h}) ⊗^{L} Z/p^{n}.

Set β_{Q} := holim_{n} β_{n} ⊗ Q and ρ_{dR} := β^{−1}α. This is a morphism of filtered E_{∞} B_{dR}^{+}-algebras, compatible with G_{K}-action.

Theorem 4.3. ([4, 3.2], [3, 3.6])For X ∈ 𝒱 ar\overline{K}, we have canonical quasi-isomorphisms

ρ_{cr} : RΓ_{cr}(X_{h}) ⊗_{A_{cr}} B_{cr} →^{∼} RΓ(X_{ét}, Q_{p}) ⊗ B_{cr}, ρ_{HK} : RΓ_{HK}^{B}(X_{h})_{B_{cr}}^{τ} →^{∼} RΓ(X_{ét}, Q_{p}) ⊗ B_{cr},
ρ_{dR} : RΓ_{dR}(X_{h}) ⊗\overline{K} B_{dR} →^{∼} RΓ(X_{ét}, Q_{p}) ⊗ B_{dR}.

Pulling back ρ_{HK} to the Fontaine-Hyodo-Kato 𝔾_{a}-torsor Spec(B_{st})/ Spec(B_{cr}) we get a canonical quasi-isomorphism of B_{st}-complexes

(47)  ρ_{HK} : RΓ_{HK}^{B}(X_{h}) ⊗_{K_{0}^{nr}} B_{st} →^{∼} RΓ(X_{ét}, Q_{p}) ⊗ B_{st}

compatible with the (φ, N)-action and with the G_{K}-action on 𝒱 ar\overline{K}.

Corollary 4.4. The period morphisms are compatible, i.e., the following diagrams commute.

\xymatrix{RΓ_{HK}^{B}(X_{h}) ⊗_{K_{0}^{nr}} B_{st} \ar[r]^{ι_{dR}^{B}⊗ι} \ar[d]^{ρ_{HK}} & RΓ_{dR}(X_{h}) ⊗\overline{K} B_{dR} \ar[d]^{ρ_{dR}} & RΓ_{cr}(X_{h}) ⊗_{A_{cr}} B_{dR} \ar[d]_{ρ_{cr}⊗Id_{B_{dR}}} & RΓ_{dR}(X_{h}) ⊗\overline{K} B_{dR} \ar[dl]^{ρ_{dR}} \ar[l]_{γ_{dR}}^{∼} \\ RΓ(X_{ét}, Q_{p}) ⊗ B_{st} \ar[r]^{1⊗ι} & RΓ(X_{ét}, Q_{p}) ⊗ B_{dR} & RΓ(X_{ét}, Q_{p}) ⊗ B_{dR}}

Proof. The second diagram commutes by [4, 3.4]. The commutativity of the first one can be reduced, by the equality ρ_{HK} = ρ_{cr}ι_{cr}^{B} and the second diagram above, to the commutativity of the right diagram in Lemma 3.24. □

We will now define the syntomic period map

ρ_{syn} : RΓ_{syn}(X_{h}, r)_{Q} → RΓ(X_{ét}, Q_{p}(r)), r ≥ 0.

Set Z/p^{n}(r)′ := (1/(p^{a}a!)Z_{p}(r)) ⊗ Z/p^{n}, where a is the largest integer ≤ r/(p − 1). Recall that we have the fundamental exact sequence [62, Theorem 1.2.4]

0 → Z/p^{n}(r)′ → J_{cr,n}^{<r>}⟶^{1−φ_{r}}A_{cr,n} → 0,

where

J_{n}^{<r>} := {x ∈ J_{n+s}^{[r]} | φ(x) ∈ p^{r}A_{cr,n+s}}/p^{n},

for some s ≥ r. Set S_{n}(r) := Cone(J_{cr,n}^{[r]}⟶^{p^{r}−φ}A_{cr,n})[−1]. There is a natural morphism of complexes S_{n}(r) → Z/p^{n}(r)′ (induced by p^{r} on J_{cr,n}^{[r]} and Id on A_{cr,n}) , whose kernel and cokernel are annihilated by p^{r}.

The Filtered Crystalline Poincaré Lemma implies easily the following Syntomic Poincaré Lemma.

Corollary 4.5. (1) For 0 ≤ r ≤ p − 2, there is a unique quasi-isomorphism Z/p^{n}(r) ⟶^{∼} 𝒮_{n}(r) of

complexes of sheaves on 𝒱 ar_{\overline{K},h} that is compatible with the Crystalline Poincaré Lemma.

(2) There is a unique quasi-isomorphism S_{n}(r) →^{∼} 𝒮_{n}(r) of complexes of sheaves on 𝒱 ar_{\overline{K},h} that is compatible with the Crystalline Poincaré Lemma.

Proof. We will prove the second claim - the first one is proved in an analogous way. Consider the following map of distinguished triangles

\xymatrix{𝒮_{n}(r) \ar[r] & 𝒥_{cr,n}^{[r]} \ar[r]^{p^{r}−φ} & 𝒜_{cr,n} \\ S_{n}(r) \ar[r] \ar@{-->}[u] & J_{cr,n}^{[r]} \ar[u]^{≀} \ar[r]^{p^{r}−φ} & A_{cr,n} \ar[u]^{≀}}

The triangles are distinguished by definition. The vertical continuous arrows are quasi-isomorphisms by the Crystalline Poincaré Lemma. They induce the dash arrow that is clearly a quasi-isomorphism. □

Consider the natural map α_{n} : RΓ(X_{h}, 𝒮 (r)) → RΓ(X_{h}, 𝒮_{n}(r)) and the zig-zag

β_{n} : RΓ(X_{h}, 𝒮_{n}(r)) ← RΓ(X_{h}, S_{n}(r)) → RΓ(X_{ét}, Z/p^{n}(r)′) ←^{∼} RΓ(X_{h}, Z/p^{n}(r)′).

Set β := (holim_{n} β_{n}) ⊗ Q; note that this is a quasi-isomorphism. Set

ρ_{syn} := p^{−r}βα : RΓ_{syn}(X_{h}, r) → RΓ(X_{ét}, Q_{p}(r)),

where α := (holim_{n} α_{n}) ⊗ Q. The period map ρ_{syn} induces a map of graded E_{∞} algebras over Q_{p} compatible with the action of the Galois group G_{K}.

The syntomic period map has a different, more global definition that we find very useful. Define the map ρ′_{syn} by the following diagram.

\xymatrix{RΓ_{syn}(X_{h}, r) \ar[r]^{∼} \ar[d]^{ρ′_{syn}} & [RΓ_{cr}(X_{h})_{Q} \ar[r]^{(1−φ_{r},γ_{r}^{−1}κ_{r}^{−1})} \ar[d]^{ρ_{cr}} & RΓ_{cr}(X_{h})_{Q} ⊕ RΓ_{dR}(X_{h})/F^{r}] \ar[d]^{ρ_{cr}+ρ_{dR}} \\ RΓ_{ét}(X, Q_{p}(r)) \ar[r]^{∼} & [RΓ_{ét}(X, Q_{p}(r)) ⊗ B_{cr} \ar[r]^{(1−φ_{r},can)} & RΓ_{ét}(X, Q_{p}(r)) ⊗ B_{cr} ⊕ RΓ_{ét}(X, Q_{p}(r)) ⊗ B_{dR}/F^{r}]}

This definition makes sense since the following diagram commutes.

\xymatrix{RΓ_{cr}(X_{h})_{Q} \ar[r]^{γ_{r}^{−1}κ_{r}^{−1}} \ar[d]^{ρ_{cr}} & RΓ_{dR}(X_{h})/F^{r} \ar[d]^{ρ_{dR}} \\ RΓ_{ét}(X, Q_{p}(r)) ⊗ B_{cr} \ar[r]^{can} & RΓ_{ét}(X, Q_{p}(r)) ⊗ B_{dR}/F^{r}}

The syntomic period morphisms ρ_{syn} and ρ′_{syn} are homotopic by a homotopy compatible with the G_{K}- action (and, unless necessary, we will not distinguish them in what follows). These two facts follow easily from the definitions.

For X ∈ 𝒱 ar_{K}, we have a quasi-isomorphism

(48)  α_{ét} : RΓ(X_{\overline{K},ét}, Q_{p}(r)) →^{∼} C(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\})

that we define as the inverse of the following composition of quasi-isomorphisms (square brackets denote complex)

C(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\})⟶_{∼}^{ρ}RΓ(X_{\overline{K},ét}, Q_{p}) ⊗_{Q_{p}} [B_{st} ⟶^{(N,1−φ_{r},ι)}B_{st} ⊕ B_{st} ⊕ B_{dR}/F^{r} ⟶^{(1−φ_{r−1})−N}B_{st}]
←^{∼} RΓ(X_{\overline{K},ét}, Q_{p}) ⊗_{Q_{p}} C(D_{st}(Q_{p}(r))) ←^{∼} RΓ(X_{\overline{K},ét}, Q_{p}(r)).

The last quasi-isomorphism is by Remark 2.7. The map ρ is defined using the period morphisms ρ_{HK} and ρ_{dR} and their compatibility (Corollary 4.4). The map α_{ét} is compatible with the action of G_{K}.

Proposition 4.6. For a variety X ∈ 𝒱 ar_{K}, we have a canonical, compatible with the action of G_{K}, quasi-isomorphism

ρ_{syn} : τ_{≤r}RΓ_{syn}(X_{\overline{K},h}, r) →^{∼} τ_{≤r}RΓ(X_{\overline{K},ét}, Q_{p}(r)).

Proof. The Bousfield-Kan spectral sequences associated to the homotopy limits defining the complexes C^{+}(H_{HK}^{j}(X_{\overline{K},h})\{r\}) and C(H_{HK}^{j}(X_{\overline{K},h})\{r\}) form the following commutative diagram

\xymatrix{^{+}E_{2}^{i,j} = H^{i}(C^{+}(H_{HK}^{j}(X_{\overline{K},h})\{r\})) \ar[d]^{can} \ar@{=>}[r] & H^{i+j}(C^{+}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\})) \ar[d]^{can} \\ E_{2}^{i,j} = H^{i}(C(H_{HK}^{j}(X_{\overline{K},h})\{r\})) \ar@{=>}[r] & H^{i+j}(C(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}))}

We have D_{j} = H_{HK}^{j}(X_{\overline{K},h})\{r\} ∈ MF_{K}^{ad}(φ, N, G_{K}). For j ≤ r, F^{1}D_{j,K} = F^{1−(r−j)}H_{dR}^{j}(X_{h})\{r\} = 0. Hence, by Corollary 2.15, we have ^{+}E_{2}^{i,j} →^{∼} E_{2}^{i,j}. This implies that τ_{≤r}C^{+}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}) →^{∼} τ_{≤r}C(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}).

Since ρ_{HK} = ρ_{cr}ι_{cr}^{B}, we check easily that we have the following commutative diagram

(49)

RΓ_{syn}(X_{\overline{K},h}, r) ⟶_{α_{syn}}^{∼} C^{+}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\})
↓^{ρ_{syn}}  ↓^{can}
RΓ(X_{\overline{K},ét}, Q_{p}(r)) ⟶_{α_{ét}}^{∼} C(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\})

It follows that ρ_{syn} : τ_{≤r}RΓ_{syn}(X_{\overline{K},h}, r) →^{∼} τ_{≤r}RΓ(X_{\overline{K},ét}, Q_{p}(r)), as wanted. □

Let X ∈ 𝒱 ar_{K}. The natural projection ε : X_{\overline{K},h} → X_{h} defines pullback maps

ε^{∗} : RΓ_{HK}^{B}(X_{h}) → RΓ_{HK}^{B}(X_{\overline{K},h}), ε^{∗} : RΓ_{dR}(X_{h}) → RΓ_{dR}(X_{\overline{K},h}).

By construction they are compatible with the monodromy operator, Frobenius, the action of the Galois group G_{K}, and filtration. It is also clear that they are compatible with the Beilinson-Hyodo-Kato morphisms, i.e., that the following diagram commutes

\xymatrix{RΓ_{HK}^{B}(X_{h}) \ar[r]^{ι_{dR}^{B}} \ar[d]^{ε^{∗}} & RΓ_{dR}(X_{h}) \ar[d]^{ε^{∗}} \\ RΓ_{HK}^{B}(X_{\overline{K},h}) \ar[r]^{ι_{dR}^{B}} & RΓ_{dR}(X_{\overline{K},h}).}

It follows that we can define a canonical pullback map

ε^{∗} : C_{st}(RΓ_{HK}^{B}(X_{h})\{r\}) → C^{+}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}).

Lemma 4.7. Let r ≥ 0. The following diagram commutes in the derived category.

