Andrei Suslin and Vladimir Voevodsky

Transcription

1 BLOCH-KATO CONJCTUR AND MOTIVIC COHOMOLOGY WITH FINIT COFFICINTS Andrei Susin and Vadimir Voevodsky Contents Introduction 0. Notations, terminoogy and genera remarks. 1. Homotopy invariant presfeaves with transfers. 2. Tensor structure on the category DM (F). 3. Motivic cohomoogy. 4. Fundamenta distinguished trianges in the category DM (F). 5. Motivic cohomoogy of non-smooth schemes and the cdhtopoogy. 6. Truncated etae cohomoogy and the Beiinson-Lichtenbaum Conjecture. 7. The Boch-Kato Conjecture. 8. B (n)-cohomoogy with Supports. 9. Shift of degrees. 10. Proof of the Main Theorem. 11. Boch-Kato Conjecture and vanishing of the Bockstein homomorphisms. 12. Appendix. cdh-cohomoogica dimension of Noetherian schemes. Introduction 0. Notations, terminoogy and genera remarks. Throughout the paper we fix a fied F and consider the category Sm/F of smooth separated schemes of finite type over F. We make Sm/F into a site using one of the foowing three topoogies: Zariski topoogy, Nisnevich topoogy or etae topoogy. For any presheaf F on Sm/F we denote by FZar, F Nis and F et the sheaf associated with F in Zariski, Nisnevich and etae topoogies respectivey. For any site C we denote by C (resp. C ) the category of abeian sheaves (resp. the category of abeian presheaves) on C. We denote by Sch/F the category of a separated schemes of finite type over F. Let F : Sm/F A be a (covariant) functor from Sm/F to an abeian category A. Let further X Sm/F be a smooth scheme and et i : Y X be a smooth Typeset by AMS-TX 1

2 2 ANDRI SUSLIN AND VLADIMIR VOVODSKY cosed subscheme provided with a retraction r : X Y (i.e. ri = 1 Y ). In this case F(r)F(i) = 1 F(Y ) and hence F(Y ) is canonicay a direct summand in F(X). We use the notation F(X/Y ) for the compementary direct summand. Thus F(X/Y ) may be identified either with the kerne of the homomorphism F(r) : F(X) F(Y ) or with the cokerne of the homomorphism F(i) : F(Y ) F(X). We use simiar notations in case F : Sch/F A is a functor from Sch/F to an abeian category A and Y X is a retract of X Sch/F. More generay, assume we are given a scheme X and n cosed subschemes i j : Y j X (1 j n) (the schemes X, Y j shoud be smooth in case of the category Sm/F). Assume further that we are given retractions r j : X Y j (1 j n) such that the morphisms ρ j = i j r j : X X pairwise commute. In this case each of F(Y j ) may be identified with a direct summand in F(X). Moreover the sum n j=1 F(Y j) F(X) is aso a canonica direct summand: the corresponding projection is given by the formua 1 F(X) (1 F(X) F(ρ 1 ))... (1 F(X) F(ρ n )) = n = ( 1) s 1 F(ρ j1... ρ js ) : F(X) s=1 1 j 1 <...<j s n n F(Y j ). The above situation arises in particuar in the foowing case. Let (Z, z 0 ) be a scheme provided with a distinguished rationa point. Set X = Z n. The point z 0 defines n embeddings of Z (n 1) in X. The corresponding subschemes Y j X consist of a points with j-th coordinate equa to z 0, the corresponding morphisms ρ j : X X are given by the formua ρ j (z 1,..., z n ) = (z 1,..., z j 0,..., z n ) and obviousy pairwise commute. We use the notation F(Z n ) for the direct summand of F(Z n ) compementary to n j=1 F(Y j). Throughout the paper this kind of notation is constanty used in the specia case when Z = G m = A 1 \ 0 and z 0 = 1 is the identity of the group scheme G m. The previous construction has an obvious anaogue for contravariant functors. In this case F(X/Y ) may be identified either with the kerne of the homomorphism F(i) : F(X) F(Y ) or with the cokerne of the homomorphism F(r) : F(Y ) F(X). Furthermore the direct summand F(Z n ) F(Z n ) may be identified with the intersection of kernes of restriction homomorphisms F(Z n ) F(i j) F(Z (n 1) ). The standart cosimpicia scheme F. Reca that n F is a cosed subscheme in An+1 F defined by the equation t t n = 1. We refer to the points v i = (0,..., 1,..., 0) n (0 i n) as vertices i of n. ach nondecreasing map φ : [m] = {0, 1,..., m} [n] = {0, 1,..., n} defines the corresponding morphism of schemes m n : (t 0,..., t m ) t 0 v φ(0) t m v φ(m). j=1

3 BLOCH-KATO CONJCTUR AND MOTIVIC COHOMOLOGY 3 There are n + 1 coface morphisms d i : n 1 n (0 i n) (corresponding to stricty increasing maps [n 1] [n]). These coface morphisms are obviousy cosed embeddings, the corresponding cosed subschemes of n are defined by equations t i = 0 and are caed the (codimension one) faces of n. For any presheaf of abeian groups F : Sm/F Ab one can consider a simpicia presheaf C (F) defined by the formua C n (F)(U) = F(U n ). It s easy to see that homoogy presheaves H i of the compex C (F) are homotopy invariant - i.e. H i (U A 1 ) = H i (U) for any U Sm/F (see [S-V 7]). In the same way one can construct the bisimpicia presheaf C, (F), defined by the formua C p,q (F)(U) = F(U p q ). Homotopy invariance of the cohomoogy presheaves of C (F) impies easiy that the natura embedding i : C (F) Tot C, (F) is a quasiisomorphism. The quasiisomorphism i has a canonica eft inverse caed the shuffe map (cf. [D] ch. 6, 12), whose construction we are going to remind. very stricty increasing map (φ, ψ) : [p + q] [p] [q] defines a inear isomorphism of schemes p+q p q (t 0,..., t p+q ) t 0 (v φ(0), v ψ(0) ) t p+q (v φ(p+q), v ψ(p+q) ) and hence gives an isomorphism of presheaves (φ, ψ) : C p,q (F) C p+q (F). Note further that stricty increasing maps (φ, ψ) : [p + q] [p] [q] are in one to one correspondence with (p, q) shuffes: each (p, q)-shuffe σ defines a map (φ, ψ) via the formua φ(x) = {1 i p : σ(i) x} ψ(x) = {p + 1 i p + q : σ(i) x}. We use the notation σ : C p,q (F) C p+q (F) for the isomorphism of presheaves defined by the stricty increasing map (φ, ψ) corresponding to σ. The shuffe map : Tot C, (F) C (F) is defined via the formua p,q = ǫ(σ)σ : C p,q (F) C p+q (F). σ (p,q) shuffe A we known and easy computation shows that is a homomorphism of compexes which is eft inverse to i (i.e. i = 1 C (F)). It s not difficut to see that moreover the composition i is homotopic to identity (so that i and are mutuay unverse homotopy equivaences), but we won t need this fact. Resoution of singuarities Definition 0.1. We be saying that resoution of singuarities hods over the fied F, provided that the foowing two conditions are satisfied. (1) For any integra separated scheme of finite type X over F there exists a proper birationa morphism Y X with Y smooth (over F). (2) For any smooth integra scheme X over F and any birationa proper morphism Y X there exists a tower of morphisms X n X n 1... X 0 = X, each stage of which is a bow up with a smooth center, and such that the composition morphism X n X may be factored through Y X.

