A lemma on vanishing cycles and its application to the tame Lubin-Tate space

J.-F. Dat

Abstract

This note is concerned with a cohomological consequence of a geometric construction due to Yoshida, which relates the tame level of the Lubin-Tate tower to some Deligne-Lusztig variety of Coxeter type. More precisely, we show that the equivariant morphism in cohomology which follows from Yoshida’s construction is an isomorphism, whatever the coefficients are. In particular, this gives a conceptual explanation to the observation that ℓ-adic cohomologies indeed were “the same”, once computed independently on each side (by Boyer, resp. Lusztig). This also gives a “simple” proof of the absence of torsion in the integral cohomology of the tame Lubin-Tate space. Our main tool is a general result on vanishing cycles for schemes with semi-stable reduction which generalizes previous results of Zheng and Illusie. In rough terms, this states that the restriction of the nearby cycles complex to a closed stratum is the push-forward of its restriction to the corresponding open stratum.

1 Main result

Let K be a non-archimedean local field with ring of integers O, uniformizer ϖ and residue field 𝔽_{q}.

(1.1) The tame Lubin-Tate space. We fix an integer d and a formal O-module ℍ_{d} of dimension 1 and height d over \overline{𝔽}_{q}. We refer the reader e.g. to [10] for the definition of such an object and related structures. Let us denote by Ŏ the completion of the maximal unramified extension of O. We consider the Ŏ-formal scheme M̂_{n} which parametrizes deformations of ℍ_{d} together with an “order n Drinfeld level structure”. It is the formal spectrum of a complete local Ŏ-algebra A_{n}. The algebra A_{0} is formally smooth over Ŏ of relative dimension d − 1, and A_{n} is finite over A_{0} with deck tranformation group GL_{d}(O/ϖ^{n}). Let us denote by M_{n} the generic fiber of M̂_{n}, an analytic space over the non-archimedean field K̆ := Ŏ[1/p]. So M_{0} is an open ball of dimension d − 1 and M_{n} is a Galois étale covering of M_{0} with group GL_{d}(O/ϖ^{n}). In this note, we will focus on M_{1} with its action by GL_{d}(𝔽_{q}).

(1.2) A Deligne-Lusztig variety. Let w = (1, 2, ⋯, d) ∈ 𝔖_{d} be the Coxeter element of the Weyl group of GL_{d}. We denote by X (w) the variety over \overline{𝔽}_{q} which parametrizes Borel subgroups of GL_{d,𝔽_{q}} which are in relative position w with their image under the Frobenius map relative to 𝔽_{q}. It carries an action of GL_{d}(𝔽_{q}). Further we denote by π : X̃(w) ⟶ X (w) the GL_{d}(𝔽_{q})-equivariant 𝔽_{q^{d}}^{×}-torsor defined in [5, p. 111]. These objects admit the following more concrete description, see [5, p.117]. First, X (w) identifies with the open subset Ω_{𝔽_{q}}^{d−1} of ℙ_{𝔽_{q}}^{d−1} defined, in projective coordinates, by the non-vanishing

of the determinant det ((x_{i}^{q^{j}})_{0⩽i,j⩽d−1}). In other words, Ω_{𝔽_{q}}^{d−1} is the complement of 𝔽_{q}-

rational hyperplanes, the so-called Drinfeld upper half-space. Second, X̃(w) identifies

with the open subset of 𝔸_{𝔽_{q}}^{d} defined by the equation det ((x_{i}^{q^{j}})_{0⩽i,j⩽d−1})^{q−1} = (−1)^{d−1}.

(1.3) A relation between these objects. Let K̆_{d} denote the ramified extension of K̆ generated by a q^{d}−1-root of ϖ. In [11], T. Yoshida constructed a GL_{d}(𝔽_{q})-equivariant integral model of M_{1} ˆ⊗K̆ K̆_{d}, whose special fiber contains an open subset GL_{d}(𝔽_{q})-isomorphic to X̃(w), and which is smooth at points of this open subset. Therefore, the general formalism of nearby/vanishing cycles provides a morphism in D^{b}(ΛGL_{d}(𝔽_{q})) :

RΓ(M_{1}^{ac}, Λ) ⟶ RΓ(X̃(w), Λ). (1)

Here ^{ac} denotes a base change to an algebraic closure of K̆ and Λ is ℤ/ℓ^{n}ℤ or ℤ_{ℓ}.