\xymatrix{RΓ_{syn}(X_{h}, r) \ar[r]^{α_{syn}} \ar[d]^{ε^{∗}} & C_{st}(RΓ_{HK}^{B}(X_{h})\{r\}) \ar[d]^{ε^{∗}} \\ RΓ_{syn}(X_{\overline{K},h}, r) \ar[r]^{α_{syn}} & C^{+}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}).}

Proof. Take a number t ≥ 2 dim X + 2 and choose a finite Galois extension (V′, K′)/(V, K) (see the proof of Proposition 3.18) such that we have an h-hypercovering Z_{•} → X_{K′} with (Z_{•})_{≤t+1} built from logschemes log-smooth over V^{′,×} and of Cartier type. Since the top map α_{syn} is compatible with base change (cf. Proposition 3.20) it suffices to show that the diagram in the lemma commutes with X replaced by (Z_{•})_{≤t+1}. By Propositions 41, 3.18, and 3.14, this reduces to showing that, for an ss-pair (U, \overline{U}) split over V, the following diagram commutes canonically in the ∞-derived category (we set Y := (U, \overline{U}), \overline{Y} := Y\overline{V}, π - a fixed uniformizer of V).

\xymatrix{RΓ_{syn}(Y, r)_{Q} \ar[r]^{α_{syn,π}^{B}} \ar[d]^{ε^{∗}} & C_{st}(RΓ_{HK}^{B}(Y)\{r\}) \ar[d]^{ε^{∗}} \\ RΓ_{syn}(Y\overline{K}, r)_{Q} \ar[r]^{α_{syn}} & C^{+}(RΓ_{HK}^{B}(Y\overline{K})\{r\}).}

From the uniqueness property of the homotopy fiber functor, it suffices to show that the following diagram commutes canonically in the ∞-derived category.

[figure: RΓ_{cr}(Y)_{Q}; RΓ_{cr}(Y/R)_{Q}^{N=0}; RΓ_{HK}^{B}(Y_{1})_{R_{Q}}^{τ,N=0}; ι_{π}; ∼; RΓ_{HK}^{B}(Y_{1})^{N=0}; β; ∼; RΓ_{cr}(\overline{Y})_{Q}; RΓ_{HK}^{B}(\overline{Y}_{1})_{B_{cr}^{+}}^{τ,N=0}; ι_{cr}^{B}; ∼; (RΓ_{HK}^{B}(\overline{Y}_{1}) ⊗_{K_{0}^{nr}} B_{st}^{+})^{N=0}]

To do that we will need the ring of periods Â_{st} [62, p.253]. Set

Â_{st,n} = H_{cr}^{0}(\overline{V} _{n}^{×}/R_{n}), Â_{st} = \underleftarrow{lim}_{n} H_{cr}^{0}(\overline{V} _{n}^{×}/R_{n}).

The ring Â_{st,n} has a natural action of G_{K}, Frobenius φ, and a monodromy operator N. It is also equipped with a PD-filtration F^{i}Â_{st,n} = H_{cr}^{0}(\overline{V} _{n}^{×}/R_{n}, 𝒥_{cr,n}^{[i]}). We have a morphism A_{cr,n} → Â_{st,n} induced by the map H_{cr}^{0}(\overline{V}_{n}/W_{n}(k)) → H_{cr}^{0}(\overline{V} _{n}^{×}/R_{n}). It is compatible with the Galois action, the Frobenius, and the filtration. The natural map R_{n} → Â_{st,n} is compatible with all the structures. We can view Â_{st,n} as the PD-envelope of the closed immersion

\overline{V} _{n}^{×} ↪ A_{cr,n} ×_{W_{n}(k)} W_{n}(k)[X]^{×}

defined by the map θ : A_{cr,n} → \overline{V}_{n} and the projection W_{n}(k)[X] → \overline{V}_{n}, X ↦ π. This makes \overline{V} _{1}^{×} ↪ Â_{st,n} into a PD-thickening in the crystalline site of \overline{V}_{1}. Set B̂_{st}^{+} := Â_{st}[1/p].

Commutativity of the last diagram will follow from the following commutative diagram

\xymatrix{RΓ_{cr}(Y)_{Q} \ar[d] \ar[rr] &  & RΓ_{cr}(\overline{Y})_{Q} \ar[dl]^{∼} \\ RΓ_{cr}(Y/R)_{Q}^{N=0} \ar[r] & RΓ_{cr}(\overline{Y} /Â_{st})_{Q}^{N=0} \\ RΓ_{HK}^{B}(Y_{1})_{R_{Q}}^{τ,N=0} \ar[u]^{ι_{π}}_{≀} \ar[r] & RΓ_{HK}^{B}(Y_{1})τ,N=0 ; B̂_{st}^{+} \ar[u]^{ιB ; B̂{+ \atop st}}_{≀} & RΓ_{HK}^{B}(Y_{1})_{B_{cr}^{+}}^{τ,N=0} \ar[l] \ar[uu]^{ι_{cr}^{B}}_{≀} \\  & RΓ_{HK}^{B}(Y_{1})^{N=0} \ar@{-->}[ur] \ar[ul]^{β}_{∼}}

as soon as we show that the map RΓ_{cr}(\overline{Y})_{Q} → RΓ_{cr}(\overline{Y} /Â_{st})_{Q}^{N=0} is a quasi-isomorphism. Notice that the map ιB ; B̂_{st}^{+} is a quasi-isomorphism by Theorem 3.6. Hence using the Beilinson-Hyodo-Kato maps ιB ; B̂_{st}^{+} and ι_{cr}^{B}

this reduces to proving that the canonical map RΓ_{HK}^{B}(Y_{1})_{B_{cr}^{+}}^{τ,N=0} → RΓ_{HK}^{B}(Y_{1})τ,N=0 ; B̂_{st}^{+} is a quasi-isomorphism.

In fact, we claim that for any (φ, N)-module M we have an isomorphism M_{B_{cr}^{+}}^{τ,N=0} →^{∼} M τ,N=0 ; B̂_{st}^{+}. Indeed, assume first that the monodromy N_{M} is trivial. We calculate

M_{B_{cr}^{+}}^{τ} = (M ⊗_{K_{0}} B_{cr}^{+,τ})^{N′=0} = M ⊗_{K_{0}} (B_{cr}^{+,τ})^{N_{τ} =0} = M ⊗_{K_{0}} B_{cr}^{+}, N′ = N_{M} ⊗ 1 + 1 ⊗ N_{τ} = 1 ⊗ N_{τ} ,
M τ ; B̂_{st}^{+} = (M ⊗_{K_{0}} B̂_{st}^{+,τ})^{N′=0} = M ⊗_{K_{0}} (B̂_{cr}^{+,τ})^{N_{τ} =0} = M ⊗_{K_{0}} B̂_{st}^{+}

Hence M_{B_{cr}^{+}}^{τ,N=0} = M ⊗_{K_{0}} B_{cr}^{+} and M τ,N=0 ; B̂_{st}^{+} = M ⊗_{K_{0}} (B̂_{st}^{+})^{N=0} = M ⊗_{K_{0}} B_{cr}^{+}, where the last equality is proved in [62, Lemma 1.6.5]. We are done in this case.

In general, we can write M ⊗_{K_{0}} B_{st}^{+} ←^{∼} M′ ⊗_{K_{0}} B_{st}^{+} for a (φ, N)-module M′ such that N_{M′} = 0 (take for M′ the image of the map M → M ⊗_{K_{0}} B_{st}^{+}, m ↦ exp(N_{M}(m)u), for u ∈ B_{st}^{+} such that B_{st}^{+} = B_{cr}^{+}[u], N_{τ} (u) = −1). Similarly, using the fact that the ring B_{st}^{+} is canonically (and compatibly with all the structures) isomorphic to the elements of B̂_{st}^{+} annihilated by a power of the monodromy operator [42, 3.7], we can write in a compatible way M ⊗_{K_{0}} B_{st}^{+} ←^{∼} M′ ⊗_{K_{0}} B̂_{st}^{+} for the same module M′. We obtained a commutative diagram

M_{B_{cr}^{+}}^{τ,N=0} ⟶ M τ,N=0 ; B̂_{st}^{+}
↓^{≀}  ↓^{≀}
M_{B_{cr}^{+}}^{′τ,N=0} ⟶^{∼} M ′τ,N=0 ; B̂_{st}^{+}

that reduces the general case to the case of trivial monodromy on M that we treated above. □

Let X ∈ 𝒱 ar_{K}, r ≥ 0. Set

C_{pst}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}) := [ \xymatrix{RΓ_{HK}^{B}(X_{\overline{K},h})^{G_{K}} \ar[r]^{(1−φ_{r},ι_{dR}^{B})} \ar[d]^{N} & RΓ_{HK}^{B}(X_{\overline{K},h})^{G_{K}} ⊕ (RΓ_{dR}(X_{\overline{K},h})/F^{r})^{G_{K}} \ar[d]^{(N,0)} \\ RΓ_{HK}^{B}(X_{\overline{K},h})^{G_{K}} \ar[r]^{1−φ_{r−1}} & RΓ_{HK}^{B}(X_{\overline{K},h})^{G_{K}}} ]

The above makes sense since the action of G_{K} on RΓ_{HK}^{B}(X_{\overline{K},h})\{r\} and RΓ_{dR}(X_{\overline{K},h}) is smooth. In particular, we have

H^{j}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}^{G_{K}}) ≃ H^{j}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\})^{G_{K}} , H^{j}(RΓ_{dR}(X_{\overline{K},h})^{G_{K}}) ≃ H^{j}(RΓ_{dR}(X_{\overline{K},h}))^{G_{K}} .

Consider the canonical pullback map

ε^{∗} : C_{st}(RΓ_{HK}^{B}(X_{h})\{r\}) →^{∼} C_{pst}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}).

By Proposition 3.22, this is a quasi-isomorphism. This allows us to construct a canonical spectral sequence (the syntomic descent spectral sequence)

(50)  \xymatrix{^{syn}E_{2}^{i,j} = H_{st}^{i}(G_{K}, H^{j}(X_{\overline{K},ét}, Q_{p}(r))) \ar@{=>}[r] & H_{syn}^{i+j}(X_{h}, r)}

Indeed, the Bousfield-Kan spectral sequences associated to the homotopy limits defining complexes C_{pst}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\})) and C_{st}(RΓ_{HK}^{B}(X_{h})\{r\})) give us the following commutative diagram

\xymatrix{^{pst}E_{2}^{i,j} = H^{i}(C_{pst}(H_{HK}^{j}(X_{\overline{K},h})\{r\})) \ar@{=>}[r] & H^{i+j}(C_{pst}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\})) \\ ^{syn}E_{2}^{i,j} = H^{i}(C_{st}(H_{HK}^{j}(X_{h})\{r\})) \ar[u]^{≀}_{ε^{∗}} \ar@{=>}[r] & H^{i+j}(C_{st}(RΓ_{HK}^{B}(X_{h})\{r\})) \ar[u]^{≀}_{ε^{∗}}}

Since, by Proposition 3.20, we have α_{syn} : H_{syn}^{i+j}(X_{h}, r) →^{∼} H^{i+j}(C_{st}(RΓ_{HK}^{B}(X_{h}){r})), we have obtained a spectral sequence

\xymatrix{E_{2}^{i,j} = H^{i}(C_{pst}(H_{HK}^{j}(X_{\overline{K},h})\{r\})) \ar@{=>}[r] & H_{syn}^{i+j}(X_{h}, r)}

It remains to show that there is a canonical isomorphism

(51)  H^{i}(C_{pst}(H_{HK}^{j}(X_{\overline{K},h})\{r\})) ≃ H_{st}^{i}(G_{K}, H^{j}(X_{\overline{K},ét}, Q_{p}(r))).

But, we have D_{j} = H_{HK}^{j}(X_{\overline{K},h})\{r\} ∈ MF_{K}^{ad}(φ, N, G_{K}), V_{pst}(D_{j}) ≃ H^{j}(X_{\overline{K},ét}, Q(r)), and D_{pst}(H^{j}(X_{\overline{K},ét}, Q(r))) ≃ D_{j}. Hence isomorphism (51) follows from Remark 2.12 and we have obtained the spectral sequence (50).

4.2. Arithmetic case. In this subsection, we define the arithmetic syntomic period map by Galois descent from the geometric case. Then we show that, via this period map, the syntomic descent spectral sequence and the étale Hochschild-Serre spectral sequence are compatible. Finally, we show that this implies that the arithmetic syntomic cohomology and étale cohomology are isomorphic in a stable range.