4 4 ANDRI SUSLIN AND VLADIMIR VOVODSKY It s we known that resoution of singuarities hods over fieds of characteristic zero, whether it hods over (perfect) fieds of positive characteristic remains one of the centra probems of agebraic geometry. The foowing important theorem due to de Jong may be used sometimes as a repacement of the first of the above properties. Theorem 0.2 [J]. For any fied F and any integra separated scheme of finite type X over F there exists a proper surjective morphism Y X with Y smooth over F. It shoud be noted that this theorem impies that a resuts of [S-V] hod over an agebraicay cosed fied of arbitrary characteristic. Hypercohomoogy. Motivic cohomoogy are defined as Zariski (or Nisnevich) hypercohomoogy with coefficients in a compex of sheaves which is not bounded beow. We find it necessary to remind the definition and properties of hypercohomoogy in this generaity. Let A be an abeian category with enough injectives. Let further F : A Ab be a eft exact additive functor. Denote the right derived functors of F by R i F. Let finay A be a compex of degree +1 in A and et A I be the Cartan- ienberg resoution of A. The hypercohomoogy groups R F(A ) are defined as cohomoogy of the tota compex corresponding to the bicompex F(I ) R F(A ) = H (Tot(F(I ))). The foowing two resuts concerning hypercohomoogy constitute a minor generaization of we-known facts. For the sake of competness we sketch the proof beow. Theorem 0.3. Assume that either the compex A is bounded beow or the functor F is of finite cohomoogica dimension (i.e. R i (F) = 0 for i >> 0). Then both spectra sequences of the bicompex F(I ) are strongy convergent. Thus we have two hypercohomoogy spectrta sequences I pq 1 = Rp F(A q ) R p+q F(A ) II pq 2 = Rq F(H p (A )) R p+q F(A ). Proof. The case when the compex A is bounded beow is we-known so we consider ony the case when the functor F is of finite cohomoogica dimension. Let 0 A ǫ I 0 d 1 d I... be the Cartan-ienberg resoution of A. Define the compex A n as the kerne of the homomorphism d : I n I n+1. The foowing properties of the compex A n are straightforward from definitions. (0.3.1). For each i we have exact sequences 0 A i I i,0... I i,n 1 A i,n 0 0 Z i (A ) Z i (I 0 )... Z i (I n 1 ) Z i (A n ) 0 0 B i (A ) B i (I 0 )... B i (I n 1 ) B i (A n ) 0 0 H i (A ) H i (I 0 )... H i (I n 1 ) H i (A n ) 0

5 BLOCH-KATO CONJCTUR AND MOTIVIC COHOMOLOGY 5 We concude immediatey from (0.3.1) the foowing further properties of the compex A n. (0.3.2). 0 A n n d I the compex A n. I n+1 d... is the Cartan-ienberg resoution of (0.3.3). Assume that n cd F then a objects A i,n, Z i (A n ), B i (A n ), H i (A n ) are F-acycic. Lemma Let F be a functor of finite cohomoogica dimension d. Let further A be a compex such that a objects A i, H i (A ) are F-acycic, and et 0 A I be the Cartan-ienberg resoution of A, then (1) The objects Z i (A ), B i (A ) are aso F-acycic for a i. (2) Z i (F(A )) = F(Z i (A )), B i (F(A )) = F(B i (A )), H i (F(A )) = = F(H i (A )) for a i. (3) F(A ) Tot F(I ) is a quasiisomorphism (F-acycicity of A i (for a i) aone is enough for this concusion). Proof. To prove (1) one shows by inverse induction on n 1 that R n F(Z i (A )) = R n F(B i (A )) = 0 for a i. The point (2) is immediate from (1). To prove (3) one notes that the i-th cooumn of the bicompex F(I ) is a resoution of F(A i ) which is enough to concude that F(A ) Tot F(I ) is a quasiisomorphism. To concude the proof of the Theorem 0.3 we denote by τ n I the subbicompex 0 I 0 I 1... I n 1 A n 0 of I. Taking n cd F we see immediatey from the resuts proved above that the embedding τ n I I induces a quasiisomorphism of compexes Tot F(τ n I ) Tot F(I ). Moreover the first (respectivey the second) spectra sequences of the bicompexes F(τ n I ) and F(I ) coincide from 1 -term on (resp. from 2 -term on). In case the functor F coincides with Hom A (X, ) the corresponding hypercohomoogy groups are ususy denoted RHom A(X, A ). They may be interpreted as appropriate Hom-groups in the derived category D(A). Proposition 0.4. Assume that either the compex A is bounded beow or xt i (X, ) = 0 for i >> 0. Then RHom i A (X, A ) = Hom D(A) (X, A [i]). Proof. The case when the compex A is bounded beow is we-known, so we consider ony the case when the cohomoogica dimension of the functor Hom A (X, ) = F is finite. Note first of a that according to definitions Hom D(A) (X, A [i]) = im Hom K(A) (X, B [i]) = im H i (Hom A (X, B )) A B A B where direct imit is taken over a quasiisomorphisms A B, K(A) denotes the homotopy category of compexes and Hom A (X, B ) is the compex of abeian groups... Hom A (X, B i ) Hom A (X, B i+1 )....

