6. Motivic cohomology 6.1. Suslin complexes. Recall that the Dold-Thom theorem asserts that for a C.W. complex T the singular homology groups of T are naturally isomorphic to the homotopy groups of the free abelian group on T , a d + H∗(T, Z) ' π∗(( S T ) ), d d d where S T = T /Σd is the d-fold symmetric power of T (given the quotient topology d d ` d via the projection T → S T ) and the group completion of the monoid d S T is ` d 2 given the topology of a quotient of the self-product ( d S T ) . If X is an algebraic variety, then SdX admits the natural structure of an algebraic variety. Even though a typical variety X may receive no non-trivial maps from affine spaces of positive dimension, Suslin recognized that Hom(∆n,SdX) carries significant information about X. Definition 6.1. Let X be a quasi-projective algebraic variety over a field k. Then the Suslin complex Sus∗(X) of X is defined to be the normalized chain complex of the simplicial abelian group obtained as the group completion of the simplicial abelian monoid Hom(∆•,SdX), a • d + Sus∗(X) ≡ N (( Hom(∆ ,S X)) ). d

The algebraic singular homology of X is defined to be the homology of Sus∗(X), alg.sing H∗ (X) ≡ H∗(Sus∗(X)). Remark 6.2. There is an equivalence of categories between simplicial abelian groups and normalized chain complexes,

N :(s.ab grps) → C∗(Ab).

Under this equivalence, the homotopy groups π∗(A•) of the underlying simplicial set of a simplical abelian group A• are naturally isomorphic to the homology of the associated chain complex H∗(N (A•)). Thus, alg.sing a • d + H∗ (X) ' π∗(( Hom(∆ ,S X)) ) d Notation If P is a contravariant functor on quasi-projective varieties over k with values in an abelian category (e.g,, abelian groups or abelian sheaves on a site), then we denote by C∗(P ) the complex • C∗(P ) ≡ N (P (∆ )). ` d + For example, C∗(( d Hom(−,S X)) ) = Sus∗(X). Suslin and Voevodsky proved the following remarkable theorem, whose proof will give us insight into their formulation of motivic complexes. Theorem 6.3. Let X be a complex quasi-projective variety. Then there is a natural isomorphism ∗ ∗ an H (Sus∗(X), Z/n) ' Hsing(X , Z/n). More generally, if X is a quasi-projective variety over an algebraically closed field k and if n is invertible in k, then ∗ ∗ H (Sus∗(X), Z/n) ' Het(X, Z/n). 1 2

We begin discussion of this theorem by discussing “transfers” for an abelian presheaf. In different papers, this notion has been given different definitions. Per- haps the simplest way to define such a presheaf with transfers is as a contravariant functor F :(SmCor(k))op → Ab, where SmCor(k) is the category whose objects are smooth, quasi-projective vari- eties and whose maps from X to Y are elements in the free abelian group generated by integral closed subschemes of X × Y finite and surjective over some connected component of X. For example, any morphism f : X → Y determines such a finite correspondence, namely the graph Γf ⊂ X ×Y . The graph of a morphism is special in that it projects isomorphically to X, whereas a general finite correspondence Z ⊂ X × Y only maps finite-to-one to X. For example, if g : Y → X is a finite map surjective onto a connected component of X, then the graph Γg ⊂ Y × X determines a finite correspondence from X to Y . Thus, a presheaf with transfers is covariant with respect to finite surjective maps between smooth varieties. A presheaf P is said to be homotopy invariant if π∗ : P (X) → P (X × A1) is an isomorphism for all X. Suslin and Voevodsky prove that if a Φ is a homotopy invariant, abelian presheaf with transfers on SmCor(k) such that nΦ = 0 for some n invertible in k, then