Let X ∈ 𝒱 ar_{K}. For r ≥ 0, we define the canonical syntomic period map

ρ_{syn} : RΓ_{syn}(X_{h}, r) → RΓ(X_{ét}, Q_{p}(r)),

as the following composition

RΓ_{syn}(X_{h}, r) = RΓ(X_{h}, 𝒮 (r))_{Q} → holim_{n} RΓ(X_{h}, 𝒮_{n}(r))_{Q} →^{ε^{∗}} holim_{n} RΓ(G_{K}, RΓ(X_{\overline{K},h}, 𝒮_{n}(r)))_{Q}

⟶^{p^{−r}β} holim_{n} RΓ(G_{K}, RΓ(X_{\overline{K},ét}, Z/p^{n}(r)′))_{Q} ←^{∼} holim_{n} RΓ(X_{ét}, Z/p^{n}(r)′)_{Q} = RΓ(X_{ét}, Q_{p}(r)).

It induces a morphism of graded E_{∞} algebras over Q_{p}.

The syntomic period map ρ_{syn} is compatible with the syntomic descent and the Hochschild-Serre spectral sequences.

Theorem 4.8. For X ∈ 𝒱 ar_{K}, r ≥ 0, there is a canonical map of spectral sequences

\xymatrix{^{syn}E_{2}^{i,j} = H_{st}^{i}(G_{K}, H^{j}(X_{\overline{K},ét}, Q_{p}(r))) \ar[d]^{can} \ar@{=>}[r] & H_{syn}^{i+j}(X_{h}, r) \ar[d]^{ρ_{syn}} \\ ^{ét}E_{2}^{i,j} = H^{i}(G_{K}, H^{j}(X_{\overline{K},ét}, Q_{p}(r))) \ar@{=>}[r] & H^{i+j}(X_{ét}, Q_{p}(r))}

Proof. We work in the (classical) derived category. The Bousfield-Kan spectral sequences associated to the homotopy limits defining complexes C(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}) and C_{pst}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}), and Theorem 2.18 give us the following commutative diagram of spectral sequences

\xymatrix{^{II}E_{2}^{i,j} = H^{i}(G_{K}, C(H_{HK}^{j}(X_{\overline{K},h})\{r\})) \ar@{=>}[r] & H^{i+j}(G_{K}, C(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\})) \\ ^{pst}E_{2}^{i,j} = H^{i}(C_{pst}(H_{HK}^{j}(X_{\overline{K},h})\{r\})) \ar[u]^{δ} \ar@{=>}[r] & H^{i+j}(C_{pst}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\})) \ar[u]^{δ}}

More specifically, in the language of Section 2.5, set X = C(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}) (hopefully, the notation will not be too confusing). Filtering complex X in the direction of the homotopy limit we obtain a Postnikov system (14) with Y^{i} = 0, i ≥ 3, and

Y^{0} = RΓ_{HK}^{B}(X_{\overline{K},h})\{r\} ⊗_{K_{0}^{nr}} B_{st},
Y^{1} = RΓ_{HK}^{B}(X_{\overline{K},h})\{r − 1\} ⊗_{K_{0}^{nr}} B_{st} ⊕ (RΓ_{HK}^{B}(X_{\overline{K},h})\{r\} ⊗_{K_{0}^{nr}} B_{st} ⊕ (RΓ_{dR}(X\overline{K}) ⊗\overline{K} B_{dR})/F^{r}),
Y^{2} = RΓ_{HK}^{B}(X_{\overline{K},h})\{r − 1\} ⊗_{K_{0}^{nr}} B_{st}.

Still in the setting of Section 2.5, take for A the abelian category of sheaves of abelian groups on the pro-étale site Spec(K)_{proét} of Scholze [58, 3].

Remark 4.9. We work with the pro-étale site to make sense of the continuous cohomology RΓ(G_{K}, ·). If the reader is willing to accept that this is possible then he can skip the tedious parts of the proof involving passage to the pro-étale site (and existence of continuous sections).

Recall that there is a projection map ν : Spec(K)_{proét} → Spec(K)_{ét} such that, for an étale sheaf ℱ, we have the quasi-isomorphism ν^{∗} : ℱ →^{∼} Rν_{∗}ν^{∗}ℱ [8, 5.2.6]. More generally, for a topological G_{K}-module M, we get a sheaf νM on Spec(K)_{proét} by setting, for a profinite G_{K}-set S, νM(S) = Hom_{cont,G_{K}} (S, M), and Scholze shows that there is a canonical quasi-isomorphism H^{∗}(Spec(K)_{proét}, νM) ≃ H_{cont}^{∗}(G_{K}, M) [58, 3.7 (iii)], [59]. In this proof we will need this kind of quasi-isomorphisms for complexes M as well and this will require extra arguments. For that observe that the functor ν is left exact. To study right exactness, it suffices to look at the global sections on profinite sets S with a free G_{K}-action of the form S = S′ × G_{K} for a profinite set S′ with trivial G_{K}-action^{8}. Then, for any G_{K}-module T, we have Γ(S, νT) = Hom_{cont}(S′, T). It follows that, for a surjective map T_{1} → T_{2} of G_{K}-modules, the pullback map νT_{1} → νT_{2} is also surjective if the original map had a continuous set-theoretical section. This is a criterium familiar from continuous cohomology and we will use it often.

We will see the complex X as a complex of sheaves on the site Spec(K)_{proét} in the following way: represent RΓ_{HK}^{B}(X_{\overline{K},h}) and RΓ_{dR}(X\overline{K}) by (filtered) perfect complexes of K_{0}^{nr}- and \overline{K}-modules, respectively, think of X as νX, and work on the pro-étale site. This makes sense, i.e., functor ν transfers (filtered) quasi-isomorphisms of representatives of RΓ_{HK}^{B}(X_{\overline{K},h}) and RΓ_{dR}(X\overline{K}) to quasi-isomorphisms of the corresponding sheaves νX. To see this look at the Postnikov system of sheaves on Spec(K)_{proét} obtained by pulling back by ν the above Postnikov system. Now, look at the global sections on profinite sets S = S′ × G_{K} as above and note that we have Γ(S, νY^{0}) = Hom_{cont}(S′, Y^{0}). Conclude that, by perfectness of the Beilinson-Hyodo-Kato complexes, quasi-isomorphisms of representatives of RΓ_{HK}^{B}(X_{\overline{K},h}) yield quasi-isomorphisms of the sheaves νY^{0}. By a similar argument, we get the analogous statement for Y^{2}. For Y^{1}, we just have to show that filtered quasi-isomorphisms of representatives of RΓ_{dR}(X\overline{K}) yield quasi-isomorphisms of the sheaves ν((RΓ_{dR}(X\overline{K}) ⊗\overline{K} B_{dR})/F^{r}). Again, we look at the global section on S = S′ × G_{K} as above. By compactness of S′ we may replace (RΓ_{dR}(X\overline{K}) ⊗\overline{K} B_{dR})/F^{r} by (t^{−i}RΓ_{dR}(X\overline{K})⊗\overline{K} B_{dR}^{+})/F^{r}, for some i ≥ 0, where, using devissage, we can again argue by (filtered) perfection of RΓ_{dR}(X\overline{K}). Observe that the same argument shows that ℋ ^{j}(νY^{i}) ≃ νH^{j}(Y^{i}), for i = 0, 1, 2.

8To see this, for a profinite G_{K}-set S′, use the covering S′ × G_{K} → S′, where the first S′ has trivial G_{K}-action, induced from the G_{K}-action on S′.

The above Postnikov system gives rise to an exact couple

D_{1}^{i,j} = ℋ ^{j}(X^{i}), E_{1}^{i,j} = ℋ ^{j}(Y^{i}) ⟹ ℋ ^{i+j}(X)

This is the Bousfield-Kan spectral sequence associated to X.

Consider now the complex X_{pst} := C_{pst}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}). We claim that the canonical map

C_{pst}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}) →^{∼} C(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\})^{G_{K}}

is a quasi-isomorphism (recall that taking G_{K}-fixed points corresponds to taking global sections on the pro-étale site). In particular, that the term on the right hand side makes sense. To see this, it suffices to show that the canonical maps

(RΓ_{dR}(X_{\overline{K},h})/F^{r})^{G_{K}} →^{∼} ((RΓ_{dR}(X_{\overline{K},h}) ⊗\overline{K} B_{dR})/F^{r})^{G_{K}} ,
RΓ_{HK}^{B}(X_{\overline{K},h})^{G_{K}} →^{∼} (RΓ_{HK}^{B}(X_{\overline{K},h}) ⊗_{K_{0}^{nr}} B_{st})^{G_{K}}

are quasi-isomorphisms and to use the fact that the action of G_{K} on RΓ_{HK}^{B}(X_{\overline{K},h}) is smooth. The fact that the first map is a quasi-isomorphism follows from the filtered quasi-isomorphism RΓ_{dR}(X) ⊗_{K} \overline{K} →^{∼} RΓ_{dR}(X_{\overline{K},h}) and the fact that B_{dR}^{G_{K}} = K. Similarly, the second map is a quasi-isomorphism because, by [28, 4.2.4], RΓ_{HK}^{B}(X_{\overline{K},h}) is the subcomplex of those elements of RΓ_{HK}^{B}(X_{\overline{K},h}) ⊗_{K_{0}^{nr}} B_{st} whose stabilizers in G_{K} are open.

Taking the G_{K}-fixed points of the above Postnikov system we get an exact couple

pstD_{1}^{i,j} = H^{j}(X_{pst}^{i}), ^{pst}E_{1}^{i,j} = H^{j}(Y_{pst}^{i}) ⟹ H^{i+j}(X_{pst})

corresponding to the Bousfield-Kan filtration of the complex X_{pst}. On the other hand, applying RΓ(Spec(K)_{proét}, ·) to the same Postnikov system we obtain an exact couple

ID_{1}^{i,j} = H^{j}(Spec(K)_{proét}, X^{i}), ^{I}E_{1}^{i,j} = H^{j}(Spec(K)_{proét}, Y^{i}) ⟹ H^{i+j}(Spec(K)_{proét}, X)

together with a natural map of exact couples (^{pst}D_{1}^{i,j},^{pst}E_{1}^{i,j}) → (^{I}D_{1}^{i,j},^{I}E_{1}^{i,j}).

We also have the hypercohomology exact couple

IID_{2}^{i,j} = H^{i+j}(Spec(K)_{proét}, τ_{≤j−1}X), ^{II}E_{2}^{i,j} = H^{i}(Spec(K)_{proét}, ℋ ^{j}(X)) ⟹ H^{i+j}(Spec(K)_{proét}, X)

Theorem 2.18 gives us a natural morphism of exact couples (^{I}D_{2}^{i,j},^{I}E_{2}^{i,j}) → (^{II}D_{2}^{i,j},^{II}E_{2}^{i,j}) – hence a natural morphism of spectral sequences ^{I}E_{2}^{i,j} → ^{II}E_{2}^{i,j} compatible with the identity map on the common abutment – if our original Postnikov system satisfies the equivalent conditions (2.17). We will check the condition (4), i.e., that the following long sequence is exact for all j

0 → ℋ ^{j}(X) → ℋ ^{j}(Y^{0}) → ℋ ^{j}(Y^{1}) → ℋ ^{j}(Y^{2}) → 0

For that it is enough to show that (1) ℋ ^{j}(νY^{i}) ≃ νH^{j}(Y^{i}), for i = 0, 1, 2;

(2) ℋ ^{j}(νX) ≃ νH^{j}(X);

(3) the following long sequence of G_{K}-modules

0 → H^{j}(X) → H^{j}(Y^{0}) → H^{j}(Y^{1}) → H^{j}(Y^{2}) → 0

is exact;

(4) the pullback ν preserves its exactness. The assertion in (1) was shown above. The sequence in (3) is equal to the top sequence in the following commutative diagram (where we set M = H_{HK}^{j}(X_{\overline{K},h}), M_{dR} = H_{dR}^{j}(X_{\overline{K},h}), E = H^{j}(X_{\overline{K},ét}, Q_{p})).

\xymatrix{H^{j}(X) \ar[d]^{α_{ét}^{−1}}_{≀} \ar[r] & M ⊗_{K_{0}^{nr}} B_{st} \ar[r]^{(N,1−φ_{r},ι)} \ar[d]^{ρ_{HK}}_{≀} & M ⊗_{K_{0}^{nr}} (B_{st} ⊕ B_{st}) ⊕ (M_{dR} ⊗\overline{K} B_{dR})/F^{r} \ar[r]^{(1−φ_{r−1})−N} \ar[d]^{ρ_{HK}+ρ_{HK}+ρ_{dR}}_{≀} & M ⊗_{K_{0}^{nr}} B_{st} \ar[d]^{ρ_{HK}}_{≀} \\ E(r) \ar@{^{(}->}[r] & E ⊗ B_{st} \ar[r]^{(N,1−φ_{r},ι)} & E ⊗ (B_{st} ⊕ B_{st}) ⊕ E ⊗ B_{dR}/F^{r} \ar@{->>}[r]^{(1−φ_{r−1})−N} & E ⊗ B_{st}}

Since the bottom sequence is just a fundamental exact sequence of p-adic Hodge Theory, the top sequence is exact, as wanted.

