Voevodsky’s Derived Category of Motives

Tamás Szamuely

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1. Introduction

Let k be a field. A previous lecture discussed the derived category DM_{gm}(k) of geometrical motives over k which is a natural extension of Grothendieck’s category of pure motives, comprising not only motives of smooth projective k-varieties but motives of arbitrary smooth k-schemes as well. Though its construction is close to geometric intuition, it seems to be a difficult task to check whether it satisfies the properties which the conjectural derived category of mixed motives is expected to enjoy (eg. localisation and Gysin sequences, blowup formula, duality results), whereas such properties are available when one is working in some (hypothetical) realisation such as étale cohomology. One of the main innovations of Voevodsky was the construction of a full embedding of the category DM_{gm}^{eff}(k) of effective geometrical motives into a larger category DM^{eff,−}(k) called the triangulated category of effective mixed motives which is, so to speak, “halfway” between geometrical motives and realisations: it is already a subcategory of the derived category of sheaves on some site which enables one to use sheaf-theoretic techniques for its study; its deeper properties, however, still repose on the geometry of algebraic cycles, in accordance with the original intuition of Weil and Grothendieck.

In this lecture we discuss how one constructs DM^{eff,−}(k) and how one embeds DM_{gm}^{eff}(k) into it, based on [8] and the first two sections of [6]. We shall also describe the tensor structure on these categories but the deeper properties mentioned above will be treated in later lectures.

A notational convention: since we shall exclusively deal with effective motives in this lecture, in order to make life easier we shall drop the superscript ”eff” and write DM_{gm}(k) for what was denoted by DM_{gm}^{eff}(k) earlier and similarly we shall simply denote by DM^{−}(k) the category of effective mixed motives we are to construct. In a later lecture non-effective mixed motives will also make their appearance but their construction will require the duality theorems yet to come.

2. The Category DM^{−}(k)

Let us assume from now on that k is perfect and all k-schemes under consideration are separated. Recall from the previous lectures that given two smooth k-schemes X, Y one defines the group Cor(X, Y) of smooth correspondences from X to Y as the free abelian group generated by closed integral subschemes Z ⊂ X × Y such that the restriction of the first projection X × Y → X to Z is finite and surjective over a component of X. As in the classical case, the intersection product induces a composition rule for correspondences α ∈ Cor(X, Y) and β ∈ Cor(Y, Z) by the formula p_{13∗}(p_{12}^{∗}α · p_{23}^{∗}β), where the p_{ij} are the projection maps from X × Y × Z. (Note that the pushforward cycle by p_{13} exists since the restriction of p_{13} to V × Z is proper for any component V of the support of α, being obtained by base change from the finite map V → X.) One thus obtains a category SmCor(k) whose objects are smooth k-schemes and whose morphisms are smooth correspondences; this is an additive category. An (abelian) presheaf with transfers is a contravariant additive functor from SmCor(k) into the category of abelian groups. The category of presheaves with transfers is abelian.

Since the graph of any morphism of schemes is a smooth correspondence, a presheaf with transfers can be viewed as an abelian presheaf on the category Sm(k) of smooth k-schemes with some additional structure. If this presheaf is actually a sheaf for some Grothendieck topology on Sm(k), we speak of a sheaf with transfers for this topology. In the sequel we shall mostly consider Nisnevich sheaves with transfers.

Lemma 2.1 The Nisnevich sheaf associated to any presheaf with transfers carries a canonical structure of a presheaf with transfers.

The proof of this lemma is not obvious. We refer the reader to [8], Theorem 3.1.4.

Corollary 2.2 The category of Nisnevich sheaves with transfers is a full abelian subcategory.

Proof: Given a morphism between Nisnevich sheaves with transfers, the sheaf kernel is the same as the presheaf kernel and hence is a presheaf with transfers but to treat the sheaf cokernel one applies the previous lemma.

A presheaf F on Sm(k) is called homotopy invariant if for all smooth kschemes X the natural map F(X) → F(X × A^{1}) is an isomorphism (where A^{1} denotes the affine line over k). We have now similar statements as for the transfer structure (cf. [9], Theorem 5.6):

Lemma 2.3 Given a homotopy invariant presheaf with transfers, the associated Nisnevich sheaf is again homotopy invariant. Thus homotopy invariant Nisnevich sheaves with transfers form an abelian subcategory of the category of Nisnevich sheaves with transfers.