However, the cohomological tools at hand only allowed him to relate the cuspidal part of the \overline{ℚ}_{ℓ}-cohomologies of both objects, as well as the alternating sum. This is kind of frustrating, because the explicit computations of Boyer [4], resp. Lusztig [7], for Λ = \overline{ℚ}_{ℓ} suggest a stronger relation, at least for ℓ-adic coefficients.

Our aim here is to improve on the cohomological tools, in order to prove :

Theorem.– The morphism (1) is an isomorphism. Moreover, by duality we also get an isomorphism

RΓ_{c}(M_{1}^{ac}, Λ) ⟶^{∼} RΓ_{c}(X̃(w), Λ).

The cohomological tool we provide is a property of nearby cycles for semi-stable schemes and is the object of Section 2, see Theorem (2.2). The application to the tame Lubin-Tate space is done in Section 3.

After completing this note, the author has learned from L. Illusie that a statement very close to, but slightly weaker, than Theorem (2.2) already appears in [12, Lemma 5.6] (and generalizes a previous particular case of [6, 1.5]). However, our enhanced version is necessary for our application to the Lubin-Tate space and probably for further similar applications such as to Drinfeld spaces.

Here is an example of a consequence of the foregoing Theorem. We will use it in a forthcoming paper on a realization of the mod-ℓ local Langlands correspondence.

Corollary.– The ℤ_{ℓ}-modules H_{c}^{i}(M_{1}, ℤ_{ℓ}) are torsion-free.

Indeed, the torsion-freeness in the case of X̃(w) follows from Lemma 3.9 and Corollary 4.3 of [3]^{1}. We note that Boyer has announced torsion-freeness results for the whole Lubin-Tate tower, but the proof here is arguably easier.

(1.4) An open question. The action of GL_{d}(O)/(1 + ϖM_{d}(O)) on the cohomology groups H^{i}(M_{1}, Λ) is the restriction of an action by “cohomological correspondences” of the Hecke algebra H_{Λ}(GL_{d}(K), (1 + ϖM_{d}(O))) of compactly supported measures on GL_{d}(K) which are bi-invariant by the compact open subgroup 1 + ϖM_{d}(O). An interesting problem would be to determine this action.

1I thank Olivier Dudas for supplying this reference to me

2 Nearby cycles

(2.1) The semi-stable setting. Let S = Spec (W) be a strict henselian trait with generic point η and closed point s of residue characteristic p. We choose a geometric point \overline{η} over η and denote by Δ the corresponding Galois (inertia) group. Let Λ be either ℤ_{ℓ}/ℓ^{k}ℤ_{ℓ} or ℤ_{ℓ}. For an S-scheme X, we denote by RΨ : D^{+}(X_{η}, Λ) ⟶ D^{+}(X_{s}, ΛΔ) the (absolute) nearby cycles functor^{2}.

We will assume that X has “generalized” semi-stable reduction, i.e. that it is flat over S, with smooth generic fiber X_{η}, and with special fiber X_{s} a “tame” normal crossings divisor. The latter condition means that each closed point of X_{s} has an étale neighborhood which admits an étale map to some Spec (W [X_{1}, ⋯, X_{n}]/(X_{1}^{m_{1}} ⋯ X_{n}^{m_{n}} − ϖ)) where ϖ is a uniformizer of W and the m_{i} are prime to p.

Write X_{s} = ∑_{i∈I} e_{i}Y_{i} as a sum of irreducible reduced divisors with multiplicities. For ∅ ≠ J ⊆ I, we put

• Y_{J} := ⋂_{i∈J} Y_{i} and i_{J} : Y_{J} ↪ X its closed immersion.