To prove assertion (4), we pass to the bottom exact sequence above and apply ν to it. It is easy to see that it enough now to show that the following surjections have continuous Q_{p}-linear sections

B_{st} →^{N} B_{st}, B_{cr} ⟶^{(1−φ_{r},can)}B_{cr} ⊕ B_{dR}/F^{r}.

For the monodromy, write B_{st} = B_{cr}[u_{s}] and take for a continuous section the map induced by bu_{s}^{i} ↦ −(b/(i + 1))u_{s}^{i+1}, b ∈ B_{cr}. For the second map, the existence of continuous section was proved in [13, 1.18]. For a different argument: observe that an analogous statement was proved in [18, Prop. II.3.1] with B_{max} in place of B_{cr} as a consequence of the general theory of p-adic Banach spaces. We will just modify it here. Write A_{i} = t^{−i}B_{cr}^{+} and B_{i} = t^{−i}B_{cr}^{+} ⊕ t^{−i}B_{dR}^{+}/t^{r} for i ≥ 1. These are p-adic Banach spaces. Observe that B_{i} ⊂ B_{i+1} is closed. Indeed, it is enough to show that tB_{cr}^{+} ⊂ B_{cr}^{+} is closed. But we have tB_{cr}^{+} = ⋂_{n≥0} ker(θ ∘ φ^{n}).

It follows [18, Prop. I.1.5] that we can find a closed complement C_{i+1} of B_{i} in B_{i+1}. Set f = (1 − φ_{r}, can) : B_{cr} → B_{cr} ⊕ B_{dR}/F^{r}. We know that f maps A_{i} onto B_{i}. Write t^{−i}B_{cr}^{+} ⊕ t^{−i}B_{dR}^{+}/t^{r} = B_{1} ⊕ (⊕_{j=2}^{i−1}C_{j}). By [18, Prop. I.1.5], we can find a continuous section s_{1} : B_{1} → A_{1} of f and, if i ≥ 2, a continuous section s_{i} : C_{i} → A_{i} of f. Define the map s : t^{−i}B_{cr}^{+} ⊕ t^{−i}B_{dR}^{+}/t^{r} → B_{cr} by s_{1} on B_{1} and by s_{i} on C_{i} for i ≥ 2. Taking inductive limit over i we get our section of f.

To prove assertion (2), take a perfect representative of the complex RΓ(X_{\overline{K},ét}, Z_{p}(r)). Consider the complex Z = RΓ(X_{\overline{K},ét}, Q_{p}(r)) as a complex of sheaves on Spec(K)_{proét}. As before, we see that this makes sense and we easily find that (canonically) ℋ ^{j}(Z) ≃ νH^{j}(X_{\overline{K},ét}, Q_{p}(r)). To prove (2), it is enough to show that we can also pass with the map α_{ét} : RΓ(X_{\overline{K},ét}, Q_{p}(r)) →^{∼} C(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}) to the site Spec(K)_{proét}. Looking at its definition (cf. (48)) we see that we need to show that the period quasi-isomorphisms ρ_{cr}, ρ_{HK}, ρ_{dR} as well as the quasi-isomorphism

Q_{p}(r) →^{∼} [B_{st} ⟶^{(N,1−φ_{r},ι)}B_{st} ⊕ B_{st} ⊕ B_{dR}/F^{r} ⟶^{(1−φ_{r−1})−N}B_{st}]

can be lifted to the pro-étale site. The last fact we have just shown. For the crystalline period map ρ_{cr} this follows from the fact that it is defined integrally and all the relevant complexes are perfect. For the Hyodo-Kato period map ρ_{HK} - it follows from the case of ρ_{cr} and from perfection of complexes involved in the definition of the Beilinson-Hyodo-Kato map. For the de Rham period map ρ_{dR} this follows from perfection of the involved complexes as well as from the exactness of holim_{n} (in the definition of ρ_{dR}) on the pro-étale site of K (cf. [58, 3.18]).

We define the map of spectral sequences δ := (δ_{D}, δ) := (^{pst}D_{2}^{i,j},^{pst}E_{2}^{i,j}) → (^{II}D_{2}^{i,j},^{II}E_{2}^{i,j}) – that we stated at the beginning of the proof – as the composition of the two maps constructed above

δ : (^{pst}D_{2}^{i,j},^{pst}E_{2}^{i,j}) → (^{I}D_{2}^{i,j},^{I}E_{2}^{i,j}) → (^{II}D_{2}^{i,j},^{II}E_{2}^{i,j}).

To get the spectral sequence from the theorem we need to pass from ^{II}E_{2} to the Hochschild-Serre spectral sequence. To do that consider the hypercohomology exact couple

étD_{2}^{i,j} = H^{i+j}(Spec(K)_{proét}, τ_{≤j−1}Z), ^{ét}E_{2}^{i,j} = H^{i}(Spec(K)_{proét}, ℋ ^{j}(Z)) ⟹ H^{i+j}(Spec(K)_{proét}, Z)

and, via α_{ét}^{−1}, a natural morphism of exact couples (^{II}D_{2}^{i,j},^{II}E_{2}^{i,j}) → (^{ét}D_{2}^{i,j},^{ét}E_{2}^{i,j}) – hence a natural morphism of spectral sequences ^{II}E_{2}^{i,j} →^{ét}E_{2}^{i,j} compatible with the map α_{ét}^{−1} on the abutment. We have a quasi-isomorphism ψ : RΓ(Spec(K)_{proét}, Z) →^{∼} RΓ(X_{ét}, Q_{p}(r)) defined as the composition

ψ : RΓ(Spec(K)_{proét}, RΓ(X_{\overline{K},ét}, Q_{p}(r))) →^{∼} Q ⊗ holim_{n} RΓ(G_{K}, RΓ(X_{\overline{K},ét}, Z/p^{n}(r)))
= Q ⊗ holim_{n} RΓ(X_{ét}, Z/p^{n}(r)) = RΓ(X_{ét}, Q(r))

We have obtained the following natural maps of spectral sequences

\xymatrix{^{syn}E_{2}^{i,j} = H_{st}^{i}(G_{K}, H^{j}(X_{\overline{K},ét}, Q_{p}(r))) \ar[d]_{≀} \ar@{=>}[r] & H_{syn}^{i+j}(X_{h}, r) \ar[d]^{α_{syn}}_{≀} \\ E_{2}^{i,j} = H^{i}(C_{st}(H_{HK}^{j}(X_{h})\{r\})) \ar[d]^{α_{ét}^{−1}δε^{∗}} \ar@{=>}[r] & H^{i+j}(C_{st}(RΓ_{HK}^{B}(X_{h})\{r\})) \ar[d]^{ψα_{ét}^{−1}δε^{∗}} \\ ^{ét}E_{2}^{i,j} = H^{i}(G_{K}, H^{j}(X_{\overline{K},ét}, Q_{p}(r))) \ar@{=>}[r] & H^{i+j}(X_{ét}, Q_{p}(r))}

It remains to show that the right vertical composition γ : H_{syn}^{i+j}(X_{h}, r) → H^{i+j}(X_{ét}, Q_{p}(r)) is equal to the map ρ_{syn}. Since we have the equality α_{syn} = ρ_{syn}α_{ét} (in the derived category) from (49) and, by Lemma 4.7, ε^{∗}α_{syn} = α_{syn}ε^{∗}, the map γ can be written as the composition

ρ̃_{syn} : H_{syn}^{i+j}(X_{h}, r) →^{ε^{∗}} H^{i+j}(Spec(K)_{proét}, νRΓ_{syn}(X_{\overline{K},h}, r)) →^{ρ_{syn}} H^{i+j}(Spec(K)_{proét}, νRΓ(X_{\overline{K},ét}, Q_{p}(r)))

→^{ψ} H^{i+j}(X_{ét}, Q_{p}(r)),

where the period map ρ_{syn} is understood to be on sheaves on Spec(K)_{proét}. There is no problem with that since we care only about the induced map on cohomology groups. It is easy now to see that ρ̃_{syn} = ρ_{syn}, as wanted. □

Remark 4.10. If X is proper and smooth, it is known that the étale Hochschild-Serre spectral sequence degenerates, i.e., that ^{ét}E_{2} =^{ét}E_{∞}. It is very likely that so does the syntomic descent spectral sequence in this case, i.e., that ^{syn}E_{2} = ^{syn}E_{∞}^{9}.

Corollary 4.11. For X ∈ 𝒱 ar_{K}, we have a canonical quasi-isomorphism

ρ_{syn} : τ_{≤r}RΓ_{syn}(X_{h}, r)_{Q} →^{∼} τ_{≤r}RΓ(X_{ét}, Q_{p}(r)).

Proof. By Theorem 4.8, the syntomic descent and the Hochschild-Serre spectral sequence are compatible. We have D_{j} = H_{HK}^{j}(X_{\overline{K},h})\{r\} ∈ MF_{K}^{ad}(φ, N, G_{K}). For j ≤ r, F^{1}D_{j,K} = F^{1−(r−j)}H_{dR}^{j}(X_{h}) = 0. Hence, by Proposition 2.16, we have ^{syn}E_{2}^{i,j} →^{∼} ^{ét}E_{2}^{i,j}. This implies that ρ_{syn} : τ_{≤r}RΓ_{syn}(X_{h}, r) →^{∼} τ_{≤r}RΓ(X_{ét}, Q_{p}(r)), as wanted. □

Remark 4.12. All of the above automatically extends to finite diagrams of K-varieties, hence to essentially finite diagrams of K-varieties (i.e., the diagrams for which every truncation of their cohomology τ_{≤n} is computed by truncating the cohomology of some finite diagram). This includes, in particular, simplicial and cubical varieties.

Proposition 4.13. Let X ∈ 𝒱 ar_{K} and i ≥ 0. The composition

H_{dR}^{q}(X)/F^{r} →^{∂} H_{syn}^{q+1}(X_{h}, r) ⟶^{ρ_{syn}} H_{ét}^{q+1}(X, Q_{p}(r)) → H_{ét}^{q+1}(X\overline{K}, Q_{p}(r))

is the zero map. The map induced by the syntomic descent spectral sequence

H_{dR}^{q}(X)/F^{r} → H^{1}(G_{K}, H_{ét}^{q}(X\overline{K}, Q_{p}(r)))

is equal to the Bloch-Kato exponential associated with the Galois representation V^{q}(r) = H_{ét}^{q}(X\overline{K}, Q_{p}(r)).

Proof. In what follows we will omit the passage to the pro-étale site. Consider the Postnikov system from the proof of Theorem 4.8, which arises from the complex X = C(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}); then Y^{p} = C^{p}(RΓ_{HK}^{B}(X_{\overline{K},h})\{r\}). The discussion from Example 2.19 then applies to the functor f(−) = (−)^{G_{K}} and yields the following four exact couples.

9This was, in fact, shown in [25].

(1) D_{1}^{p,q} = H^{q}(X^{p}), E_{1}^{p,q} = H^{q}(Y^{p}) = C^{p}(H_{HK}^{q}(X_{\overline{K},h})\{r\})) = C^{p}(H_{HK}^{q}\{r\}). The corresponding quasiisomorphism H^{q}(X) →^{∼} E_{1}^{•,q} is then identified, via the various period maps, with

V^{q}(r) →^{∼} C(H_{HK}^{q}{r}) = C(D_{pst}(V^{q}(r))).

(2) ^{f}D_{1}^{p,q} = H^{q}(f(X^{p})), ^{f}E_{1}^{p,q} = H^{q}(f(Y^{p})) = f(H^{q}(Y^{p})) = C_{st}^{p}(H_{HK}^{q}{r}) = f(E_{1}^{p,q}).

(3) ^{I}D_{1}^{p,q} = (R^{q}f)(X^{p}), ^{I}E_{1}^{p,q} = (R^{q}f)(Y^{p}).

(4) ^{II}D_{2}^{p,q} = (R^{p+q}f)(τ_{≤q−1}X), ^{II}E_{2}^{p,q} = (R^{p}f)(H^{q}(X)) = H^{p}(G_{K}, V^{q}(r)). There is a canonical morphism of exact couples (2) → (3) and a morphism (3) → (4) given by the maps (u, v) from the proof of Theorem 2.18. As observed in 2.14, the Bloch-Kato exponential for V = V^{q}(r) is obtained by applying R^{0}f to

Z^{1}C(H_{HK}^{q}{r}) = Z^{1}(E_{1}^{•,q}) ⟶^{can} (σ_{≥1}C(H_{HK}^{q}{r}))[1] = (σ_{≥1}C(E_{1}^{•,q}))[1] ⟶^{− can} C(H_{HK}^{q}{r}))[1] = E_{1}^{•,q}[1]

←^{∼} V^{q}(r)[1] = H^{q}(X)[1],

hence is equal to the composite map

f(Z^{1}(E_{1}^{•,q})) = Z^{1}(^{f}E_{1}^{•,q}) → ^{f}E_{2}^{1,q} ⟶^{can} ^{I}E_{2}^{1,q} ⟶^{−v′=v} (R^{1}f)(E_{1}^{•,q}) = ^{II}E_{2}^{p,q},

which coincides, in turn, with

Z^{1}C_{st}(H_{HK}^{q}{r}) ⟶^{can} H_{st}^{1}(G_{K}, V^{q}(r)) → H^{1}(G_{K}, V^{q}(r)).