6 6 ANDRI SUSLIN AND VLADIMIR VOVODSKY Lemma Let F be a eft exact additive functor of finite cohomoogica dimension d, then (1) For any compex A there exists a quasiisomorphism A B, where the terms B i are F-acycic for a i. (2) Let B f C be a quasiisomorphism of compexes. Assume that a terms B i and C i are F-acycic. Then the induced homomorphism of compexes of abeian groups F(f) : F(B ) F(C ) is aso a quasiisomorphism. Proof. To prove the point (1) it suffices to note that A Tot τ n I is aways a quasiisomorphism. Moreover the terms of the compex Tot τ n I are F-acycic for any eft exact additive functor F of cohomoogica dimension d provided that n d. To prove the second statement denote by co(f) the cone of f. The compex co(f) is acycic (since f is a quasiisomorphism) and consists of F-acycic terms. Lemma shows that the compex co(f(f)) = F(co(f)) is acycic as we and hence F(f) is a quasiisomorphism. Lemma shows that Hom D(A) (X, A [i]) = H i (Hom A (X, B )) for any quasiisomorphism A B from A to a compex with Hom A (X, )-acycic terms. Appying this to the compex Tot τ n I (with n >> 0) we easiy concude the proof of Proposition 0.4. xt-groups For any two compexes A, B D(A) we set xt i A (A, B ) = Hom D(A) (A, B [i]). It s we-known that in case A, B D (A) (resp. D + (A), D b (A)) the corresponding xt-groups coincide with Hom-groups in the category D (A) (resp. in D + (A), D b (A)). Infinite direct sums in the derived categories. Let A be an abeian category satisfying the Grothendieck s axiom Ab-?. Beow we discuss infinite direct sums in the category D (A). A simiar discussion appies with minima modifications to D + (A) and D b (A) as we. Lemma 0.5. Let {A i } i I be a famiy of compexes. Any quasiisomorphism C i I A i may be dominated (in the category K(A)) by a quasiisomorphism of the form i I C f i i i I A i (where each f i : C i A i is a quasiisimorphism). Proof. A we-known property of the homotopy category K(A) shows that for each i there exists a compex C i a quasiisomorphism f i : C i A i and a morphism g i : C i C which make the foowing diagram commute up to homotopy. C i f i g i C A i in i i I A i

7 BLOCH-KATO CONJCTUR AND MOTIVIC COHOMOLOGY 7 A famiy of morphisms g i : C i C defines a morphism g = (g i ) i I : C i C and according to the construction the foowing diagram commutes up to homotopy i I C i f i i I A i g C = i I A i Coroary For any compex B we have a natura isomorphism Hom D(A) ( i I A i, B ) = i I Hom D(A) (A i, B ). In other words i I A i is a direct sum of compexes A i we. in the category D(A) as Proof. In view of definitions and Lemma 0.5 we have the foowing identifications Hom D(A) ( A i, B ) = im Hom K(A) (C, B ) = i I C L i I A i = im Hom K(A) (C i, B ) = Hom D(A) (A i, B ). A i } i I i I i I {C i We be saying that the famiy of compexes {A i } i I is uniformy bounded above (resp. uniformy homoogicay bounded above) iff there exists an integer N such that A n i = 0 (resp. Hn (A i ) = 0) for a i I and a n N. Coroary A direct sum of a famiy of compexes {A i } i I D (A) exists in D (A) if and ony if this famiy is uniformy homoogicay bounded above. Let {A i } i I D (A) be a famiy of compexes. We denote by < {A i } > D (A) the minima fu trianguated subcategory cosed with respect to taking direct sums of uniformy homoogicay bounded above famiies of compexes. We be saying that the famiy {A i } i I weaky generates the category < {A i } i I >. Lemma 0.6. Let A be a bounded above bicompex in A. Then Tot(A ) < {A i } i Z > D (A). Proof. Denote by σ n A A the subbicompex of A consisting of terms A ij with j n. Denote further by i n : σ n A A and j n : σ n A σ n 1 A the obvious embeddings. The compexes Tot(σ n A ) obviousy ie in the trianguated subcategory generated by a A i. Furthermore we have a short exact sequence of bicompexes 0 n σ n A (1 σ n A j n) n σ n A (i n) A 0 which shows that Tot(A ) ies in the subcategory weaky generated by A i.

8 8 ANDRI SUSLIN AND VLADIMIR VOVODSKY 1. Homotopy invariant presheaves with transfers. For any X, Y Sm/F define Cor(X, Y ) to be the free abeian group generated by cosed integra subschemes Z X F Y which are finite and surjective over a component of X. Let X, Y, W Sm/F be smooth schemes and et Z Cor(X, Y ), T Cor(Y, W) be cyces on X Y and Y W each component of which is finite and surjective over a component of X (respectivey over a component of Y ). One checks easiy that the cyces Z W and X T intersect propery on X Y W and each component of the intersection cyce (Z W) (X T) is finite and surjective over a component of X. Thus setting T Z = (pr 1,3 ) ((Z Y ) (X T)) we get a biinear composition map Cor(Y, W) Cor(X, Y ) Cor(X, W). In this way we get a new category (denoted SmCor/F) whose objects are smooth schemes of finite type over F and Hom SmCor/F (X, Y ) = Cor(X, Y ) see [V 1] for detais. The category SmCor/F is ceary additive, the direct sum of two schemes being given by their disjoint union. A presheaf with transfers on the category Sm/F is defined as a contravariant additive functor F : SmCor/F Ab. Note that associating to each morphism f : X Y its graph Γ f Cor(X, Y ) we get a canonica map Hom Sm/F (X, Y ) Cor(X, Y ). In this way we get a canonica functor Sm/F SmCor/F, which aows to view presheaves with transfers on Sm/F as presheaves in the usua sence equipped with appropriate additiona data. For any X Sm/F we denote by Z tr (X) the corresponding representabe functor, i.e. Z tr (X) is a presheaf with transfers defined via the formua Z tr (X)(U) = Cor(U, X). One checks easiy that the pesheaf Z tr (X) is actuay a sheaf in the etae topoogy (and a fortiory in Zariski and Nisnevich topoogies as we). Direct sums of such sheaves are sometimes caed free presheaves with transfer or free Nissnevich sheaves with transfers. This terminoogy is not very consistent since sheaves Z tr (X) are obviousy projective objects in the category of presheaves with transfers, but ceary not in the category of Nisnevich sheaves with transfers. In a simiar way we may associate a presheaf with transfers Z tr (X) to each scheme of finite type X Sch/F. To be more precise for each U Sm/F we define Z tr (X)(U) to be the free abeian group generated by cosed integra subschemes Z U X which are finite and surjective over a component of U. The same construction as above defines (for any U, V Sm/F) the biinear pairing Cor(V, U) Z tr (X)(U) Z tr (X)(V ) which makes Z tr (X) into a presheaf with transfers. Moreover the presheaf with transfers Z tr (X) is an etae sheaf for any X Sch/F. Obviousy Z tr is a functor from Sch/F to the category of (etae) sheaves with transfer.