Examples of homotopy invariant presheaves are furnished by étale cohomology groups with coefficients G_{m}. For us, however, the most important ones will be those arising from the following simplicial construction.

Recall that for each n ≥ 0 the standard n-simplex Δ^{n} is the closed subscheme of A^{n+1} = Spec k[t_{0}, …, t_{n}] defined by the equation t_{0} + … + t_{n} = 1. Adding a new coordinate equal to 0 (resp. taking the sum of two successive coordinates) defines standard coface (resp. codegeneracy) maps which endow the system Δ^{•} of the Δ^{n} with the structure of a simplicial scheme. Given any abelian presheaf F on Sm(k), the rule C^{−n}F(X) = F(X × Δ^{n}) defines a simplicial presheaf on Sm(k) whose associated complex of presheaves (which is concentrated in nonpositive degrees but has cohomological indexing according to our convention) is denoted by C^{∗}(F). Now the basic fact is:

Lemma 2.4 ([5], Corollary 7.5) For any abelian presheaf F on Sm(k) the cohomology presheaves of the complex C^{∗}(F) are homotopy invariant. If F is a Nisnevich sheaf, the same holds for the cohomology sheaves of C^{∗}(F).

Proof: In view of Lemma 2.3 it suffices to prove the first statement. For this, one first shows that for any smooth k-scheme X the morphisms of complexes of abelian groups i_{0}^{∗}, i_{1}^{∗} : C^{∗}F(X × A^{1}) → C^{∗}F(X) induced by the maps i_{0}, i_{1} : X → X ×A^{1} sending X respectively to X ×{0} and X ×{1} are homotopic. Here the chain maps s_{n} : F(X × A^{1} × Δ^{n}) → F(X × Δ^{n+1}) giving the homotopy are defined by a classical simplicial trick: if ψ_{i} : Δ^{n+1} → Δ^{n} × A^{1} is the map which takes the value v_{j} × {0} on the j-th vertex v_{j} of Δ^{n+1} for j ≤ i and v_{j−1} × {1} otherwise, then s_{n} is defined by the formula

s_{n} = ∑_{i=0}^{n}(id_{X} × ψ_{i})^{∗}

where the ^{∗} indicates the corresponding restriction map of the sheaf F as above.

So in particular we get that for any smooth k-scheme X the maps i_{0}^{∗} and i_{1}^{∗} induce the same maps H^{i}C^{∗}(F)(X × A^{1}) → H^{i}C^{∗}(F)(X) on cohomology. Apply this with X = Y × A^{1} where Y is another smooth k-scheme and consider the map λ : Y × A^{1} × A^{1} → Y × A^{1} given by the identity in the first variable and by the multiplication map A^{1}×A^{1} → A^{1} in the second two. It follows that λ ∘ i_{0} and λ ∘ i_{1} induce the same maps H^{i}C^{∗}(F)(Y × A^{1}) → H^{i}C^{∗}(F)(Y × A^{1}). But the second of these maps is the identity and the first factors through H^{i}C^{∗}(F)(Y × {0}) ≅ H^{i}C^{∗}(F)(Y), whence the lemma.

One can extend the above construction to a bounded above complex F^{•} of presheaves by taking the total complex C^{∗}(F^{•}) (with the direct sum construction) of the double complex whose columns are the C^{∗}(F^{i}). The first spectral sequence of the double complex (plus Lemma 2.3 then implies:

Corollary 2.5 For the above complex F^{•} the cohomology presheaves of the complex C^{∗}(F^{•}) are homotopy invariant. If F^{•} is a complex of Nisnevich sheaves, the same holds for the cohomology sheaves of C^{∗}(F^{•}).

Now comes the main definition:

Definition 2.6 Denote by DNT^{−}(k) the derived category of bounded above complexes of Nisnevich sheaves with transfers. The category DM^{−}(k) is defined to be the full subcategory of DNT^{−}(k) spanned by objects with homotopy invariant cohomology sheaves.

Lemma 2.3 implies that DM^{−}(k) is in fact a triangulated subcategory of DNT^{−}(k). Here the latter category carries its natural triangulation as the derived category of an abelian category (cf. Corollary 2.2).

3. On the Cone of the Map F^{•} → C^{∗}(F^{•})

In this section we prove a key technical statement which lies at the heart of the proofs to follow. Since it is not particularly endearing in itself, the reader may choose to take it for granted for a while and return to it later if she wishes so.

