WEIGHT FILTRATION AND MIXED TATE MOTIVES

BRUNO KAHN

This talk reported on a slight improvement of a result of Marc Levine on the existence of a weight filtration on a suitable category of mixed Tate motives [10], by Annette Huber and the author. The reader will realise that this text is little more than a survey summarising the main results of [10], that he is warmly encouraged to consult for details and proofs.

According to [10], a Q-linear tensor triangulated category T is of Tate type if it is generated by objects Q(n), n ∈ Z, together with isomorphisms

Q(n) ⊗ Q(m) → Q(n + m)

satisfying the usual associativity and commutativity constraints, and such that

Hom_{T} (Q(n)[a], Q(m)[b]) = \begin{cases}0 & if n > m \\ 0 & if n = m and a ≠ b \\ QId & if n = m and a = b. \end{cases}

For −∞ ≤ a ≤ b ≤ +∞, let T_{[a,b]} denote the strictly full triangulated subcategory of T generated by the Q(n) for a ≤ −2n ≤ b; denote T_{[a,a]} simply by T_{a}. Then T_{a} = 0 for a odd, while for a even T_{a} is equivalent to the derived category of the category of finite dimensional Q-vector spaces.

In [10], Levine constructs exact functors

W_{≤b} : T → T_{[−∞,b]}
W^{>b} : T → T_{[b+1,+∞]}

which are respectively left and right adjoint to the corresponding inclusions of categories^{1} and such that, for any X ∈ T, there is a functorial exact triangle

W_{≤b}X → X → W^{>b}X → W_{≤b}X[1].

(This comes from the fact that T_{[−∞,0]} and T_{[1,+∞]} define a t-structure on T.) Set gr_{a}^{W} X = W^{>a−1}W_{≤a}X: This defines an exact functor T → T_{a}. The collection of these data is the weight filtration on T.

There are at least three examples of Q-linear tensor triangulated categories of Tate type in the literature, relative to a fixed base field k: those obtained by taking the strictly full subcategory generated by Tate objects in the categories of motives (with rational coefficients) defined respectively by Hanamura [5], Levine [11] and Voevodsky [12]. The above construction applies to each of them.

1This is not explicitly stated in [10].

Our contribution is to refine the definition of a weight filtration to the strictly full subcategory T DM_{gm}(k) of Voevodsky’s category DM_{gm}(k) [12] generated by the Tate objects^{2}. Since Voevodsky’s motives are with integral coefficients, this provides an integral version of Levine’s weight filtration. However, Levine’s methods are sufficient to provide such a weight filtration integrally, with a suitable modification of his axioms. Our real contribution is therefore rather to realise this weight filtration with an explicit formula, which may be useful in computations.

Recall that the subcategory DM_{gm}^{eff}(k) ⊂ DM_{gm}(k) of effective geometrical motives is fully embedded into a sheaf-theoretic category DM_{−}^{eff}(k). There is a partially defined internal Hom:

\underline{Hom}_{eff} : DM_{gm}^{eff}(k) × DM_{−}^{eff}(k) → DM_{−}^{eff}(k)

which is a partial right adjoint of the tensor product on DM_{−}^{eff}(k). It is a very interesting open question whether \underline{Hom}_{eff}(X, Y) ∈ DM_{gm}^{eff}(k) for all X, Y ∈ DM_{gm}^{eff}(k); this is obvious if X, Y ∈ T DM_{gm}^{eff}(k), the effective part of T DM_{gm}(k), and then \underline{Hom}_{eff}(X, Y) ∈ T DM_{gm}^{eff}(k). Hence the category T DM_{gm}^{eff}(k) enjoys an everywhere-defined internal Hom.

Our formula for W_{≤2n}X (X ∈ T DM_{gm}^{eff}(k), n ≥ 0) is then

W_{≤2n}X = \underline{Hom}_{eff}(Z(n), X)(n).

This extends to X ∈ T DM_{gm}(k) and n ∈ Z without difficulty, using the fact that the functor T DM_{gm}^{eff}(k) → T DM_{gm}(k) is a full embedding. The corresponding result for DM would require resolution of singularities for the ground field k, but in the case of mixed Tate motives it is not necessary [13].

For any n ∈ Z, the functor

D_{c}^{b}(Ab) → T DM_{gm}(k)_{2n}
C ↦ C ⊗ Z(n)[2n]

is an equivalence of triangulated categories, where D_{c}^{b}(Ab) is the bounded derived category of finitely generated Z-modules. In this way, one may translate the functors gr_{W}^{2n} into a collection of triangulated functors

(c_{n}) : T DM_{gm}(k) → D_{f}^{b}(gr Ab)

where gr Ab denotes the category of Z-graded abelian groups. This total functor is a tensor functor and is conservative (but not an equivalence of categories!) This formalism also yields spectral sequences

E_{2}^{p,q}(M, n) = H^{p−q}(k, c_{q}(M)^{∗} ⊗ Z(n − q)) ⇒ H^{p+q}(M, Z(n))

2Note that Voevodsky’s category is homological rather than cohomological: the functor associating its motive to a variety is covariant rather than contravariant. For this reason, the Tate object which would be denoted by Z(−n) in Levine’s setting is denoted here by Z(n).

for M ∈ T DM_{gm}(k). Here c_{q}(M)^{∗} denotes the dual of c_{q}(M) in D_{f}^{b}(Ab) and H^{p+q}(M, Z(n)) := Hom_{DM_{gm}(k)}(M, Z(n)[p + q]). This is one of the weight spectral sequences for motivic cohomology for mixed Tate motives. In [9], we give variants of these spectral sequences for étale motivic cohomology of geometrically cellular varieties over a field of characteristic 0: they turn out to be very useful for the computation of the unramified cohomology of projective homogeneous varieties.