9 BLOCH-KATO CONJCTUR AND MOTIVIC COHOMOLOGY 9 Consider the standart cosimpicia object = F in Sm/F. For any presheaf of abeian groups F on Sm/F we get a simpicia presheaf C (F), by setting C n (F)(U) = F(U n ). We use the same notation C (F) for the corresponding compex (of degree 1) of abeian presheaves. Usuay we be deaing with compexes of degree +1, in particuar, we reindex the compex C (F) (in the standart way), by setting C i (F) = C i (F). Reca that a presheaf F : Sm/F Ab is said to be homotopy invariant, provided that for any U Sm/F the natura homomorphism F(U) F(U A 1 ) is an isomorphism. It s easy to see ( [S-V 1] Coroary 7.5 ) that cohomoogy presheaves of the compex C (F) are homotopy invariant. The foowing resut sums up some of the basic properties of homotopy invariant presheaves with transfers. Theorem 1.1 [V 1]. Let F be a homotopy invariant presheaf with transfers on Sm/F, then (1) The sheaf FZar coincides with F Nis and has a natura structure of a homotopy invariant sheaf with transfers. (2) For any X Sm/F and any i 0 HZar i (X, F Zar ) = Hi Nis (X, F Nis ) (3) The presheaves X HZar i (X, F Zar ) = Hi Nis (X, F Nis ) are homotopy invariant (and have a natura structure of presheaves with transfers). Coroary Let C be a compex (of degree +1) of Nisnevich sheaves with transfers with homotopy invariant cohomoogy presheaves. Then for any scheme X Sm/F we have a natura isomorphism of hypercohomoogy groups H Zar (X, C ) = H Nis (X, C ). Proof. Note that cohomoogica dimension of X with respect to both Zariski and Nisnevich topoogy is finite (and equa to dim X). This remark and Theorem 0.3 give us two convergent hypercohomoogy spectra sequences (in which H q denotes the q-th cohomoogy presheaf of C ) ZarII pq 2 = Hp Zar (X, (Hq ) Zar ) Hp+q Zar (X, C ) NisII pq 2 = Hp Nis (X, (Hq ) Nis) H p+q Nis (X, C ) Moreover we have a natura homomorphism of spectra sequences Zar II Nis II. Theorem 1.1 shows that the map on 2 terms is an isomorphism, hence the map on the imits is an isomorphism as we. Nisnevich topoogy is much more convenient in deaing with presheaves with transfers as one sees from the foowing Lemma (which fais in case of the Zariski topoogy). Lemma 1.2 [V 1]. Let F be any presheaf with transfers on Sm/F. Then there exists the unique structure of a Nisnevich sheaf with transfers on FNis making the homomorphism F FNis into a homomorphism of presheaves with transfers.

10 10 ANDRI SUSLIN AND VLADIMIR VOVODSKY For any Nisnevich sheaf with transfers G and any homomorphism of presheaves with transfers F G the associated homomorphism FNis G is compatibe with transfers. Lemma 1.2 impies easiy that Nisnevich sheaves with transfers form an abeian category which we denote NSwT/F. Lemma 1.3. Homotopy invariant Nisnevich sheaves with transfers form a fu abeian subcategory in NSwT/F, cosed under taking kernes and cokernes of morphisms and under extensions. Proof. Let f : F G be a homomorphism of homotopy invariant sheaves with transfers. Denote by K and C the kerne and cokerne of f in the category of presheaves. Then K and C obviousy are homotopy invariant presheaves with transfers. Furthermore K is a sheaf and coincides with Ker f, whereas Coker f = CNis. Theorem 1.1 (1) shows now that Coker f is homotopy invariant. Finay et 0 F H G 0 be an extension of Nisnevich sheaves with transfers and assume that F and G are homotopy invariant. For any U Sm/F we have a commutative diagram with exact rows 0 F(U) H(U) G(U) HNis 1 (U, F) = = = 0 F(U A 1 ) H(U A 1 ) G(U A 1 ) H 1 Nis (U A1, F) Theorem 1.1 (3) shows that the right hand side vertica map is an isomorphism so that homotopy invariance of H foows from the five emma. Homotopy invariance of the cohomoogy presheaves of the compexes C (F) impies immediatey the foowing fact. Lemma 1.4. Let A be a bounded above compex of Nisnevich sheaves with transfers. A cohomoogy presheaves of the compex Tot C (A ) are homotopy invariant. Hence a cohomoogy sheaves of Tot C (A ) are homotopy invariant as we. A convenient framework for deaing with homotopy invariant sheaves with transfers is provided by the tensor trianguated category DM (F). Reca (see [V 2]) that DM (F) is defined as the fu subcategory of the derived category D (NSwT/F) of bounded above compexes of Nisnevich sheaves with transfers comprising the compexes with homotopy invariant cohomoogy sheaves. Lemma 1.4 shows that for any bounded above copmpex of Nisnevich sheaves with transfers A the compex C (A ) is in DM (F). For any X Sm/F we define its motive M(X) as the eement C (Z tr (X)) DM (F). The foowing theorem reates DM (F) to Nisnevich cohomoogy. Theorem 1.5[V 2]. For any compex A DM (F) and any X Sm/F we have natura isomorphisms H i Nis(X, A ) = Hom DM (F)(M(X), A [i]).