Notice that for any presheaf with transfers there is a natural map of complexes F → C^{∗}(F) induced by the identity map of F in degree 0. This obviously extends to complexes of presheaves with transfers giving maps of complexes F^{•} → C^{∗}(F^{•}) for each complex F^{•}. Denote by C^{•} the cone of this map.

Assume now further that F^{•} is a complex of Nisnevich sheaves cohomologically bounded from above. We may then consider C^{•} as an object of the derived category DNT^{−}(k).

Proposition 3.1 Let C^{•} be the complex defined above. Then for any bounded above complex of Nisnevich sheaves with transfers G^{•} with homotopy invariant cohomology sheaves we have Hom_{DNT^{−}(k)}(C^{•}, G^{•}) = 0.

Note the corollary:

Corollary 3.2 If F has homotopy invariant cohomology sheaves, then the map F^{•} → C^{∗}(F^{•}) is a quasi-isomorphism.

Indeed, in this case C^{•} is an object of DM^{−}(k), hence by the proposition the identity map C^{•} → C^{•} is 0.

We begin by sorting out the property of C^{•} which will be exploited during the proof. Following [6], let us call a presheaf with transfers F contractible if there is a morphism of presheaves ϕ : F → C^{−1}(F) (i.e. a collection of maps ϕ_{X} : F(X) → F(X × A^{1}) for each smooth k-scheme X, contravariant in X) such that for each section s ∈ F(X) the image of ϕ_{X}(s) by the natural restriction map F(X × A^{1}) → F(X × {0}) ≅ F(X) is 0 but the image by F(X × A^{1}) → F(X × {1}) ≅ F(X) is s. Now as promised we have:

Lemma 3.3 The complex C^{•} is isomorphic in DNT^{−}(k) to a complex with contractible terms.

Proof: Each term F^{i} has an acyclic resolution of the form

F^{i•} : … ⟶ F^{i} ⟶^{id} F^{i} ⟶^{0} F^{i} ⟶^{id} F^{i} ⟶^{0} F^{i}.

This is none but the chain complex associated to the constant simplicial sheaf given by to F^{i} and hence there is a natural map of complexes F^{i•} → C^{∗}(F^{i}) induced by the projections X × Δ^{n} → X. Denote by F_{c}^{•} the total complex associated to the bicomplex whose columns are the F^{i•}. By construction, the map F^{•} → C^{∗}(F^{•}) considered above factors as F^{•} → F_{c}^{•} → C^{∗}(F^{•}) where the first map is a quasi-isomorphism. Since we are interested in C^{•} only as an object of DNT^{−}(k) we may replace it by the cokernel of the second map.

Then we can reason for each term separately so we may assume we are dealing with the cokernel of a morphism of sheaves F → C^{−n}(F). For this proof consider Δ^{n} embedded into A^{n+1} as a hyperplane passing through the origin O. Then for any X the restriction F(X) → F(X × Δ^{n}) is split by the map λ_{X}^{n} : F(X × Δ^{n}) → F(X × {O}) ≅ F(X) so it is enough to show this map has contractible kernel. Indeed, the map X ×A^{1} ×Δ^{n} → X ×Δ^{n} which is the identity in the first variable and the multiplication of coordinates in the second induces a restriction ϕ_{X} : F(X × Δ^{n}) → F(X × A^{1} × Δ^{n}) with ker(λ_{X×A^{1}}^{n} ∘ ϕ_{X}) ⊃ ker λ_{X}^{n}. The restriction of the ϕ_{X} to ker λ_{X}^{n} satisfy the requirement in the definition of contractibility.

Remark 3.4 Notice that in the above proof we did not use the fact that the terms of the complex were Nisnevich sheaves. Thus it also holds in the derived category of bounded above complexes of presheaves with transfers.

In view of the previous lemma it will suffice to show:

Proposition 3.5 ([6], Corollary 1.11.1) Let F^{•}, G^{•} be bounded above complexes of Nisnevich sheaves with transfers such that the terms of F^{•} are contranctible and the cohomology sheaves of G^{•} are homotopy invariant. Then we have Hom_{DNT^{−}(k)}(F^{•}, G^{•}) = 0.