• Y_{J}^{0} := smooth part of (Y_{J} \ ⋃_{i∉J} Y_{i}) and j_{J} : Y_{J}^{0} ↪ X the corresponding locally closed immersion.

We extend the notation to J = ∅ by putting Y_{∅} := X, so that Y_{∅}^{0} = X_{η}. For J ⊆ K we have Y_{K} ⊆ Y_{J}, and if Y_{K} is not the empty scheme, then it is has pure codimension codim(Y_{K}, Y_{J}) = |K \ J|.

(2.2) Theorem.– For ∅ ≠ J ⊆ I, the canonical morphism

i_{J,∗}i_{J}^{∗}(RΨ(Λ)) ⟶ Rj_{J,∗}j_{J}^{∗}(RΨ(Λ))

is an isomorphism.

Remark.– The main differences with Zheng’s result in [12, Lemma 5.6] is that we allow generalized semi-stable models and deeper strata. Also our proof is different, and it is not cleat how to get deeper strata with Zheng’s approach.

Proof. Note that this statement is equivalent to the following one :

(∗) For any closed point x ∈ Y_{J} \ Y_{J}^{0}, we have i_{x}^{!}i_{J}^{∗}(RΨ(Λ)) = 0.

Therefore it is local in the étale topology, and we may assume that

X = Spec (W [x_{1}, ⋯, x_{n}]/(x_{1}^{e_{1}} ⋯ x_{n}^{e_{n}} − ϖ)).

More precisely, all we have to do is to prove statement (∗) for x the point with ideal (ϖ, x_{1}, ⋯, x_{n}) in X as above. Further, by invariance of nearby cycles under smooth morphisms, we may assume that all e_{i} are non-zero. In this case, I = {1, ⋯, n} and Y_{J} is non-empty, smooth and irreducible for all J ⊆ I. In particular Y_{J}^{0} = Y_{J} \ ⋃_{i∉J} Y_{i}. Note also that Y_{I} = {x} = Y_{I}^{0}, so our statement is non-empty only for J ≠ I. In fine we are left to prove that for our particular X above, the following holds :

(2) for any J ⊊ I we have , i_{I}^{!}i_{J}^{∗}(RΨ(Λ) = 0

2Sometimes also called “vanishing” cycles functor

Step 1. Our next task is to reduce further the latter statement to the case where e_{i} = 1 for all i ∈ I. To do this, we will follow the idea of a similar reduction step in the proof of [9, Prop. 6]. Let e be a common multiple of the e_{i}’s which is prime to p, and put

X̃ := Spec (W [y_{1}, ⋯, y_{n}]/(y_{1}^{e} ⋯ y_{n}^{e} − ϖ)).

The map x_{i} ↦ y_{i}^{e/e_{i}} defines a flat covering X̃ ⟶^{f} X with deck transformation group H = ∏_{i} μ_{e/e_{i}} (roots of unity) acting by (ζ_{1}, ⋯, ζ_{n}) · (y_{1}, ⋯, y_{n}) = (ζ_{1}y_{1}, ⋯, ζ_{n}y_{n})and which is Galois étale over η. Therefore, the sheaf f_{η,∗}(ΛX̃_{η}) on X is a sheaf of Λ[H]-modules and the adjunction map Λ_{X_{η}} ⟶ f_{η,∗}(ΛX̃_{η}) is an isomorphism Λ_{X_{η}} ⟶^{∼} f_{η,∗}(ΛX̃_{η})^{H}. Note that taking H-invariants is a finite projective limit, hence commutes with pull-back functors. On the other hand, the derived H-invariant functor R_{H} on complexes of Λ[H]- sheaves coincides with the functor RHom_{Λ[H]}(Λ, −), which commutes with direct images and extraordinary inverse images (see SGA 4, Exp. XVIII, Cor. 3.1.12.2 for the latter). Therefore we get an isomorphism

i_{I}^{!}i_{J}^{∗}RΨ(Λ) ⟶^{∼} R_{H} (i_{I}^{!}i_{J}^{∗}RΨ(f_{η,∗}Λ)) .