Φ(Sr) = Φ(Spec k) r where Sr is the henselization at 0 of affine r-space A . This theorem is proved by induction on r, reducing one to the consideration of relative divisors in X/S smooth of relative one, and thus to finite correspondences. The fact that Φ is a homotopy invariant presheaf with transfers, enables one to see that one has an action which factors through a relative Picard group which modulo n can be studied using elementary etale cohomology. This is an abstraction of an argument (“Suslin rigidity”) used by Suslin to show determine K∗(F, Z/n) for any algebraically closed field F and any n invertible in F . The next step is the incorporation of sheaves, the interpretation of Suslin co- homology as sheaf cohomology. The Grothendieck site that they use is one that voevodsky had earlier introduced, the “qfh-topology”, the quasi-finite h-topology. This topology has more open sets than the etale topology. (A qfh-covering {pi : ` Ui → U} is a collection of quasi-finite maps with the property that i Ui → U is universal topological epimorphism. A typical qfh-covering is of the form {Ui → U → U} for U normal, where U → U is the normalization of U in a finite normal extension of k(U) and each Ui → U is a Zariski open inclusion.) The key property of the qfh-topology is that qfh-sheaves admit a unique structure of a presheaf with transfers. This enables Suslin-Voevodsky to prove that

∗ ∗ ∗ ext (6.3.1) Extet(Φet, Z/n) ' Extqfh(Φqfh, Z/n) ' ExtAb(Φ(Spec k), Z/n) for any homotopy invariant abelian presheaf with transfers satisfying nΦ = 0, where Φet (repsectively, Φqfh) is the sheaf in the etale (resp., qfh) topology associated to the presheaf Φ, the Ext-groups on the left and middle are taken in the categories of sheaves for the etale and qfh topologies and the Ext-groups on the right are in the category of abelian groups. 3

Finally, there is a clever hypercohomology argument. If P is an abelian presheaf, then the homology presheaves U 7→ Hn(C∗(P (U)) are homotopy invariant. Suslin and Voevodsky consider the maps of complexes of presheaves

C∗(Spec k) → C∗(P )qfh ← Pqfh in the qfh-topology, where the complex on the left is of constant presheaves. (We have abused notation by using the same notation for a simplicial abelian group and its associated normalized chain complex.) One hypercohomology spectral sequence is of the form p,q p q p+q E1 = Ext (P (∆ × −)qfh, Z/n) ⇒ Ext (C∗(P )qfh, Z/n), q ∗ which collapses because Pqfh → P (∆ ×−)qfh induces an isomorphism on Ext (−, Z/n) by an elementary general argument. The second hypercohomology spectral se- quence has the form 0 p,q p • p+q E2 = Ext (Hq(P (∆ )qfh), Z/n) ⇒ Ext (C∗(P )qfh, Z/n). We apply (6.3.1) to conclude that p • p Ext (Hq(P (∆ )qfh), Z/n) ' Ext (Sus∗(P ), Z/n).

This should motivate the context in which Voevodsky and Suslin-Voevodsky work. Perhaps we begin by brutally giving the following definition.

Definition 6.4. Denote by ShvNis(SmCor(k)) the category of Nisnevich sheaves with transfers (on smooth, quasi-projective varieties over k) and let D (ShvNis(SmCor(k))) denote the derived category of complexes of ShvNis(SmCor(k)) which are bounded above. We define

DMk ⊂ D (ShvNis(SmCor(k))) to be the full subcategory of those complexes with homotopy invariant cohomology sheaves. So, our world of algebraic geometry has become the derived category of Nisnevich sheaves with transfers. The role of transfers will be to insure Suslin rigidity, the role of the Nisnevich topology will be to facilitate general results especially by incorporating resolution of singularities. Given an algebraic variety, we define its Voevodsky motive as follows, once again inspired by the previous discussion of Suslin homology. Definition 6.5. Let X be a quasi-projective variety over k and let

cequi(X, 0) : (Sm/k) → Ab be the presheaf taking a smooth variety U to cequi(X, 0)(U), the abelian group of finite correspondences from U to X. We define the motive of X, M(X) by op M(X) ≡ C∗(cequi(X, 0)) : (Sm/k) → C∗(Ab).

Remark 6.6. cequi(X, 0) is actually a sheaf for the qfh topology; in particular, it is a presheaf with transfers. If X is normal, then the Suslin complex Sus∗(X) equals the evaluation of M(X) on Spec k. 4

We mention a fundamental theorem of Voevodsky which Suslin-Voevodsky apply to cequi(X, 0) which enables them to prove that motivic cohomology has many good properties. nis-eval Theorem 6.7. If k is a perfect field and if

0 → F1 → F2 → F3 → 0 is a short exact sequence of Nisnevich sheaves with transfers, then

C∗(F1)(Spec k) → C∗(F2)(Spec k) → C∗(F3)(Spec k) → C∗(F1)(Spec k)[1] is a distinguished triangle.