Proof: It is enough to treat the case where F^{•} and G^{•} have a single nonzero term F and G, respectively; the general case then follows by a descending double induction using the stupid filtration of the complex in the first variable and the canonical one for that in the second. Since the category of Nisnevich sheaves with transfers has enough injectives (proof standard), we may choose an injective resolution I^{•} of G and thus it is enough to show that the complex Hom(F, I^{•}) is nullhomotopic. Consider the complex C^{−1}(I^{•}). According to Voevodsky ([8], Proposition 3.1.8, which is not deep), the nth cohomology presheaf of C^{−1}(I^{•}) is just the usual cohomology presheaf X ↦ H_{Nis}^{n}(X × A^{1}, G). The group on the right equals H_{Nis}^{n}(X, G) by another result of Voevodsky (deep this time, see [9], Theorem 5.6) and hence vanishes Nisnevich locally for n > 0. Thus C^{−1}(I^{•}) is an acyclic resolution of C^{−1}(G). The maps i_{0}, i_{1} introduced in the proof of Lemma 2.4 induce morphisms of complexes i_{0}^{∗}, i_{1}^{∗} : C^{−1}(I^{•}) → I^{•} which both lift the inverse of the natural isomorphism G → C^{−1}(G) (existing by the homotopy invariance of F), and hence by a basic lemma of homological algebra they are chain homotopic by a chain map s (of degree −1). Now is the moment where contractibility of F comes in: we may use the map ϕ : F → C^{−1}(F) to define a contracting homotopy for the identity map of Hom(F, I^{•}). Indeed, given a map f ∈ Hom(F, I^{n}), define u(f) ∈ Hom(F, I^{n−1}) as the composite sC^{−1}(f)ϕ. Now since s is a contracting homotopy for i_{1}^{∗} − i_{0}^{∗}, we have ud(f) + du(f) = (i_{1}^{∗} − i_{0}^{∗})C^{−1}(f)ϕ = f(i_{1}^{∗} − i_{0}^{∗})ϕ. But by contractibility of ϕ the map (i_{1}^{∗} − i_{0}^{∗})ϕ is the identity and we are done.

4. A Second Construction of DM^{−}(k)

Now we can describe a second construction of DM^{−}(k) which will be more convenient when comparing DM^{−}(k) with DM_{gm}(k). It is based on:

Proposition 4.1 The functor C^{∗} : DNT^{−}(k) → DM^{−}(k) is left adjoint to the natural embedding DM^{−}(k) → DNT^{−}(k).

Proof: For any bounded above complex of Nisnevich sheaves with transfers F^{•} the natural map of complexes ϕ : F^{•} → C^{∗}(F^{•}) induces a map

Hom_{DNT^{−}(k)}(C^{∗}(F^{•}), G^{•}) → Hom_{DNT^{−}(k)}(F^{•}, G^{•})

where by virtue of Corollary 2.5, the Hom-group on the left is actually a Hom-group in DM^{−}(k). Now it suffices to note that for the cone C^{•} of ϕ we have the annulations Hom_{DNT^{−}(k)}(C^{•}, G^{•}) = Hom_{DNT^{−}(k)}(C^{•}[−1], G^{•}) = 0 according to Proposition 3.1.

Corollary 4.2 The functor C^{∗} : DNT^{−}(k) → DM^{−}(k) is an exact functor of triangulated categories.

Proof: In fact, C^{∗} is exact on the category of (bounded above complexes of) presheaves with transfers, hence it is left exact on the category of bounded above complexes of Nisnevich sheaves with transfers. The above adjunction statement implies that after passing to the corresponding derived category the functor becomes right exact as well.

Now we may embark on the promised second construction of DM^{−}(k). We begin by the observation that although Corollary 3.2 does not necessarily hold for arbitrary F^{•}, by applying it to the map C^{∗}(F^{•}) → C^{∗}(C^{∗}(C^{•})) we get the acyclicity of C^{∗}(C^{•}) by exactness of C^{∗}. This suggests that if we localise DNT^{−}(k) so that the complexes C^{∗}(C^{•}) become isomorphic to zero, the resulting category must be close to DM^{−}(k). Now recall the following formal statement encountered previously:

Lemma 4.3 Let C be a triangulated category and D a full triangulated subcategory such that the natural inclusion functor ι : D → C admits an exact left adjoint F.

Denote by A the full subcategory of C spanned by objects A ∈ C with F (A) ≅ 0. Then the system of morphisms X → Y which can be inserted into a distinguished triangle X → Y → Z → X[1] with Z in A is multiplicative and cofinally small. Moreover, F extends to the localisation C/A and induces an equivalence of triangulated categories between C/A and D.