11 BLOCH-KATO CONJCTUR AND MOTIVIC COHOMOLOGY 11 The proof spits naturay into two parts. In the first part we compare Nisnevich cohomoogy with Hom-groups in the category D (NSwT/F). We start with the foowing Lemma. Lemma 1.6 [V 1]. Let f : Y X be a Nisnevich covering of a (not necessariy smooth) scheme X. Then the foowing sequence of Nisnevich sheaves with transfers is exact 0 Z tr (X) f Z tr (Y ) (p 2) (p 1 ) Ztr (Y X Y ) (p 23) (p 13 ) +(p 12 )... Proof. It suffices to show that if S is a smooth henseian scheme then the sequence of abeian groups A (S) = (0 Z tr (X)(S) f Z tr (Y )(S) (p 2) (p 1 ) Ztr (Y X Y )(S)...) is exact. Fix a cosed integra subscheme Z X S finite and surjective over S and denote by A Z n(s) the subgroup of A n (S) generated by cosed integra subschemes T Y X... X Y S (finite and surjective over S) whose set theoretica image in X S coincides with Z. Obviousy A Z (S) is a subcompex of A (S) and moreover A (S) = Z A Z (S). Thus it suffices to show that every compex A Z (S) is contractibe. The scheme Z being finite and surjective over a henseian scheme S is henseian itsef, which impies that the projection p 1 : Z X admits a ifting f : Z Y. Let T Y X... X Y S be a generator of A Z n (S) and et g : T Z be the corresponding morphism. The morphism }{{} n T pr 1 X fg pr 2 Y X... X Y }{{} X Y S is obviousy a cosed embedding and hence n defines a cosed integra subscheme T = u n (T) Y X... X Y S (finite and }{{} n+1 surjective over S). In this way a get a homomorphism u n : A Z n (S) AZ n+1 (S), and one checks immediatey that u is a contracting homotopy for the compex A Z (S). Coroary 1.7. Let I NSwT/F be an injective Nisnevich sheaf with transfers. Then for any X Sm/F HNis i (X, I) = 0 for a i > 0. Proof. We first compute Chech cohomoogy of I with respect to a Nisnevich covering f : Y X. These cohomoogy are the cohomoogy of the compex I(Y ) I(Y X Y )... = = Hom NSwT (Z tr (Y ), I) Hom NSwT (Z tr (Y X Y ), I)... Injectivity of I in NSwT/F and Lemma 1.6 show that H i (Y/X, I) = 0 for i > 0. Now the standard argument invoving the Cartan-Leray spectra sequence ends up the proof.

12 12 ANDRI SUSLIN AND VLADIMIR VOVODSKY Coroary For any Nisnevich sheaf with transfers F we have natura identifications xt i NSwT (Z tr(x), F) = HNis i (X, F). In particuar xt i NSwT(Z tr (X), ) = 0 for i > dimx. Proposition 1.8. Let A D (NSwT/F) be a bounded above compex of Nisnevich sheaves with transfers. Then for any X Sm/F we have a natura isomorphism. H i Nis (X, A ) = Hom D (NSwT/F)(Z tr (X), A [i]). Proof. Let A J and A I be the Cartan-ienberg resoutions of A in the categories of Nisnevich sheaves with transfers and a Nisnevich sheaves respectivey. It s easy to see from the defining property of the Cartan-ienberg resoution that there exists a unique (up to a homotopy of bidegree (0, 1)) homomorphism of resoutions J I. Furthermore Proposition 0.4 aows us to identify Hom D (NSwT/F)(Z tr (X), A [i]) to H i (Tot(J (X))) whereas H i Nis (X, A ) identifies canonicay to H i (Tot(I (X))). What we have to verify is that the above homomorphism of resoutions induces an isomorphism H (Tot(J (X))) H (Tot(I (X))). Theorem 0.3 provides us with two spectra sequences converging to cohomoogy in question and Coroary shows that the induced map on the 1 -terms of the first spectra sequences is an isomorphism. Thus the map on imits is an isomorphism as we. In the second part of the proof we compare Hom D (NSwT/F)(Z tr (X), A ) and Hom DM (F)(M(X), A ) = Hom D (NSwT/F)(C (Z tr (X)), A ). A presheaf (resp. a presheaf with transfers) F is said to be contractibe provided that there exists a homomorphism of presheaves (resp. of presheaves with transfers) φ : F C 1 (F) = C 1 (F) with the property that 0 φ = 0, 1 φ = 1 F. Here 0, 1 : C 1 (F) C 0 (F) = F are the face operations of the simpicia presheaf C (F). In other words to each section s F(U) one can associate in a natura way a section φ(s) C 1 (F)(U) = F(U A 1 ) such that φ(s) U 0 = 0, φ(s) U 1 = s. It foows easiy from the definition that Zariski (Nisnevich, etae) sheaf associated to a contractibe presheaf is again contractibe. Typica exampes of contractibe presheavs are given by the foowing Lemma. Lemma 1.9. For any presheaf (resp. presheaf with transfers) F the kerne of any iterated face operation : C n (F) C 0 (F) = F is contractibe. Proof. We consider ony the specia case K = Ker( 0 : C 1 (F) F). The proof in the genera case is simiar. For any X Sm/F we have: K(X) = {s F(X A 1 ) : s X 0 = 0}. Denote by m : A 1 A 1 A 1 the mutipication morphism (m(a, b) = ab) and finay set φ(s) = (1 X m) (s) F(X A 1 A 1 ). One checks immediatey that φ(s) X A 1 0 = 0, i.e. φ(s) K(X A 1 ) and further 0 φ(s) = φ(s) X 0 A 1 = 0, 1 φ(s) = φ(s) X 1 A 1 = s.