A t-structure on a tensor triangulated category of Tate type T is motivic if it is nondegenerate and if

X ∈ A ⟺ X ⊗ Q(1) ∈ A

where A is its heart. Levine shows that there is a (unique) motivic tstructure on T provided the following condition holds:

(1)  Hom_{T} (Q(n), Q(m)[a]) = 0 for n < m and a ≤ 0.

The heart A of this t-structure inherits a weight filtration and admits a perfect duality which is compatible with this weight filtration; in particular, it is a Tannakian category.

In the case where T is a category of (triangulated) mixed Tate motives defined from [5], [11] or [12], there are isomorphisms

Hom_{T} (Q(n), Q(m)[a]) ≃ K_{2(m−n)−a}(k)^{(m−n)}

for m ≥ n, where K_{i}(k) is the i-th algebraic K-group of k and K_{i}(k)^{(r)} is the weight r part of K_{i}(k) ⊗ Q for the action of the Adams operations. (In fact, in the cases of [10] and [12], one even has Hom_{T} (Z(n), Z(m)[a]) ≃ CH^{m−n}(k, 2(m − n) − a) where CH^{p}(k, q) denotes Bloch’s q-th p-codi-mensional higher Chow group of k [2], and the latter is known to coincide with the weight m − n part of K-theory after tensoring with Q.) In this case, condition (1) is the Beilinson-Soulé conjecture.

The Beilinson-Soulé conjecture is known for global fields, in particular for number fields. Hence Levine’s theorem applies in this case. Moreover, Levine then constructs isomorphisms

Ext_{A}^{p}(M, N) ⟶^{∼} Hom_{T} (M, N[p])

for M, N ∈ A.

One knows a lot more on the groups of (1) however, thanks to Borel’s computation of the algebraic K-theory of number fields [3]: the group Ext_{A}^{p}(Q(n), Q(m)) is 0 for p > 1, and the dimension of Ext_{A}^{1}(Q(n), Q(m)) is

• 0 for m = n
• r_{1} + r_{2} for m − n odd > 1
• r_{2} for m − n even > 0

where r_{1} and r_{2} are respectively the number of real and complex embeddings of k. Unfortunately, for m − n = 1, we have Ext_{A}^{1}(Q(n), Q(m)) ≃ k^{∗} ⊗ Q, which is infinite-dimensional. To have a workable theory of extensions of mixed Tate motives, one should have a definition of them over a ring of S-integers – something which is lacking at the present stage.

Finally, let us mention that there are now realisation functors from the various triangulated categories of motives to suitable categories, via cohomological functors. In the case of Voevodsky’s category, see [6] for the construction of a tensor realisation functor from DM_{gm}(k) to Huber’s category of mixed realisations, when k is of characteristic 0.

For the relationships between polylogarithms and extensions of Tate motives over Q, see [4], [1], [8], [11, pp. 303–310] and [7, Appendix].

References

[1] A. Beilinson, P. Deligne Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs, in Motives (Seattle, WA, 1991), 97–121, Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994.

[2] S. Bloch Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986), 267–304.

[3] A. Borel Stable real cohomology of arithmetic groups, Ann. Sci. Éc. norm. sup. 7 (1974), 235–272.

[4] P. Deligne Le groupe fondamental de la droite projective moins trois points, in Galois groups over Q (Berkeley, CA, 1987), 79–297, Math. Sci. Res. Inst. Publ., 16, Springer, New York-Berlin, 1989.

[5] M. Hanamura Mixed motives and algebraic cycles, I, Math. Res. Lett. 2 (1995), 811– 821.

[6] A. Huber Realisation of Voevodsky’s motives, J. Alg. Geometry 9 (2000), 755–799.

[7] A. Huber, G. Kings Degeneration of l-adic Eisenstein classes and of the elliptic polylog, Invent. Math. 135 (1999), no. 3, 545–594.

[8] A. Huber, J. Wildeshaus Classical motivic polylogarithm according to Beilinson and Deligne, Doc. Math. 3 (1998), 27–133. Correction in Doc. Math. 3 (1998), 297–299

[9] B. Kahn Motivic cohomology of smooth geometrically cellular varieties, Proc. Symp. Pure Math. 67 (1999), AMS, 149–174.

[10] M. Levine Tate motives and the vanishing conjectures for algebraic K-theory, Algebraic K-theory and algebraic topology (Lake Louise, AB, 1991), 167–188, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 407, Kluwer Acad. Publ., Dordrecht, 1993.

[11] M. Levine Mixed motives, Mathematical Surveys and Monographs 57, AMS, Providence, 1998.

[12] V. Voevodsky Triangulated categories of motives over a field, in Cycles, transfers, and motivic homology theories. Annals of Math. Studies, 143, Princeton University Press, Princeton, 2000, 188–238.

[13] V. Voevodsky Motivic cohomology [groups] are isomorphic to higher Chow groups, preprint, 1999.

Institut de Mathématiques de Jussieu, 175–179 rue du Chevaleret, 75013 Paris, France

E-mail address: kahn@math.jussieu.fr