Proof: Even without supposing F exact, we see that the assumptions of [11], Proposition 16 are satisfied which implies that the localisation exists and we get an equivalence of abelian categories. Now the exactness of F implies that the objects of A form a null system, so by Proposition 35 of [11] C/A carries a canonical triangulated structure compatible with that of D.

Now take A to be the full subcategory A of DNT^{−}(k) spanned by objects coming from complexes F^{•} with C^{∗}(F^{•}) acyclic. Combining the lemma with the preceding proposition and corollary finally gives:

Theorem 4.4 The functor C^{∗} extends to the localisation DNT^{−}(k)/A and induces an equivalence of categories between DNT^{−}(k)/A and DM^{−}(k).

In the sequel, we shall need a bit stronger property of the subcategory A. To formulate it precisely, recall from page 266 of [3] that a full triangulated subcategory A of a triangulated category C is called thick if for any two objects X, Y of C the existence of a morphism f : X → Y which can be factored as a composite X → W → Y with W an object of A and which inserts into a distinguished triangle X → Y → Z → X[1] with Z an object of A implies that X, Y are objects of A as well. As stated in loc. cit., this notion agrees with what is called a saturated subcategory in Section 2.1 of [10], hence it is closed under taking direct summands of objects. Another important property of thick subcategories that we shall use later is that they are closed under taking direct sums (even infinite sums if they exist in C). Now we leave the readers the pleasure to check:

Lemma 4.5 The subcategory A defined in Lemma 4.3 is thick.

Of course, this applies in particular to the subcategory A defined before the theorem.

5. From DM_{gm}(k) to DM^{−}(k)

In the remaining sections we discuss Theorem 3.2.6 of [8] which may be considered as the heart of the paper. As a first step we construct a functor from the triangulated category DM_{gm}(k) of geometrical motives into DM^{−}(k).

For this we first recall the construction of DM_{gm}(k). We start from the category SmCor(k) which is additive and denote by H^{b}(SmCor(k)) the associated homotopy category of complexes. Now take the localisation of H^{b}(SmCor(k)) with respect to the minimal thick subcategory containing the complexes of the form

[X × A^{1}] → [X]  (1)

and the Mayer-Vietoris sequences

[U ∩ V] → [U] ⊕ [V] → [X]  (2)

associated to any covering of X by two open subsets U and V, the first map being induced by the natural inclusions and the second by the difference of the inclusion maps. Finally one gets DM_{gm}(k) by taking the pseudo-abelian envelope of the localised category (i.e. by formally adjoining kernels and cokernels of projectors). There is a natural inclusion functor H^{b}(SmCor(k)) → DM_{gm}(k).

On the other hand, there is a natural functor L : H^{b}(SmCor(k)) → DNT^{−}(k) constructed as follows. First for any smooth k-scheme X one defines L(X) as the presheaf with transfers represented by X, i.e. as the contravariant functor Y → Cor(Y, X). The rule X → L(X) defines a covariant functor L from SmCor(k) to the category of presheaves with transfers which is defined on morphisms by composition of correspondences. As we have seen in the previous lecture, the presheaves L(X) are sheaves for the Nisnevich topology, so L in fact takes its values in the category of Nisnevich sheaves with transfers. As such, it extends canonically to a functor from H^{b}(SmCor(k)) to the homotopy category of bounded above Nisnevich sheaves with transfers and thus finally to DNT^{−}(k).

Proposition 5.1 There exists a functor i : DM_{gm}(k) → DM^{−}(k) making the diagram

H^{b}(SmCor(k)) ⟶^{L} DNT^{−}(k)
↓  ↓^{C^{∗}}
DM_{gm}(k) ⟶^{i} DM^{−}(k)

commute.

Proof: In view of the easy fact that DM^{−}(k) is pseudo-abelian, for the existence of i it is enough to show that the functor L maps the complexes (1) and (2) into the thick subcategory A defined at the end of the last section, i.e. that of complexes with acyclic C^{∗}. For the complexes (1) this is clear from Lemma 2.4 and for the complexes (2) it is enough to show by left exactness of L and exactness of C^{∗} (cf. Corollary 4.2) that the Nisnevich sheaf associated to the presheaf L(X)/(L(U) + L(V)) is 0. Since the functor L is additive and the local rings for the Nisnevich topology are the henselian rings, this follows from the more general lemma below applied with Y = U ∐ V.