Now, since f is finite, we have a canonical isomorphism RΨ(f_{η,∗}Λ) ⟶^{∼} Rf_{s,∗}(RΨ(X̃, Λ)), which induces in turn an isomorphism

i_{I}^{!}i_{J}^{∗}RΨ(f_{η,∗}Λ) ⟶^{∼} Rf_{s,∗} (ĩ_{I}^{!}ĩ_{J}^{∗}RΨ(X̃, Λ))

in which ĩ_{I}^{∗} and ĩ_{J}^{∗} are analogous to i_{I}^{∗} and i_{J}^{∗} but relative to X̃. Finally, let W̃ := W [t]/(t^{e} − ϖ), a strict henselian local extension of W with uniformizer t and inertia group Δ̃. We may see X̃ as a W̃-scheme by sending t to y_{1} ⋯ y_{n} and thus we get a nearby cycles complex RΨ̃(X̃, Λ) ∈ D^{+}(X̃_{s}, ΛΔ̃) relative to W̃, and related to the previous one by the formula

RΨ(X̃, Λ) ≃ IndΔ ; Δ̃ (RΨ̃(X̃, Λ)) .

On the other hand, as a W̃-scheme, X̃ is equal to X̃[y_{1}, ⋯, y_{n}]/(y_{1} ⋯ y_{n} − t). Therefore, the three displayed isomorphisms above reduce our statement to the case where all multiplicities are 1.

Step 2. We now assume all multiplicities are 1 and will reduce the vanishing of i_{I}^{!}i_{J}^{∗}(RΨ(Λ)) to its analogue for rational nearby cycles. More precisely, let us put i : X_{s} ↪ X and j = j_{∅} : X_{η} ↪ X, and let us assume that for J ⊊ I we have

i_{I}^{!}i_{J}^{∗}(i^{∗}Rj_{∗}(Λ)) = 0. (3)

Note that, by construction, i^{∗}Rj_{∗}(Λ) ⟶^{∼} R_{Δ}RΨ(Λ), where R_{Δ} is the derived functor of the Δ-invariant functor. Now, recall the following results from [8] (only the second one requires the multiplicity 1 assumption) :

i) The action of Δ on RΨ(Λ) factors through the tame quotient Δ_{t}, [8, Satz 2.23].

ii) The action of Δ on each R^{p}Ψ(Λ) is trivial, [8, Korollar 2.25] or SGA 7, I, Thm 3.3.

Let T be a progenerator of the tame inertia group Δ_{t}. As a consequence of i) and [8, Satz 1.1] we have a distinguished triangle

R_{Δ}RΨ(Λ) ⟶ RΨ(Λ) ⟶^{T − 1} RΨ(Λ) ⟶^{+1},

and as a consequence of ii), the endomorphism T −1 of RΨ(Λ) is nilpotent. Then, applying i_{I}^{!}i_{J}^{∗} to the above triangle yields, assuming (3), a triangle

0 ⟶ i_{I}^{!}i_{J}^{∗}RΨ(Λ) ⟶^{T − 1} i_{I}^{!}i_{J}^{∗}RΨ(Λ) ⟶^{+1}

which means that T − 1 is an automorphism of i_{I}^{!}i_{J}^{∗}RΨ(Λ). Since it is also nilpotent, we get i_{I}^{!}i_{J}^{∗}RΨ(Λ) = 0.

Step 3 : Proof of (3). In fact it is more convenient to prove the dual (and equivalent) statement that i_{I}^{∗}i_{J}^{!}j_{!}(Λ) = 0. Let us start with the simplicial resolution :

j_{!}(Λ) ⟶^{∼} (Λ ⟶ ⨁_{i∈I} i_{i,∗}(Λ) ⟶ ⋯ ⟶ ⨁_{K⊂I,|K|=p} i_{K,∗}(Λ) ⟶ ⋯ ⟶ i_{I,∗}(Λ))

For each subset K ⊆ I, we have Y_{J} ∩ Y_{K} = Y_{J∪K}, which is smooth of codimension |J \ K| in Y_{K}. Therefore by purity, we have an isomorphism

i_{J}^{!}i_{K,∗}(Λ) ⟶^{∼} i_{J}^{∗}i_{K,∗}(Λ)[−2|J \ K|]

and thus an isomorphism

i_{I}^{∗}i_{J}^{!}i_{K,∗}(Λ) ⟶^{∼} Λ[−2|J \ K|].