After restricting to the de Rham part of Z^{1}C(H_{HK}^{q}{r}) we obtain the desired statement about H_{dR}^{q}(X)/F^{r}. □

In more concrete terms, the above proposition says that the following diagram commutes

\xymatrix{H_{syn}^{q+1}(X_{h}, r)_{0} \ar[r]^{ρ_{syn}} & H_{ét}^{q+1}(X, Q_{p}(r))_{0} \ar[d] \\ H_{dR}^{q}(X)/F^{r} \ar[u]^{∂} \ar[r]^{_{exp_{BK}}} & H^{1}(G_{K}, H_{ét}^{q}(X\overline{K}, Q_{p}(r))),}

where the subscript 0 refers to the classes that vanish in H_{syn}^{q+1}(X_{\overline{K},h}, r) and H_{ét}^{q+1}(X\overline{K}, Q_{p}(r)), respectively.

Remark 4.14. Assume that r > q. Then in the above diagram all the maps are isomorphisms. Indeed, we have F^{r}H_{dR}^{q}(X) = 0. By [6, Theorem 6.8], the map exp_{BK} is an isomorphism. By Proposition 4.6 and Corollary 4.11, so is the period map ρ_{syn}. Since, by Theorem A.1, H^{2}(G_{K}, H_{ét}^{q}(X\overline{K}, Q_{p}(r))) = H^{2}(G_{K}, H_{ét}^{q−1}(X\overline{K}, Q_{p}(r))) = 0, the vertical map is an isomorphism as well. Hence so is the map ∂.

5. Syntomic regulators

In this section we prove that Soulé’s étale regulators land in the semistable Selmer groups. This will be done by constucting syntomic regulators that are compatible with the étale ones via the period map and by exploiting the syntomic descent spectral sequence.

5.1. Construction of syntomic Chern classes. We start with the construction of syntomic Chern classes. This will be standard once we prove that syntomic cohomology satisfies projective space theorem and homotopy property.

In this subsection we will work in the (classical) derived category. For a fine log-scheme (X, M), log-smooth over V^{×}, we have the log-crystalline and log-syntomic first Chern class maps of complexes of sheaves on X_{ét} [62, 2.2.3]

c_{1}^{cr} : j_{∗}𝒪_{X_{tr}}^{∗} →^{∼} M^{gp} → M_{n}^{gp} → Rε_{∗}𝒥_{X_{n}/W_{n}(k)}^{[1]}[1], c_{1}^{st} : j_{∗}𝒪_{X_{tr}}^{∗} →^{∼} M^{gp} → M_{n}^{gp} → Rε_{∗}𝒥_{X_{n}/R_{n}}^{[1]}[1],
c_{1}^{HK} : j_{∗}𝒪_{X_{tr}}^{∗} →^{∼} M^{gp} → M_{0}^{gp} → Rε_{∗}𝒥_{X_{0}/W_{n}(k)^{0}}^{[1]}[1], c_{1}^{syn} : j_{∗}𝒪_{X_{tr}}^{∗} →^{∼} M^{gp} → 𝒮 (1)_{X,Q}[1].

Here ε is the projection from the corresponding crystalline site to the étale site. The maps c_{1}^{cr}, c_{1}^{st}, and c_{1}^{syn} are clearly compatible. So are the maps c_{1}^{st} and c_{1}^{HK}. For ss-pairs (U, \overline{U}) over K, we get the induced functorial maps

c_{1}^{cr} : Γ(U, 𝒪_{U}^{∗}) ←^{∼} Γ(\overline{U}, j_{∗}𝒪_{U}^{∗}) → RΓ_{cr}(U, \overline{U}, 𝒥^{[1]})[1],  c_{1}^{st} : Γ(U, 𝒪_{U}^{∗}) → RΓ_{cr}((U, \overline{U})/R, 𝒥^{[1]})[1],
c_{1}^{HK} : Γ(U, 𝒪_{U}^{∗}) → RΓ_{cr}((U, \overline{U})_{0}/W_{n}(k)^{0}, 𝒥^{[1]})[1], c_{1}^{syn} : Γ(U, 𝒪_{U}^{∗}) → RΓ_{syn}(U, \overline{U}, 1)_{Q}[1].

For X ∈ 𝒱 ar_{K} we can glue the absolute log-crystalline and log-syntomic classes to obtain the absolute crystalline and syntomic first Chern class maps

c_{1}^{cr} : 𝒪_{X_{h}}^{∗} → 𝒥_{cr,X}[1], c_{1}^{syn} : 𝒪_{X_{h}}^{∗} → 𝒮 (1)_{X,Q}[1].

They induce (compatible) maps

c_{1}^{cr} : Pic(X) = H^{1}(X_{ét}, 𝒪_{X}^{∗}) → H^{1}(X_{h}, 𝒪_{X}^{∗}) →^{c_{1}^{cr}} H^{2}(X_{h}, 𝒥_{cr}),
c_{1}^{syn} : Pic(X) = H^{1}(X_{ét}, 𝒪_{X}^{∗}) → H^{1}(X_{h}, 𝒪_{X}^{∗}) →^{c_{1}^{syn}} H_{syn}^{2}(X_{h}, 1).

Recall that, for a log-scheme (X, M) as above, we also have the log de Rham first Chern class map

c_{1}^{dR} : j_{∗}𝒪_{X_{tr}}^{∗} →^{∼} M^{gp} → M_{n}^{gp} →^{dlog} Ω(X,M)_{n}/V_{n}^{×}^{•}[1].

For ss-pairs (U, \overline{U}) over K, it induces maps

c_{1}^{dR} : Γ(U, 𝒪_{U}^{∗}) ←^{∼} Γ(\overline{U}, j_{∗}𝒪_{U}^{∗}) → RΓ(\overline{U}, Ω(U,\overline{U})/V^{×}^{•})[1].

By the map RΓ_{cr}(U, \overline{U}, 𝒥^{[1]}) → RΓ_{cr}(U, \overline{U}) → RΓ(\overline{U}, Ω(U,\overline{U})/V^{×}^{•}) they are compatible with the absolute log-crystalline and log-syntomic classes [62, 2.2.3].

Lemma 5.1. For strict ss-pairs (U, \overline{U}) over K, the Hyodo-Kato map and the Hyodo-Kato isomorphism

ι : H_{HK}^{2}(U, \overline{U})_{Q} → H_{cr}^{2}((U, \overline{U})/R)_{Q}, ι_{dR,π} : H_{HK}^{2}(U, \overline{U})_{Q} ⊗_{K_{0}} K →^{∼} H^{2}(\overline{U}_{K}, Ω(U,\overline{U}_{K})/K^{•})

are compatible with first Chern class maps.

Proof. Since ι_{dR,π} = i_{π}^{∗}ι⊗Id and the map i_{π}^{∗} is compatible with first Chern classes, it suffices to show the compatibility for the Hyodo-Kato map ι. Let ℒ be a line bundle on U. Since the map ι is a section of the map i_{0}^{∗} : H_{cr}^{2}((U, \overline{U})/R)_{Q} → H_{HK}^{2}(U, \overline{U})_{Q} and the map i_{0}^{∗} is compatible with first Chern classes, we have that the element ζ ∈ H_{cr}^{2}((U, \overline{U})/R)_{Q} defined as ζ = ι(c_{1}^{HK}(ℒ))−c_{1}^{st}(ℒ) lies in T H_{cr}^{2}((U, \overline{U})/R)_{Q}. Hence ζ = T γ. Since the map ι is compatible with Frobenius and φ(c_{1}^{HK}(ℒ)) = pc_{1}^{HK}(ℒ), φ(c_{1}^{st}(ℒ)) = pc_{1}^{st}(ℒ), we have φ(ζ) = pζ. Since φ(T γ) = T^{p}φ(γ) this implies that γ ∈ ⋂_{n=1}^{∞} T^{n}H_{cr}^{2}((U, \overline{U})/R)_{Q}, which is not possible unless γ (and hence ζ) are zero. But this is what we wanted to show. □

We have the following projective space theorem for syntomic cohomology.

Proposition 5.2. Let ℰ be a locally free sheaf of rank d+1, d ≥ 0, on a scheme X ∈ 𝒱 ar_{K}. Consider the associated projective bundle π : ℙ(ℰ) → X. Then we have the following quasi-isomorphism of complexes of sheaves on X_{h}

⨁_{i=0}^{d} c_{1}^{syn}(𝒪(1))^{i} ∪ π^{∗} : ⨁_{i=0}^{d} 𝒮 (r − i)_{X,Q}[−2i] →^{∼} Rπ_{∗}𝒮 (r)_{ℙ(ℰ),Q}, 0 ≤ d ≤ r.

Here, the class c_{1}^{syn}(𝒪(1)) ∈ H_{syn}^{2}(ℙ(ℰ)_{h}, 1) refers to the class of the tautological bundle on ℙ(ℰ).

Proof. By (tedious) checking of many compatibilities we will reduce the above projective space theorem to the projective space theorems for the Hyodo-Kato and the filtered de Rham cohomologies.

To prove our proposition it suffices to show that for any ss-pair (U, \overline{U}) over K and the projective space π : ℙd ; \overline{U} → \overline{U} of dimension d over \overline{U} we have a projective space theorem for syntomic cohomology (a ≥ 0)

⨁_{i=0}^{d} c_{1}^{syn}(𝒪(1))^{i} ∪ π^{∗} : ⨁_{i=0}^{d} H_{syn}^{a−2i}(U_{h}, r − i) →^{∼} H_{syn}^{a}(ℙ_{U,h}^{d}, r), 0 ≤ d ≤ r.

By Proposition 3.18 and the compatibility of the maps H_{syn}^{∗}(U, \overline{U}, j)_{Q} →^{∼} H_{syn}^{∗}(U_{h}, j)_{Q} with products and first Chern classes, this reduces to proving a projective space theorem for log-syntomic cohomology, i.e., a quasi-isomorphism of complexes

⨁_{i=0}^{d} c_{1}^{syn}(𝒪(1))^{i} ∪ π^{∗} : ⨁_{i=0}^{d} H_{syn}^{a−2i}(U, \overline{U}, r − i)_{Q} →^{∼} H_{syn}^{a}(ℙ_{U}^{d}, ℙd ; \overline{U}, r)_{Q}, 0 ≤ d ≤ r,

where the class c_{1}^{syn}(𝒪(1)) ∈ H_{syn}^{2}(ℙ_{U}^{d}, ℙd ; \overline{U}, 1) refers to the class of the tautological bundle on ℙd ; \overline{U}. By the distinguished triangle

RΓ_{syn}(U, \overline{U}, r)_{Q} → RΓ_{cr}(U, \overline{U}, r)_{Q} → RΓ_{dR}(U, \overline{U}_{K})/F^{r}

and its compatibility with the action of c_{1}^{syn}, it suffices to prove the following two quasi-isomorphisms for the twisted absolute log-crystalline complexes and for the filtered log de Rham complexes (0 ≤ d ≤ r)

⨁_{i=0}^{d} c_{1}^{cr}(𝒪(1))^{i} ∪ π^{∗} : ⨁_{i=0}^{d} H_{cr}^{a−2i}(U, \overline{U}, r − i)_{Q} →^{∼} H_{cr}^{a}(ℙ_{U}^{d}, ℙd ; \overline{U}, r)_{Q},

⨁_{i=0}^{d} c_{1}^{dR}(𝒪(1))^{i} ∪ π^{∗} : ⨁_{i=0}^{d} F^{r−i}H_{dR}^{a−2i}(U, \overline{U}_{K}) →^{∼} F^{r}H_{dR}^{a}(ℙ_{U}^{d}, ℙ_{\overline{U}_{K}}^{d}).

For the log de Rham cohomology, notice that the above map is quasi-isomorphic to the map [3, 3.2]

⨁_{i=0}^{d} c_{1}^{dR}(𝒪(1))^{i} ∪ π^{∗} : ⨁_{i=0}^{d} F^{r−i}H_{dR}^{a−2i}(U) →^{∼} F^{r}H_{dR}^{a}(ℙ_{U}^{d}).

Hence well-known to be a quasi-isomorphism.