Lemma 5.2 Assume given a smooth k-scheme X and a Nisnevich covering Y → X. Then the natural map Cor(S, Y) → Cor(S, X) is surjective for any henselian local scheme S.

Proof: Take first an element Z ∈ Cor(S, X) which is the graph of a morphism ϕ : S → X. It suffices to lift ϕ to a map S → Y for we may then take its graph. Now the pullback Y_{S} of Y → X by ϕ is a Nisnevich covering of the henselian scheme S and thus has a section s : S → Y_{S}. The composite S →^{s} Y_{S} → Y gives then a desired lifting. For an arbitrary generator Z of Cor(S, X) note first that since it is integral and finite over the henselian local scheme S, it is itself a henselian local scheme by [4], I.4.2, so we may repeat the above argument with Z in place of S and the natural projection Z → X in place of ϕ. We obtain an element of Cor(Z, Y) which by composition with the graph of the finite projection Z → S (viewed as an element of Cor(S, Z)) yields a lifting of Z.

6. The Embedding Theorem

The whole of this section is devoted to the proof of:

Theorem 6.1 The functor i : DM_{gm}(k) → DM^{−}(k) of Proposition 5.1 is a full embedding.

We begin the proof by some dévissage. First observe that by the Yoneda Lemma the functor X → L(X) is fully faithful on SmCor(k), hence extends to a fully faithful functor L from H^{b}(SmCor(k)) to the homotopy category HP T^{−}(k) of bounded above complexes of presheaves with transfers. Since for all X the presheaves L(X) are projective objects in the category of presheaves with transfers, by a basic lemma of homological algebra ([12], Corollary 10.4.7) the composition of L with the natural inclusion of HP T^{−}(k) into the associated derived category DP T^{−}(k) is still fully faithful.

Now notice that thanks to Lemma 2.1 the functor F^{•} → F_{Nis}^{•} of taking the associated Nisnevich sheaf induces an exact left adjoint to the natural inclusion DNT^{−}(k) → DP T^{−}(k) so we may apply Lemma 4.3 (and Lemma 4.5) to conclude that DNT^{−}(k) is equivalent to the localisation of DP T^{−}(k) with respect to the thick subcategory whose objects are complexes F^{•} with F_{Nis}^{•} acyclic. To sum up, we get that DM^{−}(k) is equivalent to the localisation of DP T^{−}(k) with respect to the thick subcategory B whose objects are complexes F^{•} with C^{∗}(F_{Nis}^{•}) acyclic. On the other hand, denote by B′ the minimal thick subcategory containing the images of (1) and (2) by L. By the proof of Proposition 5.1, we have an inclusion B′ ⊂ B. Thus to prove the theorem it will suffice to show:

Proposition 6.2 The above inclusion B′ ⊂ B is an equivalence of categories.

Before embarking the proof here are two auxiliary results.

Lemma 6.3 Any object of DP T^{−}(k) is isomorphic to a bounded above complex whose terms are are (in general infinite) direct sums of presheaves of the form L(X).

Proof: For any presheaf with transfers F we have a surjection

⨁_{X/k smooth} ⨁_{s∈F(X)} L(X) → F → 0

in the category of presheaves with transfers. Over a smooth k-scheme Y it is given in the component indexed by (X, s) by mapping an element ϕ ∈ Cor(Y, X) to the image of s by the restriction map F(ϕ). By repeated application of this fact we get a resolution; for complexes we take a resolution of each term and then take the total complex (with the direct sum convention, of course).

Lemma 6.4 For any bounded above complex of presheaves with transfers F^{•} the cone of the natural map F^{•} → C^{∗}(F^{•}) is in B′.

Proof: Remember that a thick subcategory is closed under taking direct sums of objects. Thus considering a complex as an infinite direct sum of complexes with a single nonzero term we may assume we are dealing with a single presheaf F. Now by Remark 3.4 the cone under question is contractible so it is enough to show that any contractible presheaf with transfers C is in B′. It follows immediately from the definition of contractibility that to give a section of C over a smooth k-scheme X is equivalent to giving a morphism of presheaves with transfers from the cokernel K(X) of the map L(X) ⟶^{L(i_{0})} L(X × A^{1}) to C, i_{0} being induced by the inclusion of 0 as usual. But K(X) is none but the kernel of the natural map L(X × A^{1}) → L(X) and hence is in B′ as a direct summand of an object in a thick subcategory; moreover it is contractible (same argument as in Lemma 3.3). Thus as in the previous lemma we get that C is the quotient of a direct sum of presheaves K(X). Since the category of contractible presheaves with transfers is obviously closed under taking direct sums and kernels of maps, we may continue the procedure to get a resolution. Viewing it as an infinite direct sum we get the result.