The first page of the spectral sequence attached to the above resolution then reads :

E_{1}^{p,q} = ( ⨁_{K⊆I,|K|=p,q=2|J\K|} Λ) ⇒ R^{p+q} (i_{I}^{∗}i_{J}^{!}j_{!}(Λ)) .

More precisely, the line E_{1}^{•,q} is zero unless q is even. In this case, put k_{2} := |J|− \frac{1}{2}q. Then E_{1}^{•,q} decomposes as a sum of complexes E_{1}^{•,q} = ∑_{K_{2}⊆J,|K_{2}|=k_{2}} C_{K_{2}}^{•} where

C_{K_{2}}^{•} = (Λ ⟶ ⋯ ⟶ ⨁_{K_{1}⊆I\J,|K_{1}|=p−k_{2}} Λ ⟶ ⋯ ⟶ Λ)

which is, up to a schift by −k_{2}, the chain complex associated to the simplicial power set of I \ J. Since I \ J ≠ ∅, this chain complex is acyclic. Therefore E_{2}^{pq} = 0 for all p, q, and finally i_{I}^{∗}i_{J}^{!}j_{!}(Λ) = 0, as desired.

(2.3) Reinterpretation in terms of analytic tubes and formal schemes. We continue with the setting of (2.1) but we assume further that W is ϖ-adically complete and write K := W [1/p] for its field of fractions. Let X_{η}^{an} denote the K-analytic space (in the sense of Berkovich) attached to X_{η} via the GAGA functor. There is a specialisation map |X_{η}^{an}| ⟶^{sp} |X_{s}| between the underlying topological spaces. The inverse image sp^{−1}(Z) of a locally closed subset Z ⊂ |X_{s}| is locally closed in the Berkovich topology and inherits a canonical analytic structure. It is called the analytic tube of Z in X_{η}^{an}. Berkovich has defined the étale site of such an analytic space in [1]. According to [2, Cor. 3.5], the cohomology of the tube is related to Z by a canonical isomorphism in D^{b}(ΛΔ)

(4) RΓ(sp^{−1}(Z)^{ac}, Λ) ⟶^{∼} RΓ(Z, RΨ(X, Λ)),

where ^{ac} denotes a base change to an algebraic closure of K. Theorem (2.2) then implies the following statement.

Corollary.– With the notation of (2.1), the canonical restriction morphism

RΓ(sp^{−1}(Y_{J})^{ac}, Λ) ⟶ RΓ(sp^{−1}(Y_{J}^{0})^{ac}, Λ)

is an isomorphism.

For the application we have in mind, it is convenient to recast this in terms of formal schemes. Take up the notation above the Corollary and let X̂_{Z} denote the formal completion of X along Z. This belongs to a class of formal schemes 𝔛 over W for which Berkovich has defined in [2] the notion of “analytic generic fiber” 𝔛_{η}, together with a nearby cycles functor^{3} RΨ_{η} : D^{+}(𝔛_{η}, Λ) ⟶ D^{+}(𝔛_{s}, Λ) which computes the cohomology of 𝔛_{η} via the usual formula

RΓ(𝔛_{η}^{ac}, F) ⟶^{∼} RΓ(𝔛_{s}, RΨ_{η}(𝔛, F)).

In the case 𝔛 = X̂_{Z}, we have 𝔛_{s} = Z and 𝔛_{η} ≃ sp^{−1}(Z), and the main theorem of [2] exhibits a canonical isomorphism RΨ_{η}(𝔛, Λ) ⟶^{∼} RΨ(X, Λ)_{|Z} which explains isomorphism (4).