This enables us to conclude the following properties of the qfh-sheaf cequi(X, 0) as well as related qfh-sheaves zequi(X, r). The latter are defined to send a smooth quasi-projective variety U to the the abelian group generated by integral closed subvarieties of X × U which are equi-dimensional of relative dimension r over U. property Proposition 6.8. Let X be a smooth variety over k and let {U, V ⊂ X} be a Zariski open covering. Then the short exact sequence of sheaves

0 → cequi(U ∩ V, 0) → cequi(U, 0) ⊕ cequi(V, 0) → cequi(X, 0) → 0 is exact in the Nisnevich topology (as well as the cdh-topology) , thereby determining a Mayber-Vietoris long exact sequence in Zariski sheaf cohomology. If Y ⊂ X is a closed subvariety of the smooth variety X, then for any r ≥ 0 there is an exact sequence of sheaves in the cdh-topology

0 → zequi(Y, r)cdh → zequi(X, r)cdh → zequi(X − Y )cdh → 0 thereby determining a localization long exact sequence in cdh sheaf cohomology. Finally, if X is a quasi-projective variety over k and Z ⊂ X is a closed subvariety and if f : X0 → X is a proper morphism whose restriction f −1(X − Z) → X − Z is an isomorphism, then there are short exact sequences of sheaves in the cdh topology for any r ≥ 0 −1 0 0 → cequi(f (Z), 0)cdh → cequi(X , 0)cdh ⊕ cequi(Z, 0)cdh → cequi(X, 0) → 0 −1 0 0 → zequi(f (Z), r)cdh → zequi(X , r)cdh ⊕ zequi(Z, r)cdh → zequi(X, r) → 0 thereby giving blow-up long exact sequences in cdh sheaf cohomology. These short exact sequences of sheaves and the fact that these sheaves are presheaves with transfers provide distinguished triangles of complexes of presheaves thanks to the following theorem (an extension of Theorem 6.7. sheaf-acyclic Theorem 6.9. Let P be a presheaf with transfers over a field of characteristic 0 with the property that Pcdh = 0. Then the complex of sheaves in the Zariski topology, C∗(P )Zar, is acyclic. We next define the Suslin-Voevodsky motivic complexes Z(n) which serve as our best candidate for Beilinson’s conjectured ΓZar(n).

Definition 6.10. For a given n > 0, define Fn to be the sum of the n natural embeddings 1 n−1 1 n cequi((A − {0}) , ) → cequi((A − {0}) , 0) and define 1 n Z(n) = C∗(cequi((A − {0}) , 0)/Fn)[−n]. (we view Z(n) as the motive of the “n-fold smash product of Gm”). 5

We now define the Suslin-Voevodsky motivic cohomology groups. Definition 6.11. If X is a smooth quasi-projective variety over k, then we define the motivic cohomology of X by p p H (X, Z(q)) ≡ HZar(X, Z(q)) ' HomDMk (M(X), Z(q)[p]). For X not necessarily smooth, we consider the sheafification of Z(q) in the cdh- topology (to incorporate resolution of singularities) and define motivic cohomology by p p H (X, Z(q)) ≡ Hcdh(X, Z(q)cdh). Proposition 6.8 and Theorem ?? enables us to prove various basic properties of Hp(X, Z(n)). As mentioned previously, these motivic cohomology groups agree with Bloch’s higher Chow groups for smooth varieties, and therefore the Suslin-Voevodsky com- plexes satisfy most of the conjectures of Beilinson. Indeed, the key property of localization for Bloch’s higher Chow groups is best proved using motivic coho- mology. This comparison is achieved by means of a moving lemma argument of Suslin’s, showing that cycles in Bloch’s zq(X, n) ⊂ Zq(X ×∆n) can be moved to be equidimensional over X provided that n ≥ q. This, coupled with a duality result of Friedlander-Voevodsky, suffices to establish this comparison.

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Department of Mathematics, Northwestern University, Evanston, IL 60208 E-mail address: [email protected]