13 BLOCH-KATO CONJCTUR AND MOTIVIC COHOMOLOGY 13 A Nisnevich sheaf F is said to be strongy homotopy invariant provided that a cohomoogy presheaves X HNis i (X, F) are homotopy invariant. According to Theorem 1.1 each homotopy invariant Nisnevich sheaf with transfers is strongy homotopy invariant. Proposition Let G and F be Nisnevich sheaves ( resp. Nisnevich sheaves with transfers). Assume that G is contractibe and F is strongy homotopy invariant. Then xt i Nis(G, F) = 0 for a i 0 (resp. xt i NSwT(G, F) = 0 for a i 0). Proof. We give a proof for Nisnevich sheaves, the case of Nisnevich sheaves with transfers is treated simiary. Let 0 F I be an injective resoution of F. The cohomoogy presheaves of the compex C 1 (I ) are given by the formua X HNis i (X A1, F) = HNis i (X, F). Since the sheaf associated to the presheaf X HNis i (X, F) is trivia for i > 0 we concude that C 1(I ) is a resoution of C 1 (F) = F. The face operations 0, 1 : C 1 (I ) I give two maps of resoutions over the identity endomorphism of F and hence are homotopic. Thus there exists a famiy of sheaf homomorphisms s n : C 1 (I n ) I n 1 such that ds n +s n+1 C 1 (d) = 1 0. Let now φ : G C 1 (G) be the homomorphism from the definition of a contractibe sheaf. One checks immediatey now that associating to each homomorphism f Hom Nis (G, I n ) the homomorphism u n (f) = s n C 1 (f)φ Hom Nis (G, I n 1 ) we get a contracting homotopy for the compex Hom Nis (G, I ). Coroary Let G and F be Nisnevich sheaves with transfers. Assume that F is homotopy invariant (and hence strongy homotopy invariant) and G admits a finite resoution 0 G G 0... G n 0 in which a G i are contractibe Nisnevich sheaves with transfers. Then xt i NSwT/F (G, F) = 0 for a i 0. Coroary Let A be a bounded above compex of Nisnevich sheaves with transfers. Assume that a A i are contractibe and a the cohomoogy sheaves H i = H i (A ) are homotopy invariant. Then the compex A is acycic. Proof. We prove that H i = 0 by inverse induction on i. For i >> 0 H i = 0 since A is bounded above. Assume now that H j = 0 for a j > i and denote by Z i the kerne of d : A i A i+1. According to our inductive assumption the sheaf Z i has a finite resoution 0 Z i A i A i+1... with contractibe terms. Since the sheaf H i is homotopy invariant we concude from Coroary that Hom NSwT (Z i, H i ) = 0. Thus the natura epimorphism Z i H i is zero and hence H i = 0. For any presheaf F and any n 0 we have a natura monomorphism F C n (F) spit by any of the iterated face maps : C n (F) C 0 (F) = F. Denoting by C 0 (F) the constant simpicia presheaf (which has F in a dimensions and a face and degeneracy maps are identities) we get a natura monomorphism (spit in each dimension) of simpicia presheaves C 0 (F) C (F) and hence aso the associated

14 14 ANDRI SUSLIN AND VLADIMIR VOVODSKY monomorphism of compexes (of degree +1) C 0 (F) C (F). The cokerne of this monomorphism consists of contractibe presheaves according to Lemma 1.9. This construction generaizes immediatey to compexes of presheaves. In particuar, for each bounded above compex A of Nisnevich sheaves with transfers we have a natura monomorphism of compexes C 0(A ) C (A ) whose cokerne consists of contractibe sheaves. Note further that we aso have a natura embedding A C 0(A ) whose cokerne is contractibe (in the usua sence of homoogica agebra) and hence acycic. Proposition Let A be a bounded above compex of Nisnevich sheaves with transfers. (1) If each A n is contractibe then the compex C (A ) is acycic. (2) If the cohomoogy sheaves H i (A ) are homotopy invariant then the natura embedding A C (A ) is a quasiisomorphism. Proof. If each A n is contractibe then each C i (A n ) is contractibe as we, so that a terms of the compex C (A ) are contractibe. Since the cohomoogy sheaves of this compex are homotopy invariant we concude from Coroary that this compex is acycic. Assume now that the cohomoogy sheaves of A are homotopy invariant. Lemma 1.3 impies immediatey that a the cohomoogy sheaves of the compex C (A )/C 0 (A ) are homotopy invariant. Since the terms of this compex are contractibe we concude from Coroary that this compex is acycic. Thus the natura embedding C 0 (A ) C (A ) is a quasiisomorphism and hence the embedding A C 0 (A ) C (A ) is a quasiisomorphism as we. Coroary Let B, A be bounded above compexes of Nisnevich sheaves with transfers. Assume that the sheaves B n are contractibe and the cohomoogy sheaves H n (A ) are homotopy invariant. Then Hom D (NSwT/F)(B, A ) = 0. Proof. It suffices to show that each homomorphism of compexes f : B A gives zero in Hom D (NSwT/F)(B, A ). This foows immediatey from the commutative diagram B f A C (B ) C (f) C (A ) in which the right vertica arrow is a qasiisomorphism and the compex C (B ) is acycic. Coroary Let B, A be bounded above compexes of Nisnevich sheaves with transfers. Assume that the cohomoogy sheaves H n (A ) are homotopy invariant. Then the natura embedding B C (B ) induces an isomorphism Hom D (NSwT/F)(C (B ), A ) Hom D (NSwT/F)(B, A ).

15 BLOCH-KATO CONJCTUR AND MOTIVIC COHOMOLOGY 15 Proof. Once again we decompose the embedding B C (B ) into a composition B C 0 (B ) C (B ). The first arrow is a quasiisomorphism and hence induces isomorphisms on Hom D (NSwT/F)(, A ). The second embedding has cokerne consisting of contractibe sheaves and hence aso induces an isomorphism on Hom D (NSwT/F)(, A ) according to coroary Coroary shows that if A DM (F) is a bounded above compex of Nisnevich sheaves with transfers with homotopy invariant cohomoogy sheaves then for any scheme X Sm/F the natura embedding Z tr (X) C (Z tr (X)) = M(X) induces an isomorphism Hom D (NSwT/F)(M(X), A [i]) Hom D (NSwT/F)(Z tr (X), A [i]) which together with Coroary 1.8 ends up the proof of the Theorem 1.5. Proposition 1.11 shows immediatey that if A is an acycic bounded above compex of Nisnevich sheaves with transfers then the compex C (A ) is acycic as we. This impies further that if f : A B is a quasiisomorphism of bounded above compexes of Nisnevich sheaves with transfers then C (f) is aso a quasiisomorphism. Thus we get a functor C : D (NSwT/F) DM (F). Denote by A the thick trianguated subcategory of D (NSwT/F) comprising those compexes A for which the compex C (A ) is acycic. The previous discussion proves immediatey the foowing resut. Theorem 1.12 [V 2]. (1) The functor C takes distinguished trianges to distinguished trianges and commutes with direct sums (of homoogicay bounded above famiies). (2) The functor C is eft adjoint to the embedding functor DM (F) D (NSwT/F) and estabishes an equivaence of DM (F) with the ocaization of D (NSwT/F) with respect to the thick trianguated subcategory A. We finish this section by a more detaied description of the category A. Lemma A compex A D (NSwT) is in A if and ony if it is quasiisomorphic to a (bounded above) compex of contractibe Nisnevich sheaves with transfers. Proof. If a the entries of the compex A are contractibe then the compex C (A ) is acycic according to Proposition Assume now that the compex C (A ) is acycic. The distinguished triange C 0(A ) C (A ) C (A )/C 0(A ) C 0(A )[1]... shows that the compex C 0 (A ) (which is quasiisomorphic to A ) is quasiisomorphic to a compex C (A )/C 0(A )[ 1] with contractibe entries. Let X Sch/F be a scheme with a distinguished rationa point x 0 X. We be saying that the scheme X is (agebraicay) contractibe (to the point x 0 ) iff there exists a morphism f : X A 1 X such that f X 1 = 1 X, f X 0 = x 0 and f x0 A 1 = x 0.