Corollary.– With the notation of (2.1), there is a canonical isomorphism

RΓ((X̂_{Y_{J}})_{η}^{ac}, Λ) ⟶^{∼} RΓ(Y_{J}^{0}, RΨ_{η}(X̂_{Y_{J}} , Λ)).

3 Application to the “tame” Lubin-Tate deformation space

Here, we take up the notation of the introduction. Our former W now becomes Ŏ.

(3.1) We focus on M̂_{1} = Spf(A_{1}) and recall some of the constructions of Yoshida’s in [11]. Note that the latter rather works with the spectrum X := Spec(A_{1}). We will stick to his notation by writing X̂ := Spf(A_{1}). In [11, Sec. 4], a “semi-stable” GL_{d}(𝔽_{q})- equivariant modification of X is constructed. It is a Ŏ-scheme Z_{st}, equiped with an action of GL_{d}(𝔽_{q}), endowed with an equivariant proper morphism Z_{st} ⟶^{f} X which induces an isomorphism of generic fibers, and whose completed local rings at points of the special fiber are of the usual “semi-stable” form, up to replacing polynomials by formal series. The closed fiber of Z_{st} over X (meaning the fiber over the closed point of X), denoted by Y_{ℙ,st} in loc. cit., is equivariantly isomorphic to a projective space ℙ_{\overline{𝔽}_{q}}^{d−1} blown up at all 𝔽_{q}-rational linear subspaces. As a component of the special fiber, it has multiplicity q^{d} − 1 and meets the other components along the exceptional locus of this blow up. In other words, the open complement in Y_{ℙ,st} of the other irreducible components, denoted by Y_{ℙ,st}^{0}, identifies with the Drinfeld upper half space Ω_{𝔽_{q}}^{d−1} (the complement in ℙ_{\overline{𝔽}_{q}}^{d−1} of all 𝔽_{q}-rational hyperplanes).

Corollary.– Let us denote by Ẑ_{st} the completion of Z_{st} along its closed fiber Y_{ℙ,st}. There is a GL_{d}(𝔽_{q})-equivariant isomorphism

RΓ (M_{1}^{ac}, Λ) ⟶^{∼} RΓ (Ω_{𝔽_{q}}^{d−1}, RΨ_{η}(Ẑ_{st}, Λ)) .

Moreover RΨ_{η}(Ẑ_{st}, Λ)_{|Ω_{𝔽q}^{d−1}} is concentrated in cohomological degree 0 and R^{0}Ψ_{η}(Ẑ_{st}, Λ)_{|Ω_{𝔽q}^{d−1}}

is a locally constant scheaf in free Λ-modules of rank q^{d} − 1.

3called “vanishing cycles” in loc.cit

Proof. In section 4.2 of [11], it is recalled how X̂ can be naturally identified to the completion Ŝh\overline{s} of some Shimura variety Sh defined over O at some “supersingular” geometric point \overline{s} of its special fiber. Then Yoshida exhibits a sequence of blow up’s resulting in a (generalized) semi-stable model Sh_{st} ⟶^{f} Sh of the Shimura variety over O. The relation with the local construction above is that Ẑ_{st} identifies with the formal completion Ŝh_{st,f^{−1}(\overline{s})} of Sh_{st} at the closed subscheme f^{−1}(\overline{s}) of its special fiber. In particular, f^{−1}(\overline{s}) ≃ Y_{ℙ,st}. Therefore, the claimed isomorphism follows from the second corollary in (2.3), and the remaining statements follow from [9, Prop. 6].

Remark.– In [11, Prop. 6.14 i)], Yoshida uses a result of Saito’s to get an analogue of the above statement in the Grothendieck group of ΛGL_{d}(𝔽_{q})-modules (and involving compactly supported cohomology on the open stratum).

(3.2) Now the task of describing the sheaf R^{0}Ψ_{η}(Ẑ_{st}, Λ)_{|Y_{ℙ,st}^{0}} was already achieved

by Yoshida. Recall that Ω_{𝔽_{q}}^{d−1} is naturally equivariantly isomorphic to the Deligne-Lusztig base variety X (w) described in the introduction. Recall also the covering map π : X̃(w) ⟶ X (w).