Proof of Proposition 6.2. It suffices to show that for any object A of DP T^{−}(k) and any object B of B we have Hom_{DP T^{−}(k)/B′}(A, B) = 0. We shall prove that more generally for any n ≥ 0 we have Hom_{DP T^{−}(k)/B′}(A, B[n]) = 0.

By lemma 6.3 and the compatibility of Hom with infinite direct sums we may assume A = L(X). Now consider the family of presheaves with transfers

H^{n} : X → Hom_{DP T^{−}(k)/B′}(L(X), B[n]).

Since B′ contains the complexes coming from the sequences (2), for each open covering of a smooth k-scheme X by Zariski open subsets U, V the groups H^{n}(X) are connected by Mayer-Vietoris long exact sequences. Furthermore the underlying topological space of X is Noetherian of finite Krull dimension, so an old theorem of Brown and Gersten ([1], Theorem 1’) applies to conclude that the presheaves H^{n} are zero provided the associated Zariski sheaves are. On the other hand, since B′ contains the L-image of the complexes (1) the H^{n} are homotopy invariant presheaves with transfers so by a theorem of Voevodsky ([9], Proposition 5.5) their associated Zariski and Nisnevich sheaves are isomorphic (as presheaves); therefore it is enough to show that the Nisnevich sheaves associated to the H^{n} are zero. This means that any morphism L(X) → B[n] in DP T^{−}(k)/B′ becomes trivial on some Nisnevich covering. To give such a morphism is to give a pair of morphisms L(X) → M, B[n] → M in DP T^{−}(k) such that the cone of the latter is in B′. Note that since B′ is included in B and C^{∗} ∘ ( )_{Nis} is an exact functor, M is also an object of B, i.e. C^{∗}(M_{Nis}) ≅ 0. To prove that the map L(X) → M (this time in DP T^{−}(k)) becomes trivial on some Nisnevich covering, it is enough to prove the same for the composite L(X) → M → C^{∗}(M) as the second map is an isomorphism in DP T^{−}(k)/B′ according to Lemma 6.4. This amounts to proving H^{0}(X, C^{∗}(M)_{Nis}) ≅ 0 which in turn is a consequence of C^{∗}(M_{Nis}) ≅ 0, a property we have just seen.

7. Tensor Structure

Recall that the category DM_{gm}(k) has a tensor structure coming from the direct product X ×_{k} Y of two smooth k-schemes X and Y; patient readers will verify that this operation equips DM_{gm}(k) with the structure of a tensor category in the sense of the first section of [2]. We now define a tensor product operation on the category DM^{−}(k).

Theorem 7.1 The category DM^{−}(k) can be equipped with the structure of a tensor category such that the functor i of Proposition 5.1 preserves tensor products.

Proof: First we define a “primitive” tensor product for two preshaves with transfers of the form L(X), L(Y): we set (surprisingly)

L(X) ⊗ L(Y) = L(X ×_{k} Y).

As in the classical case of modules over a ring, we can unambigously extend this tensor product to the derived category DP T^{−}(k) using the resolutions in Lemma 6.3. This product satisfies all properties of the usual tensor product operation; in particular, it commutes with direct sums. Now in order to descend this tensor structure to DM^{−}(k) it is enough by Proposition 6.2 and the paragraph preceding it to verify that for any objects A of DP T^{−}(k) and B of B′ their product A ⊗ B lies in B′. By dévissage it suffices to check this for A of the form L(X) and B the L-image of a sequence (1) or (2), but this is obvious by definition of the “primitive” tensor product. Again we omit checking that DM^{−}(k) becomes a tensor category with this operation. The second statement of the theorem, however, is true by construction.

Explicitly, the product of two objects A and B of DM^{−}(k) is given by

A ⊗ B = C^{∗}((L^{•} ⊗ M^{•})_{Nis})

where L^{•}, M^{•} are bounded above complexes of presheaves with transfers whose terms are sums of presheaves of the form L(X) and which are isomorphic in DP T^{−}(k) to A and B, respectively. Their tensor product is to be taken in the “primitive” sense.