For the twisted log-crystalline cohomology, notice that since Frobenius behaves well with respect to c_{1}^{cr}, it suffices to prove a projective space theorem for the absolute log-crystalline cohomology H_{cr}^{∗}(U, \overline{U})_{Q}.

⨁_{i=0}^{d} c_{1}^{cr}(𝒪(1))^{i} ∪ π^{∗} : ⨁_{i=0}^{d} H_{cr}^{a−2i}(U, \overline{U})_{Q} →^{∼} H_{cr}^{a}(ℙ_{U}^{d}, ℙd ; \overline{U})_{Q}

Without loss of generality we may assume that the pair (U, \overline{U}) is split over K. By the distinguished triangle

RΓ_{cr}(U, \overline{U}) → RΓ_{cr}((U, \overline{U})/R) →^{N} RΓ_{cr}((U, \overline{U})/R))

and its compatibility with the action of c_{1}^{cr}(𝒪(1)) (cf. [62, Lemma 4.3.7]), it suffices to prove a projective space theorem for the log-crystalline cohomology H_{cr}^{∗}((U, \overline{U})/R)_{Q}. Since the R-linear isomorphism ι : H_{HK}^{∗}(U, \overline{U})_{Q}⊗R_{Q} →^{∼} H_{cr}^{∗}((U, \overline{U})/R)_{Q} is compatible with products [62, Prop. 4.4.9] and first Chern classes (cf. Lemma 5.1) we reduce the problem to showing the projective space theorem for the Hyodo-Kato cohomology.

⨁_{i=0}^{d} c_{1}^{HK}(𝒪(1))^{i} ∪ π^{∗} : ⨁_{i=0}^{d} H_{HK}^{a−2i}(U, \overline{U})_{Q} →^{∼} H_{HK}^{a}(ℙ_{U}^{d}, ℙd ; \overline{U})_{Q}

Tensoring by K and using the isomorphism ι_{dR,π} : H_{HK}^{∗}(U, \overline{U})_{Q} ⊗_{K_{0}} K →^{∼} H_{dR}^{∗}(U, \overline{U}_{K}) that is compatible with products [62, Cor. 4.4.13] and first Chern classes (cf. Lemma 5.1) we reduce to checking the projective space theorem for the log de Rham cohomology H_{dR}^{∗}(U, \overline{U}_{K}). And we have done this above. □

The above proof proves also the projective space theorem for the absolute crystalline cohomology.

Corollary 5.3. Let ℰ be a locally free sheaf of rank d + 1, d ≥ 0, on a scheme X ∈ 𝒱 ar_{K}. Consider the associated projective bundle π : ℙ(ℰ) → X. Then we have the following quasi-isomorphism of complexes of sheaves on X_{h}

⨁_{i=0}^{d} c_{1}^{cr}(𝒪(1))^{i} ∪ π^{∗} : ⨁_{i=0}^{d} 𝒥_{X,Q}^{[r−i]}[−2i] →^{∼} Rπ_{∗}𝒥_{ℙ(ℰ),Q}^{[r]}, 0 ≤ d ≤ r.

Here, the class c_{1}^{cr}(𝒪(1)) ∈ H^{2}(ℙ(ℰ)_{h}, 𝒥_{cr}) refers to the class of the tautological bundle on ℙ(ℰ).

For X ∈ 𝒱 ar_{K}, using the projective space theorem (cf. Theorem 5.2) and the Chern classes

c_{0}^{syn} : Q_{p} →^{can} 𝒮 (0)_{X_{Q}}, c_{1}^{syn} : 𝒪_{X_{h}}^{∗} → 𝒮 (1)_{X_{Q}}[1],

we obtain syntomic Chern classes c_{i}^{syn}(ℰ), for any locally free sheaf ℰ on X.

Syntomic cohomology has homotopy invariance property.

Proposition 5.4. Let X ∈ 𝒱 ar_{K} and f : 𝔸_{X}^{1} → X be the natural projection from the affine line over X to X. Then, for all r ≥ 0, the pullback map

f^{∗} : RΓ_{syn}(X_{h}, r)⟶^{∼}RΓ_{syn}(𝔸_{X,h}^{1}, r)

is a quasi-isomorphism.

Proof. Localizing in the h-topology of X we may assume that X = U - the open set of an ss-pair (U, \overline{U}) over K. Consider the following commutative diagram.

RΓ_{syn}(U, \overline{U}, r)_{Q} ⟶^{f^{∗}} RΓ_{syn}(𝔸_{U}^{1}, ℙ1 ; \overline{U}, r)_{Q}
↓^{≀}  ↓^{≀}
RΓ_{syn}(U_{h}, r) ⟶^{f^{∗}} RΓ_{syn}(𝔸_{U,h}^{1}, r)

The vertical maps are quasi-isomorphisms by Proposition 3.18. It suffices thus to show that the top horizontal map is a quasi-isomorphism. By Proposition 3.8, this reduces to showing that the map

C_{st}(RΓ_{HK}(U, \overline{U})_{Q}\{r\}) →^{f^{∗}} C_{st}(RΓ_{HK}(𝔸_{U}^{1}, ℙ1 ; \overline{U})_{Q}\{r\})

is a quasi-isomorphism. Or, that the map f : (𝔸_{U}^{1}, ℙ1 ; \overline{U}) → (U, \overline{U}) induces a quasi-isomorphism on the Hyodo-Kato cohomology and a filtered quasi-isomorphism on the log de Rham cohomology:

RΓ_{HK}(U, \overline{U})_{Q} →^{f^{∗}} RΓ_{HK}(𝔸_{U}^{1}, ℙ1 ; \overline{U})_{Q}, RΓ_{dR}(U, \overline{U}_{K}) →^{f^{∗}} RΓ_{dR}(𝔸_{U}^{1}, ℙ_{\overline{U}_{K}}^{1} )

Without loss of generality we may assume that the pair (U, \overline{U}) is split over K. Tensoring with K and using the Hyodo-Kato quasi-isomorphism we reduce the Hyodo-Kato case to the log de Rham one. The latter follows easily from the projective space theorem and the existence of the Gysin sequence in log de Rham cohomology. □

Remark 5.5. The above implies that syntomic cohomology is a Bloch-Ogus theory. A proof of this fact was kindly communicated to us by Frédéric Déglise and is contained in Appendix B, Proposition B.4.

Proposition 5.6. For a scheme X, let K_{∗}(X) denote Quillen’s higher K-theory groups of X. For X ∈ 𝒱 ar_{K}, i, j ≥ 0, there are functorial syntomic Chern class maps

c_{i,j}^{syn} : K_{j}(X) → H_{syn}^{2i−j}(X_{h}, i).

Proof. Recall the construction of the classes c_{i,j}^{syn}. First, one constructs universal classes C_{i,l}^{syn} ∈ H_{syn}^{2i}(B_{•}GL_{l,h}, i). By a standard argument, projective space theorem and homotopy property show that

H_{syn}^{∗}(B_{•}GL_{l,h}, ∗) ≃ H_{syn}^{∗}(K, ∗)[x_{1}^{syn}, …, x_{l}^{syn}],

where the classes x_{i}^{syn} ∈ H_{syn}^{2i}(B_{•}GL_{l,h}, i) are the syntomic Chern classes of the universal locally free sheaf on B_{•}GL_{l} (defined via a projective space theorem). For l ≥ i, we define

C_{i,l}^{syn} = x_{i}^{syn} ∈ H_{syn}^{2i}(B_{•}GL_{l,h}, i).

The classes C_{i,l}^{syn} ∈ H_{syn}^{2i}(B_{•}GL_{l,h}, i) yield compatible universal classes (see [36, p. 221]) C_{i,l}^{syn} ∈ H_{syn}^{2i}(X, GL_{l}(𝒪_{X}), i), hence a natural map of pointed simplicial sheaves on X_{ZAR}, C_{i}^{syn} : B_{•}GL(𝒪_{X}) → 𝒦 (2i, 𝒮′(i)_{X}), where 𝒦 is the Dold–Puppe functor of τ_{≥0}𝒮′(i)_{X}[2i] and 𝒮′(i)_{X} is an injective resolution of 𝒮 (i)_{X} := Rε_{∗}𝒮 (i)_{Q}, ε : X_{h} → X_{Zar}. The characteristic classes c_{i,j}^{syn} are now defined [36, 2.22] as the composition

K_{j}(X) → H^{−j}(X, Z × B_{•}GL(𝒪_{X})^{+}) → H^{−j}(X, B_{•}GL(𝒪_{X})^{+})

⟶^{C_{i}^{syn}} H^{−j}(X, 𝒦 (2i, 𝒮′(i)_{X})) →^{h_{j}} H_{syn}^{2i−j}(X_{h}, i),

where B_{•}GL(𝒪_{X})^{+} is the (pointed) simplicial sheaf on X associated to the + - construction [60, 4.2]. Here, for a (pointed) simplicial sheaf ℰ_{•} on X_{ZAR}, H^{−j}(X, ℰ_{•}) = π_{j}(RΓ(X_{ZAR}, ℰ_{•})) is the generalized sheaf cohomology of ℰ_{•} [36, 1.7]. The map h_{j} is the Hurewicz map:

H^{−j}(X, 𝒦 (2i, 𝒮′(i)_{X})) = π_{j}(𝒦 (2i, 𝒮′(i)(X)))→^{h_{j}}H_{j}(𝒦 (2i, 𝒮′(i)(X)))
= H_{j}(𝒮′(i)(X)[2i]) = H_{syn}^{2i−j}(X_{h}, i).
□

Proposition 5.7. The syntomic and the étale Chern classes are compatible, i.e., for X ∈ 𝒱 ar_{K}, j ≥ 0, 2i − j ≥ 0, the following diagram commutes

\xymatrix{ & K_{j}(X) \ar[dl]_{c_{i,j}^{syn}} \ar[dr]^{c_{i,j}^{ét}} \\ H_{syn}^{2i−j}(X_{h}, i) \ar[rr]^{ρ_{syn}} &  & H_{ét}^{2i−j}(X, Q_{p}(i))}

Proof. We can pass to the universal case (X = B_{•}GL_{l} := B_{•}GL_{l}/K, l ≥ 1). We have

H_{syn}^{∗}(B_{•}GL_{l,h}, ∗) ≃ H_{syn}^{∗}(K, ∗)[x_{1}^{syn}, …, x_{l}^{syn}],
H_{ét}^{∗}(B_{•}GL_{l}, ∗) ≃ H_{ét}^{∗}(K, ∗)[x_{1}^{ét}, …, x_{l}^{ét}]

By the projective space theorem and the fact that the syntomic period map commutes with products it suffices to check that ρ_{syn}(x_{1}^{syn}) = x_{1}^{ét} and that the syntomic period map ρ_{syn} commutes with the classes c_{0}^{syn} : Q_{p} → 𝒮 (0)_{Q} and c_{0}^{ét} : Q_{p} → Q_{p}(0). The statement about c_{0} is clear from the definition of ρ_{cr}; for c_{1} consider the canonical map f : B_{•}GL_{l} → B_{•}GL_{l,\overline{K}} and the induced pullback map

f_{ét}^{∗} : H_{ét}^{∗}(B_{•}GL_{l}, ∗) = H_{ét}^{∗}(K, ∗)[x_{1}, …, x_{l}] → H_{ét}^{∗}(B_{•}GL_{l,\overline{K}}, ∗) = Q_{p}[\overline{x}_{1}, …, \overline{x}_{l}]

that sends the Chern classes x_{i}^{ét} of the universal vector bundle to the classes \overline{x}_{i}^{ét} of its pullback. It suffices to show that f_{ét}^{∗}ρ_{syn}(C_{1,1}^{syn}) = C_{1,1}^{ét}. But, by definition, f_{ét}^{∗}ρ_{syn} = ρ_{syn}f_{syn}^{∗} and, by construction, we have the following commutative diagram

\xymatrix{H_{syn}^{2}(B_{•}𝔾_{m,h}, 1) \ar[r]^{can} \ar[d]^{ρ_{syn}} & H_{cr}^{2}(B_{•}𝔾_{m,\overline{K},h}) \ar[d]^{ρ_{cr}} \\ H_{ét}^{2}(B_{•}𝔾_{m,\overline{K}}, Q_{p}(1)) \ar[r] & H_{ét}^{2}(B_{•}𝔾_{m,\overline{K}}, B_{cr}^{+}) = H_{ét}^{2}(B_{•}𝔾_{m,\overline{K}}, Q_{p}(1)) ⊗ B_{cr}^{+}}

where the bottom map sends the generator of Q_{p}(1) to the element t ∈ B_{cr}^{+} associated to it. Since the syntomic and the crystalline Chern classes are compatible, it suffices to show that, for a line bundle ℒ, ρ_{cr}(c_{1}^{cr}(ℒ)) = c_{1}^{ét}(ℒ) ⊗ t. But this is [4, 3.2]. □

Remark 5.8. If 𝒳 is a scheme over V and X = 𝒳_{K}, we can consider the syntomic Chern classes c_{i,j}^{syn} : K_{j}(𝒳 ) → H_{syn}^{2i−j}(X_{h}, i) defined as the composition

K_{j}(𝒳 ) → K_{j}(X)⟶^{c_{i,j}^{syn}}H_{syn}^{2i−j}(X_{h}, i).