16 16 ANDRI SUSLIN AND VLADIMIR VOVODSKY Proposition (1) Assume that the scheme (X, x 0 ) is contractibe. Then for any Y Sch/F the sheaf Z tr (Y X/Y x 0 ) is contractibe. (2) The category A is weaky generated by the contractibe sheaves Z tr (Y A 1 /Y 0) (Y Sm/F). Proof. Let f : X A 1 X be the contraction of X. We define the homomorphism of sheaves φ : Z tr (Y X) C 1 (Z tr (Y X)) as the composition Z tr (Y X) Z Z A 1 C 1 (Z tr (Y X A 1 )) C 1(1 Y f) C 1 (Z tr (Y X)) Here the first arrow sends a section of Z tr (Y X) over a smooth scheme U (i.e. a cyce Z Y X U each component of which is finite and surjective over a component of U) to the section of Z tr (Y X A 1 ) over U A 1 given by the cyce Z A 1 Y X A 1 U A 1 (each component of which is obviousy finite and surjective over a component of U A 1 ). One checks immediatey that φ takes Z tr (Y x 0 ) to C 1 (Z tr (Y x 0 )) and hence defines a homomorphism φ : Z tr (Y X/Y x 0 ) C 1 (Z tr (Y X/Y x 0 )) and furthermore 1 φ = Id, 0 φ = 0. To prove the second statement it suffices (in view of Lemmas 0.6 and 1.13) to show that every contractibe sheaf F N SwT admits a eft resoution by the direct sums of sheaves of the form Z tr (Y A 1 /Y ). We show more generay that such resoutions exist for a sheaves F NSwT for which the compex C (F) is acycic (i.e. F A). Note first of a that to give a homomorphism of sheaves with transfers Z tr (Y A 1 /Y ) F is the same as to give a section s F(Y A 1 ) such that s Y 0 = 0. Surjectivity of the homomorphism C 1 (F) 1 0 F impies that ocay in the Nisnevich topoogy each section t F(Y ) may be written in the form t = s Y 1, where s F(Y A 1 ) is a section for which s Y 0 = 0. This remark shows that the natura homomorphism Y Sm/F s F(Y A 1 ):s Y 0 =0 Z tr (Y A 1 /Y ) F is surjective. The kerne of this homomorphism is again in A, so that the construction may be repeated. 2 Tensor structure on the category DM (F). To define the tensor structure on the category DM (F) we start with the definition of the tensor product of presheaves with transfers. The definition we are about to give is dictated by the foowing (expected) properties of the tensor product operation: it shoud commute with any direct sums and moreover shoud commute with tensoring (in the usua sence) with arbitrary abeian groups, be right exact and satisfy the property that (here the superscript pr stands for the tensor product in the category of presheaves) Z tr (X) pr tr Z tr(y ) = Z tr (X Y ). Note that for any presheaf with transfers F we have a natura exact sequence: 0 F X Sm/F F(X) Z tr (X) f Cor(X,Y ) F(Y ) Z tr (X)

17 BLOCH-KATO CONJCTUR AND MOTIVIC COHOMOLOGY 17 tensoring such resoutions for F and G we come immediatey to the foowing formua for F pr tr G F pr tr G = Coker( Y Sm/F,f Cor(X,X) X Sm/F,g Cor(Y,Y ) F(X) G(Y ) Z tr (X Y ) F(X) G(Y ) Z tr (X Y ) X,Y Sm/F F(X) G(Y ) Z tr (X Y )). In other words the presheaf with transfers F pr tr G is given by the formua: (F pr tr G)(Z) = X,Y Sm/F F(X) G(Y ) Cor(Z, X Y )/Λ where Λ is the subgroup generated by eements of the foowing form φ ψ (f 1 Y ) h f (φ) ψ h : f Cor(X, X), φ F(X) ψ G(Y ), h Cor(Z, X Y ) φ ψ (1 X g) h φ g (ψ) h : g Cor(Y, Y ), φ F(X) ψ G(Y ), h Cor(Z, X Y ) With this definition one verifies easiy that pr tr has a the expected properties. In particuar, the functor pr tr is right exact in each variabe, commutative and associative (up to a natura isomorphism) and Z tr (X) pr tr Z tr(y ) = Z tr (X Y ). Next one defines the tensor product of Nisnevich sheaves with transfers as the sheaf associated with their tensor product in the category of presheaves. We denote the ast operation by tr. The bifunctor tr : NSwT/F NSwT/F NSwT/F is sti right exact, commutative and associative, commutes with arbitrary direct sums and satisfies the identity Z tr (X) tr Z tr (Y ) = Z tr (X Y ). The foowing universa mapping property of the tensor product of Nisnevich sheaves with transfers is obvious from the definition. Lemma 2.1. Let F, G, H be Nisnevich sheaves with transfers. To give a homomorphism of sheaves with transfers F tr G p H is the same as to give biinear maps p X,Y : F(X) G(Y ) H(X Y ) (X, Y Sm/F ) which satisfy the foowing properties (1) For any f Cor(X, X) the foowing diagram commutes F(X) G(Y ) f 1 G(Y ) F(X ) G(Y ) p X,Y H(X Y ) (f 1 Y ) p X,Y H(X Y )