Remark 7.2 It is our duty to warn the readers that the inclusion functor DM^{−}(k) → DP T^{−}(k) does not preserve tensor products in general.

We conclude by giving an application to motivic complexes. Recall that these are complexes Z(n) of Nisnevich sheaves with transfers defined as follows. Denoting by G_{m} the open subscheme A^{1} \ {0} of A^{1}, one defines L(G_{m}^{∧n}) to be the quotient presheaf of L(G_{m}^{×n}) by the subpresheaf generated by the images of the morphisms λ_{i} : L(G_{m}^{×(n−1)}) → L(G_{m}^{×n}) induced by mapping a point (x_{1}, …, x_{n−1}) to (x_{1}, …, 1, …, x_{n−1}) with 1 at the i-th place for i = 1, …, n. Actually, the quotient map L(G_{m}^{×n}) → L(G_{m}^{∧n}) admits a splitting (see [6], Section 0 for details), so L(G_{m}^{∧n}) is canonically a direct summand of L(G_{m}^{×n}) and as such a Nisnevich sheaf with transfers. One then defines Z(n) as the complex C^{∗}(L(G_{m}^{∧n}))[−n]. Since this is a bounded above complex of Nisnevich sheaves with transfers by construction, it defines an object of DM^{−}(k) taking Lemma 2.4 into account as usual.

Proposition 7.3 In the category DM^{−}(k) we have natural isomorphisms

Z(n) ⊗ Z(m) ≅ Z(n + m)

for each n, m ≥ 0. In particular, for n > 0 the motivic complex Z(n) is isomorphic to the n-th tensor power of Z(1).

Proof: By construction it is enough to establish isomorphisms of the form L(G_{m}^{∧n}) ⊗ L(G_{m}^{∧m}) ≅ L(G_{m}^{∧(n+m)}) in the category DP T^{−}(k). This in turn follows by looking at the primitive tensor product of complexes of the shape

⨁ L(G_{m}^{×(n−1)}) ⟶^{Σλ_{i}} L(G_{m}^{×n}).

Remark 7.4 The importance of the proposition lies in the fact that it allows one to define cohomological symbols Z(n) → τ_{≤n}Rπ_{∗}μ_{m}^{⊗n} for each natural integer m invertible in k, where μ_{m} is the étale sheaf of n-th roots of unity and π is the projection from the big étale site to the big Nisnevich site of k (restricted to smooth k-schemes). For n = 1 these come from the natural maps G_{m}[−1] → μ_{m} on the étale site using the adjointness of the functors π^{∗} and Rπ_{∗} and the quasi-isomorphism Z(1) ≃ G_{m}[−1]. The cohomological symbols are useful for the definition of l-adic realisations of motivic cohomology groups; in particular they provide a natural framework for the study of cycle maps from (higher) Chow groups into étale cohomology. See [6] and [7] for details and examples of applications.

References

[1] K. S. Brown, S. Gersten, Algebraic K-theory as generalized sheaf cohomology, in H. Bass (ed.) Algebraic K-Theory I: Higher K-Theories, Springer LNM 341 (1973), pp. 266–292.

[2] P. Deligne, J. S. Milne, Tannakian categories, in Hodge Cycles, Motives and Shimura Varieties, Springer LNM 900 (1982), pp. 101–228.

[3] S. I. Gelfand, Yu. I. Manin, Methods of Homological Algebra, Springer-Verlag, Berlin–Heidelberg–New York, 1996.

[4] J. S. Milne, Étale Cohomology, Princeton University Press, 1980.

[5] A. Suslin, V. Voevodsky, Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), 61–94.

[6] A. Suslin, V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coeffiecients, preprint, revised version (1999).

[7] T. Szamuely, Sur la théorie des corps de classes pour les variétés sur les corps p-adiques, to appear in J. reine angew. Math.

[8] V. Voevodsky, Triangulated categories of motives over a field, to appear in V. Voevodsky, A. Suslin, E. M. Friedlander, Cycles, Transfers and Motivic Homology Theories, Annals of Math. Studies, vol. 143, Princeton University Press.

[9] V. Voevodsky, Cohomological theory of presheaves with transfers, to appear in V. Voevodsky, A. Suslin, E. M. Friedlander, op. cit.

[10] J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 1996.

[11] I. Waschkies, Localization and derived categories, this volume.

[12] C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, 1995.

Tamás Szamuely Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences, PO Box 127, H-1364 Budapest, Hungary

e-mail : szamuely@renyi.hu