Fact. (Yoshida)– There is a GL_{d}(𝔽_{q})-equivariant isomorphism

R^{0}Ψ_{η}(Ẑ_{st}, Λ)_{|Ω_{𝔽q}^{d−1}} ≃ π_{∗}(Λ).

Proof. We summarize Yoshida’s argument for the convenience of the reader. The scheme Z_{st} is obtained by a sequence of blow up’s. The first stage simply consists in blowing up the closed point of X, see [11, 3.4]. Denote by Z_{1} ⟶^{f_{1}} X this blow up. Its closed fiber, denoted by Y_{ℙ}, is a projective space ℙ_{\overline{𝔽}_{q}}^{d−1}. It is also the unique proper irreducible

component of Z_{1,s}, has multiplicity q^{d} − 1 and meets the other irreducible components at 𝔽_{q}-rational hyperplanes. Then Z_{st} is obtained by blowing up Z_{1} at all rational subspaces of Y_{ℙ}. We thus have a factorization f : Z_{st} ⟶^{f^{1}} Z_{1} ⟶^{f_{1}} X with f^{1} a proper morphism which is an isomorphism above the tube of Ω_{𝔽_{q}}^{d−1}. Therefore f_{s,∗}^{1} induces an isomorphism

RΨ_{η}(Ẑ_{st}, Λ)_{|Ω_{𝔽q}^{d−1}} ⟶^{∼} RΨ_{η}(Ẑ_{1}, Λ)_{|Ω_{𝔽q}^{d−1}}

with Ẑ_{1} denoting the completion of Z_{1} along its closed special fiber. Now, recall the extension Ŏ_{d} := Ŏ[ϖ^{1/(q^{d}−1)}]. If one forgets about the action of inertia, we have by definition an equality

RΨ_{η}(Ẑ_{1}, Λ) = RΨ_{η}(Ẑ_{1} ×Ŏ Ŏ_{d}, Λ).

Consider now the normalization morphism Z_{d} ⟶^{f} Z_{1} ×Ŏ Ŏ_{d}. Since normalization is a local process which takes the algebra Ŏ_{d}[X]/(X^{q^{d}−1} − ϖ) to Ŏ_{d}^{q^{d}−1}, the unique proper irreducible component of Z_{d} has now multiplicity one, and the restriction of f to the inverse image U_{d} := f^{−1}(Ω_{\overline{𝔽}_{q}}^{d−1}) of Ω_{\overline{𝔽}_{q}}^{d−1} is étale of degree q^{d} − 1. Since f is a finite morphism, we thus get

R^{0}Ψ_{η}(Ẑ_{1}, Λ)_{|Ω_{𝔽q}^{d−1}} ≃ f_{∗}(Λ_{U_{d}}).

The computation of the normalization U_{d} together with its action by GL_{d}(𝔽_{q}) is done in Section 5 of [11]. According to Prop 6.15 of loc. cit., the covering f : U_{d} ⟶ Ω_{𝔽_{q}}^{d−1} is equivariantly isomorphic to the Deligne-Lusztig covering π.

Together with the previous Corollary, this proves that the morphism (1) of the introduction is an isomorphism. Since X (w) is a smooth variety, and M_{1} is a smooth analytic space, we may apply Poincaré duality to get the isomorphism for compactly supported cohomology, as claimed in the main Theorem.

Remark.– The tube of Ω_{𝔽_{q}}^{d−1} in M_{1} is an affinoid subset of this non quasi-compact analytic space, which was explicitly described by J. Weinstein in the case K has characteristic p. The first Corollary in (2.3) shows that it captures all the cohomology of M_{1}, in the sense that the restriction morphism

RΓ (M_{1}^{ac}, Λ) ⟶ RΓ (sp^{−1}(Ω_{𝔽_{q}}^{d−1})^{ac}, Λ)

is an isomorphism.

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