By the above proposition, these classes are compatible with the étale Chern classes. Recall that analogous results were proved earlier for 𝒳 smooth and projective [50], for 𝒳 – a complement of a divisor with relative normal crossings in such, and for 𝒳 - a semistable scheme over V [54].

5.2. Image of étale regulators. In this subsection we show that Soule’s étale regulators factor through the semistable Selmer groups.

Let X ∈ 𝒱 ar_{K}. For 2r − i − 1 ≥ 0, set

K_{2r−i−1}(X)_{0} := ker(K_{2r−i−1}(X)⟶^{c_{r,i+1}^{ét}}H^{0}(G_{K}, H_{ét}^{i+1}(X\overline{K}, Q_{p}(r))))

Notice that, for 2r − i − 1 > 0, we have K_{2r−i−1}(X)_{0} = K_{2r−i−1}(X). Write r_{r,i}^{ét} for the map

r_{r,i}^{ét} : K_{2r−i−1}(X)_{0} → H^{1}(G_{K}, H_{ét}^{i}(X\overline{K}, Q_{p}(r)))

induced by the Chern class map c_{r,i+1}^{ét} and the Hochschild-Serre spectral sequence map δ : H_{ét}^{i+1}(X, Q_{p}(r))_{0} → H^{1}(G_{K}, H_{ét}^{i}(X\overline{K}, Q_{p}(r))), where we set H_{ét}^{i+1}(X, Q_{p}(r))_{0} := ker(H_{ét}^{i+1}(X, Q_{p}(r)) → H_{ét}^{i+1}(X\overline{K}, Q_{p}(r))).

Theorem 5.9. The map r_{r,i}^{ét} factors through the subgroup

H_{st}^{1}(G_{K}, H_{ét}^{i+1}(X\overline{K}, Q_{p}(r))) ⊂ H^{1}(G_{K}, H_{ét}^{i+1}(X\overline{K}, Q_{p}(r))).

Proof. By Proposition 5.7, we have the following commutative diagram

\xymatrix{K_{2r−i−1}(X) \ar[d]^{c_{r,i+1}^{syn}} \ar[dr]^{c_{r,i+1}^{ét}} \\ H_{syn}^{i+1}(X_{h}, r) \ar[r]^{ρ_{syn}} & H_{ét}^{i+1}(X, Q_{p}(r)) \ar[r] & H_{ét}^{i+1}(X\overline{K}, Q_{p}(r))}

Hence the Chern class map c_{r,i+1}^{syn} : K_{2r−i−1}(X) → H_{syn}^{i+1}(X_{h}, r) factors through H_{syn}^{i+1}(X_{h}, r)_{0} :=

ker(H_{syn}^{i+1}(X_{h}, r) →^{ρ_{syn}} H_{ét}^{i+1}(X\overline{K}, Q_{p}(r))). Compatibility of the syntomic descent and the Hochschild-Serre spectral sequences (cf. Theorem 4.8) yields the following commutative diagram

\xymatrix{K_{2r−i−1}(X)_{0} \ar[d]^{c_{r,i+1}^{syn}} \ar[dr]^{c_{r,i+1}^{ét}} \\ H_{syn}^{i+1}(X_{h}, r)_{0} \ar[d]^{δ} \ar[r]^{ρ_{syn}} & H_{ét}^{i+1}(X, Q_{p}(r))_{0} \ar[d]^{δ} \\ H_{st}^{1}(G_{K}, H_{ét}^{i}(X\overline{K}, Q_{p}(r))) \ar[r]^{can} & H^{1}(G_{K}, H_{ét}^{i}(X\overline{K}, Q_{p}(r)))}

Our theorem follows. □

Remark 5.10. The question of the image of Soulé’s regulators r_{r,i}^{ét} was raised by Bloch-Kato in [13] in connection with their Tamagawa Number Conjecture. Theorem 5.9 is known to follow from the constructions of Scholl [57]. The argument goes as follows. Recall that for a class y ∈ K_{2r−i−1}(X)_{0} he constructs an explicit extension E_{y} ∈ Ext_{ℳ ℳ_{K}}^{1} (Q(−r), h^{i}(X)) in the category of mixed motives over K. The association y ↦ E_{y} is compatible with the étale cycle class and realization maps. By the de Rham Comparison Theorem, the étale realization r_{r,i}^{ét}(y) of the extension class E_{y} in

Ext_{G_{K}}^{1} (Q_{p}(−r), H^{i}(X\overline{K}, Q_{p})) = H^{1}(G_{K}, H_{ét}^{i}(X\overline{K}, Q_{p}(r)))

is de Rham, hence potentially semistable by [6], as wanted.

Appendix A. Vanishing of H^{2}(G_{K}, V) by Laurent Berger

Let V be a Q_{p}-linear representation of G_{K}. In this appendix we prove the following theorem.

Theorem A.1. If V is semistable and all its Hodge-Tate weights are ≥ 2, then H^{2}(G_{K}, V) = 0.

Let D(V) be Fontaine’s (φ, Γ)-module attached to V [33]. It comes with a Frobenius map φ and an action of Γ_{K}. Let H_{K} = Gal(\overline{K}/K(μ_{p^{∞}})) and let I_{K} = Gal(\overline{K}/K^{nr}). The injectivity of the restriction map H^{2}(G_{K}, V) → H^{2}(G_{L}, V) for L/K finite allows us to replace K by a finite extension, so that we can assume that H_{K}I_{K} = G_{K} and that Γ_{K} ≃ Z_{p}. Let γ be a topological generator of Γ_{K}. Recall (§I.5 of [16]) that we have a map ψ : D(V) → D(V).

Ideally, our proof of this theorem would go as follows. We use the Hochschild-Serre spectral sequence

H^{i}(G_{K}/I_{K}, H^{j}(I_{K}, V |_{I_{K}})) ⇒ H^{i+j}(G_{K}, V)

and, interpreting Galois cohomology in terms of (φ, Γ)-modules, we compute that H^{2}(I_{K}, V |_{I_{K}}) = 0 and H^{1}(I_{K}, V |_{I_{K}}) = K̂^{nr} ⊗_{K} D_{dR}(V). We conclude since, by Hilbert 90, H^{1}(G_{K}/I_{K}, H^{1}(I_{K}, V |_{I_{K}})) = 0. However, we do not, in general, have Hochschild-Serre spectral sequences for continuous cohomology. We mimic thus the above argument with direct computations on continuous cocycles (again using (φ, Γ)- modules). Laurent Berger is grateful to Kevin Buzzard for discussions related to the above spectral sequence.

Lemma A.2. (1) If V is a representation of G_{K}, then there is an exact sequence

0 → D(V)^{ψ=1}/(γ − 1) → H^{1}(G_{K}, V) → (D(V)/(ψ − 1))^{Γ_{K}} → 0;
(2) We have H^{2}(G_{K}, V) = D(V)/(ψ − 1, γ − 1).

Proof. See I.5.5 and II.3.2 of [16]. □

Lemma A.3. We have D(V |_{I_{K}})/(ψ − 1) = 0

Proof. Since V |_{I_{K}} corresponds to the case when k is algebraically closed, see the proof of Lemma VI.7 of [5]. □

Let γ_{I} denote a generator of ΓK̂^{nr}.

Lemma A.4. The natural map D(V |_{I_{K}})^{ψ=1}/(γ_{I} − 1) → (D(V |_{I_{K}})/(γ_{I} − 1))^{ψ=1} is an isomorphism if V^{I_{K}} = 0.

Proof. This map is part of the six term exact sequence that comes from the map γ_{I} − 1 applied to 0 → D(V |_{I_{K}})^{ψ=1} → D(V |_{I_{K}}) ⟶^{ψ−1} D(V |_{I_{K}}) → 0. Its kernel is included in D(V |_{I_{K}})^{γ_{I}=1} which is 0, since V^{I_{K}} = 0 (note that the inclusion (K̂^{nr} ⊗ V)^{G_{K}} ⊆ (ℰ̂^{nr} ⊗ V)^{G_{K}} = D(V)^{G_{K}} is an isomorphism). □

Suppose that x ∈ D(V)/(ψ − 1, γ − 1). If x̃ ∈ D(V) lifts x, then Lemma A.3 gives us an element y ∈ D(V |_{I_{K}}) such that (ψ − 1)y = x̃. Define a cocycle δ(x) ∈ Z^{1}(G_{K}/I_{K}, D(V |_{I_{K}})^{ψ=1}/(γ_{I} − 1)) by δ(x) : \overline{g} ↦ (g − 1)(y) if g ∈ G_{K} lifts \overline{g} ∈ G_{K}/I_{K}.

Proposition A.5. If V^{I_{K}} = 0, then the map

δ : D(V)/(ψ − 1, γ − 1) → H^{1}(G_{K}/I_{K}, (D(V |_{I_{K}})/(γ_{I} − 1))^{ψ=1})

is well-defined and injective.

Proof. We first check that δ(x)(g) ∈ (D(V |_{I_{K}})/(γ_{I} − 1))^{ψ=1}. We have (ψ − 1)(g − 1)(y) = (g − 1)(x). If we write g = ih ∈ I_{K}H_{K}, then (g − 1)x = (ih − 1)x = (i − 1)x ∈ (γ_{I} − 1)D(V |_{I_{K}}) since γ_{I} − 1 divides the image of i − 1 in Z_{p}[[ΓK̂^{nr}]]. This implies that δ(x)(g) ∈ (D(V |_{I_{K}})/(γ_{I} − 1))^{ψ=1}.

We now check that δ(x) does not depend on the choices. If we choose another lift g′ ∈ G_{K} of \overline{g} ∈ G_{K}/I_{K}, then g′ = ig for some i ∈ I_{K} and (g′ − 1)y − (g − 1)y = (i − 1)gy ∈ (γ_{I} − 1)D(V |_{I_{K}}) since γ_{I} − 1 divides the image of i − 1 in Z_{p}[[ΓK̂^{nr}]]. If we choose another y′ such that (ψ − 1)y′ = x̃, then y − y′ ∈ D(V |_{I_{K}})^{ψ=1} so that δ and δ′ are cohomologous. Finally, if x̃′ is another lift of x, then x̃′ − x̃ = (γ − 1)a + (ψ − 1)b with a, b ∈ D(V). We can then take y′ = y + b + (γ_{G} − 1)c where (ψ − 1)c = a. We then have (g − 1)y′ = (g − 1)y + (g − 1)b + (γ_{G} − 1)(g − 1)c. Since G_{K} = I_{K}H_{K}, we can write g = ih and (g − 1)b = (i − 1)b. Using G_{K} = I_{K}H_{K} once again, we see that I_{K} → G_{K}/H_{K} is surjective, so that we can identify γ_{I} and γ_{G}. The resulting cocycle is then cohomologous to δ(x). This proves that δ is well-defined.

We now prove that δ is injective. If δ(x) = 0, then using Lemma A.4 there exists z ∈ D(V |_{I_{K}})^{ψ=1} such that δ(x)(\overline{g}) is the image of (g − 1)(z) in D(V |_{I_{K}})^{ψ=1}/(γ_{I} − 1). This implies that (g − 1)(y − z) ∈ (γ_{I} − 1)D(V |_{I_{K}})^{ψ=1}. Applying ψ − 1 gives (g − 1)x̃ = 0 so that x̃ ∈ D(V)^{G_{K}} ⊂ V^{I_{K}} = 0. The map δ is therefore injective. □

Lemma A.6. If V is semistable and the weights of V are all ≥ 2, then exp_{V} : D_{dR}(V |_{I_{K}}) → H^{1}(I_{K}, V) is an isomorphism.

Proof. Apply Thm. 6.8 of [6] to V |_{I_{K}}. □

Proof of Theorem A.1. We can replace K by K_{n} for n ≫ 0 and use the fact that if H^{2}(G_{K_{n}}, V) = 0, then H^{2}(G_{K}, V) = 0 since the restriction map is injective. In particular, we can assume that H_{K}I_{K} = G_{K} and that Γ_{K} is isomorphic to Z_{p}. By item (2) of Lemma A.2, we have H^{2}(G_{K}, V) = D(V)/(ψ − 1, γ − 1), and so by Proposition A.5 above, it is enough to prove that

H^{1}(G_{K}/I_{K}, (D(V |_{I_{K}})/(γ_{I} − 1))^{ψ=1}) = 0.

Lemma A.4 tells us that (D(V |_{I_{K}})/(γ_{I} − 1))^{ψ=1} = D(V |_{I_{K}})^{ψ=1}/(γ_{I} − 1). Since D(V |_{I_{K}})/(ψ − 1) = 0 by Lemma A.3, item (1) of Lemma A.2 tells us that D(V |_{I_{K}})^{ψ=1}/(γ − 1) = H^{1}(I_{K}, V).

The map exp_{V} : D_{dR}(V |_{I_{K}}) → H^{1}(I_{K}, V) is an isomorphism by Lemma A.6, and this isomorphism commutes with the action of G_{K} since it is a natural map. We therefore have H^{1}(I_{K}, V) = K̂^{nr}⊗_{K}D_{dR}(V) as G_{K}-modules. It remains to observe that the cocycle δ(x) ∈ Z^{1}(G_{K}/I_{K}, K̂^{nr} ⊗_{K} D_{dR}(V)) is continuous and that H^{1}(G_{K}/I_{K}, K̂^{nr}) = 0 by taking a lattice, reducing modulo a uniformizer of K, and applying Hilbert 90. □

Appendix B. The Syntomic ring spectrum by Frédéric Déglise

In this appendix, we explain why syntomic cohomology as defined in this paper is representable by a motivic ring spectrum in the sense of Morel and Voevodsky’s homotopy theory. More precisely, we will exhibit a monoid object 𝒮 of the triangulated category of motives with Q_{p}-coefficients (see below), DM, such that for any variety X and any pair of integers (i, r),

H_{syn}^{i}(X_{h}, r) = Hom_{DM}(M(X), 𝒮 (r)[i]).

In fact, it is possible to apply directly [24, Th. 1.4.10] to the graded commutative dg-algebra RΓ_{syn}(X, ∗) of Theorem A in view of the existence of Chern classes established in Section 5.1. However, the use of h-topology in this paper makes the construction of 𝔼_{synt} much more straightforward and that is what we explain in this appendix. Reformulating slightly the original definition of Voevodsky (see [63]), we introduce:

Definition B.1. Let PSh(K, Q_{p}) be the category of presheaves of Q_{p}-modules over the category of varieties.

Let C be a complex in PSh(K, Q_{p}). We say:

(1) C is h-local if for any h-hypercovering π : Y_{•} → X, the induced map:

C(X) → π_{∗}Tot^{⊕}(C(Y_{•}))

is a quasi-isomorphism;

(2) C is 𝔸^{1}-local if for any variety X, the map induced by the projection:

H^{i}(X_{h}, C) → H^{i}(𝔸_{X,h}^{1}, C)

is an isomorphism. We define the triangulated category DM_{h}^{eff}(K, Q_{p}) of effective h-motives as the full subcategory of the derived category D(PSh(K, Q_{p})) made by the complexes which are h-local and 𝔸^{1}-local.

Equivalently, we can define this category as the 𝔸^{1}-localization of the derived category of h-sheaves on K-varieties (see [17], Sec. 5.2 and more precisely Prop. 5.2.10, Ex. 5.2.17(2)). Recall also from loc. cit., that there are derived tensor products and internal Hom on DM_{h}^{eff}(K, Q_{p}).

For any integer r ≥ 0, the syntomic sheaf 𝒮 (r) is both h-local (by definition) and 𝔸^{1}-local (Prop. 5.4). Thus it defines an object of DM_{h}^{eff}(K, Q_{p}) and for any variety X, one has an isomorphism:

Hom_{DM_{h}^{eff} (K,Q_{p})}(Q_{p}(X), 𝒮 (r)[i]) = Hom_{D(PSh(K,Q_{p}))}(Q_{p}(X), 𝒮 (r)[i]) = H_{syn}^{i}(X_{h}, r)

where Q_{p}(X) is the presheaf of Q_{p}-vector spaces represented by X. Thus, the representability assertion for syntomic cohomology is obvious in the effective setting.

Recall that one defines the Tate motive in DM_{h}^{eff}(K, Q_{p}) as the object Q_{p}(1) := Q_{p}(ℙ_{K}^{1})/Q_{p}({∞})[−2]. Given any complex object C of DM_{h}^{eff}(K, Q_{p}), we put: C(n) := C ⊗ Q_{p}(1)^{⊗,n}. One should be careful that this notation is in conflict with that of 𝒮 (r) considered as an effective h-motive, as the natural twist on syntomic cohomology is unrelated to the twist of h-motives. To solve this matter, we are led to consider the following notion of Tate spectrum, borrowed from algebraic topology according to Morel and Voevodsky.

Definition B.2. A Tate h-spectrum (over K with coefficients in Q_{p}), is a sequence 𝔼 = (E_{i}, σ_{i})_{i∈ℕ} such that:

• for each i ∈ ℕ, E_{i} is a complex of PSh(K, Q_{p}) equipped with an action of the symmetric group Σ_{i} of the set with i-element,

• for each i ∈ ℕ, σ_{i} : E_{i}(1) → E_{i+1} is a morphism of complexes – called the suspension map in degree i,

• For any integers i ≥ 0, r > 0, the map induced by the morphisms σ_{i}, ⋯, σ_{i+r}:

E_{i}(r) → E_{i+r}

is compatible with the action of Σ_{i} × Σ_{r}, given on the left by the structural Σ_{i}-action on E_{i} and the action of Σ_{r} via the permutation isomorphism of the tensor structure on C(PSh(K, Q_{p})), and on the right via the embedding Σ_{i} × Σ_{r} → Σ_{i+r}.

A morphism of Tate h-spectra f : 𝔼 → 𝔽 is a sequence of Σ_{i}-equivariant maps (f_{i} : E_{i} → F_{i})_{i∈ℕ} compatible with the suspension maps. The corresponding category will be denoted by Sp_{h}(K, Q_{p}).

There is an adjunction of categories:

(52)  Σ^{∞} : C( PSh(K, Q_{p})) ⇆ Sp_{h}(K, Q_{p}) : Ω^{∞}

such that for any complex K of h-sheaves, Σ^{∞}C is the Tate spectrum equal in degree n to C(n), equiped with the obvious action of Σ_{n} induced by the symmetric structure on tensor product and with the obvious suspension maps.

Definition B.3. A morphism of Tate spectra (f_{i} : E_{i} → F_{i})_{i∈ℕ} is a level quasi-isomorphism if for any i, f_{i} is a quasi-isomorphism.

A Tate spectrum 𝔼 is called a Ω-spectrum if for any i, E_{i} is h-local and 𝔸^{1}-local and the map of complexes

E_{i} → \underline{Hom}(Q_{p}(1), E_{i+1})

is a quasi-isomorphism.

We define the triangulated category DM_{h}(K, Q_{p}) of h-motives over K with coefficients in Q_{p} as the category of Tate Ω-spectra localized by the level quasi-isomorphisms.

The category of h-motives notably enjoys the following properties: (DM1) The adjunction of categories (52) induces an adjunction of triangulated categories:

Σ^{∞} : DM_{h}^{eff}(K, Q_{p}) ⇆ DM_{h}(K, Q_{p}) : Ω^{∞}

such that for a Tate Ω-spectrum 𝔼, and any integer r ≥ 0, Ω^{∞}(𝔼(r)) = E_{r} (see [17, Sec. 5.3.d, and Ex. 5.3.31(2)]).

Given any variety X, we define the (stable) h-motive of X as M(X) := Σ^{∞}Q_{p}(X).

(DM2) There exists a symmetric closed monoidal structure on DM(K, Q_{p}) such that Σ^{∞} is monoidal and such that Σ^{∞}Q_{p}(1) admits a tensor inverse (see [17, Sec. 5.3, and Ex. 5.3.31(2)]). By abuse of notations, we put: Q_{p} = Σ^{∞}Q_{p}.

(DM3) The triangulated monoidal category DM_{h}(K, Q_{p}) is equivalent to all known version of triangulated categories of mixed motives over Spec(K) with coefficients in Q_{p} (see [17, Sec. 16, and Th. 16.1.2]). In particular, it contains as a full subcategory the category DM_{gm}(K) ⊗ Q_{p} obtained from the category of Voevodsky geometric motives ([35, chap.5]) by tensoring Hom-groups with Q_{p} (see [17, Cor. 16.1.6, Par. 15.2.5]).

With that definition, the construction of a Tate spectrum representing syntomic cohomology is almost obvious. In fact, we consider the sequence of presheaves

𝒮 := (𝒮 (r), r ∈ ℕ),

where each 𝒮 (r) with the trivial action of Σ_{r}. According to the first paragraph of Section 5.1, we can consider the first Chern class of the canonical invertible sheaf ℙ^{1}: c̄ ∈ H_{syn}^{2}(ℙ_{K}^{1}, 1) = H^{2}(ℙ_{K,h}^{1}, 𝒮 (1)). Take any lift c : Q_{p}(ℙ_{K}^{1}) → 𝒮 (1)[2] of this class. By the definition of the Tate twist, it defines an element Q_{p}(1) → 𝒮 (1) still denoted by c. We define the suspension map:

𝒮 (r) ⊗ Q_{p}(1)⟶^{Id ⊗c}𝒮 (r) ⊗ 𝒮 (1) ⟶^{μ} 𝒮 (r + 1)

where μ is the multiplication coming from the graded dg-structure on 𝒮 (∗). Because this dg-structure is commutative, we obtain that these suspension maps induce a structures of a Tate spectrum on 𝒮. Moreover, 𝒮 is a Tate Ω-spectrum because each 𝒮 (r) is h-local and 𝔸^{1}-local, and the map obtained by adjunction from σ_{r} is a quasi-isomorphism because of the projective bundle theorem for ℙ^{1} (an easy case of Proposition 5.2).

Now, by definition of DM_{h}(K, Q_{p}) and because of property (DM1) above, for any variety X, and any integers (i, r), we get:

Hom_{DM_{h}(K,Q_{p})}(M(X), 𝒮 (r)[i]) = Hom_{DM_{h}^{eff} (K,Q_{p})}(Q_{p}(X), Ω^{∞}(𝒮 (r))[i]) = H_{syn}^{i}(X_{h}, r).

Moreover, the commutative dg-structure on the complex 𝒮 (∗) induces a monoid structure on the associated Tate spectrum. In other words, 𝒮 is a ring spectrum (strict and commutative). This construction is completely analogous to the proof of [24, Prop. 1.4.10]. In particular, we can apply all the constructions of [24, Sec. 3] to the ring spectrum 𝒮. Let us summarize this briefly:

Proposition B.4. (1) Syntomic cohomology is covariant with respect to projective morphisms of smooth varieties (Gysin morphisms in the terminology of [24]). More precisely, to a projective morphism of smooth K-varieties f : Y → X one can associate a Gysin morphism in syntomic cohomology

f_{∗} : H_{syn}^{n}(Y_{h}, i) → H_{syn}^{n−2d}(X_{h}, i − d),

where d is the dimension of f.

(2) The syntomic regulator over Q_{p} is induced by the unit η : Q_{p} → 𝒮 of the ring spectrum 𝒮:

r_{syn} : H_{M}^{r,i}(X) ⊗ Q_{p} = Hom_{DM_{h}(K,Q_{p})}(M(X), Q_{p}(r)[i])
⟶ Hom_{DM_{h}(K,Q_{p})}(M(X), 𝒮 (r)[i]) = H_{syn}^{i}(X_{h}, r).

It is compatible with product, pullbacks and pushforwards.

(3) The syntomic cohomology has a natural extension to h-motives^{10}:

DM_{h}(K, Q_{p})^{op} → D(Q_{p}), M ↦ Hom_{DM_{h}(K,Q_{p})}(M, 𝒮)

and the syntomic regulator r_{syn} can be extended to motives.

(4) There exists a canonical syntomic Borel-Moore homology H_{∗}^{syn}(−, ∗) such that the pair of functor (H_{syn}^{∗}(−, ∗), H_{∗}^{syn}(−, ∗)) defines a Bloch-Ogus theory.

(5) To the ring spectrum 𝒮 there is associated a cohomology with compact support satisfying the usual properties.

For points (1) and (2), we refer the reader to [24, Sec. 3.1] and for the remaining ones to [24, Sec. 3.2].

Remark B.5. Note that the construction of the syntomic ring spectrum 𝒮 in DM_{h}(K, Q_{p}) automatically yields the general projective bundle theorem (already obtained in Prop. 5.2). More generally, the ring spectrum 𝒮 is oriented in the terminology of motivic homotopy theory. Thus, besides the theory of Gysin morphisms, this gives various constructions – symbols, residue morphisms – and yields various formulas – excess intersection formula, blow-up formulas (see [23] for more details).

10and in particular to the usual Voevodsky geometrical motives by (DM3) above.

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 E-mail address: jan.nekovar@imj-prg.fr, wieslawa.niziol@ens-lyon.fr