18 18 ANDRI SUSLIN AND VLADIMIR VOVODSKY (2) For any g Cor(Y, Y ) the foowing diagram commutes F(X) G(Y ) p X,Y H(X Y ) 1 F(X) g (1 X g) F(X) G(Y ) p X,Y H(X Y ) To extend this operation on the category D (NSwT/F) we need a few auxiiary resuts. For any X Sm/F denote by L X i the eft derived functors of the right exact functor Z tr (X) pr tr : Presheaves with transfers Presheaves with transfers. Note that the sheaves Z tr (X) are projective in the category of presheaves with transfer, but we don t know whether they are fat (probaby not). Lemma 2.2. Let F be a presheaf with transfers such that FNis L X i (F) Nis = 0 for a i 0 and a X Sm/F. = 0. Then Proof. We first construct certain specia presheaves with transfers F for which F Nis = 0. Let f : Y Y be a Nisnevich covering of Y Sm/F. Denote by H i (Y /Y ) the homoogy presheaves of the compex Č (Y /Y ) : 0 Z tr (Y ) f Z tr (Y ) Z tr (Y Y Y ).... Lemma 1.6 shows that H i (Y /Y ) Nis = 0 for a i. Moreover it s cear from the definition that for each presheaf with transfers F for which FNis = 0 there exists an epimorphism onto F from a direct sum of presheaves of the form H 0 (Y /Y ). We now start proving our statement by induction on i. For i < 0 everything is obvious. Assume now that the statement hods for a i < j. The previous remark shows that to prove the vanishing of L X j (F) Nis for any presheaf F with F Nis = 0 it suffices to consider the specia case F = H 0 (Y /Y ). Tensoring the compex Č (Y /Y ) with Z tr (X) we get a spectra sequence 2 pq = L X p (H q (Y /Y )) H p+q (Z tr (X) pr tr Č (Y /Y )). Note further that the compex Z tr (X) pr tr Č (Y /Y )) is equa to Č (X Y /X Y ), so that the imit of our spectra sequence coincides with H (X Y /X Y ). This shows that appying to the above spectra sequence the exact functor Nis we get a new spectra sequence which converges to zero: 2 pq = LX p (H q(y /Y )) Nis 0 Appying the induction hypothesis we get immediatey the desired concusion.

19 BLOCH-KATO CONJCTUR AND MOTIVIC COHOMOLOGY 19 Coroary 2.3. Let A be a bounded beow compex (of degree -1) of free presheaves with transfers. Assume that H i (A ) Nis = 0 for a i. Then H i(z tr (X) pr tr A ) Nis = 0 for a i and a X Sm/F. Proof. This statement foows immediatey from Lemma 2.2 in view of the spectra sequence 2 pq = LX p (H q(a )) H p+q (Z tr (X) pr tr A ). We return back to compexes of degree +1 and cohomoogica notations. Coroary 2.3 impies easiy the foowing resut Coroary 2.4. Let A and B be bounded above compexes (of degree +1) of free Nisnevich sheaves with transfers. Assume that either A or B is acycic, then the compex A tr B is acycic as we. Coroary 2.5. Let A 1 A 2 be a quasiisomorphism of bounded above compexes of free Nisnevich sheaves with transfers. Then for any bounded above compex B of free Nisnevich sheaves with transfers the induced homomorphism A 1 tr B A 2 tr B is a quasiisomorphism as we. For any Nisnevich sheaf with transfers F we have a natura epimorphism onto F from a free Nisnevich sheaf with transfers: X Sm/F,φ F(X) Z tr(x) F. Repeating this construction we get a canonica free resoution X (F) F. As aways we reindex X in cohomoogica terms, so that X (F) is a compex of degree +1 concentrated in nonpositive degrees. Appying the functor X to a bounded above compex A of Nisnevich sheaves with transfers we get a free bounded above compex X (A ) and a natura quasiisomorphism X (A ) A. We define the tensor product operation on the category D (NSwT/F) via the formua A L B = X (A ) tr X (B ). Coroary 2.5 shows that this construction is we defined and moreover A L B = A tr B for any compexes A, B of free Nisnevich sheaves with transfers (more generay for any compexes whose entries are direct summands in appropriate free sheaves, i.e. are projective in the category of presheaves with transfers). The foowing properties of the bifunctor L are now straightforward. Lemma 2.6. (1) The functor L is commutative and associative (up to a natura isomorphism) (2) The functor L takes distinguished trianges in each variabe to distinguished trianges. (3) The functor L takes direct sums (of homoogicay bounded famiies) to direct sums. To pass from the category D (NSwT/F) to the category DM (F) we have to verify that the tensor product A L B of two compexes, one of which is in the ocaizing subcategory A is again in A.

20 20 ANDRI SUSLIN AND VLADIMIR VOVODSKY Proposition 2.7. Let A, B be bounded above compexes of Nisnevich sheaves with transfers. Assume that A A, then A L B A. Proof. Since the category A is weaky generated by sheaves of the form Z tr (Y A 1 /Y ) (Y Sm/F) and the category D (NSwT) is weaky generated by sheaves Z tr (X) (X Sm/F) it suffices to note that Z tr (Y A 1 /Y ) L Z tr (X) = Z tr (Y A 1 /Y ) tr Z tr (X) = = Z tr (Y X A 1 /Y X) A. Proposition 2.7 together with Theorem 1.12 show that the tensor structure on the category D (NSwT/F) induces (via the ocaizing functor C ) a tensor structure on the category DM (F). xpicity the tensor product operation on the category DM (F) is given by the formua A B = C (A L B ). The foowing resut sums up the main properties of this tensor product operation. Proposition 2.8. (1) The functor : DM (F) DM (F) DM (F) is commutative and associative (up to a natura isomorphism). (2) The functor takes distinguished trianges in each variabe to distinguished trianges. (3) The functor takes direct sums (of homoogicay bounded famiies) to direct sums. (4) For any A, B D (NSwT) we have a natura isomorphism C (A L B ) C (A ) C (B ). In particuar for a schemes X, Y Sm/F we have a natura isomorphism M(X Y ) = M(X) M(Y ). 3. Motivic Cohomoogy The concept of motivic cohomoogy (as we understand it today) goes back to the beginning of the 80 s when A. Beiinson conjectured that there shoud exist compexes of Zariski sheaves Z(n) on Sm/F which have (among others) the foowing properties: Be1 (Normaization) The compex Z(0) is quasiisomorphic to the constant sheaf Z positioned in degree 0, the compex Z(1) is quasiisomorphic to the sheaf O of invertibe functions positioned in degree 1. Be2 (The Beiinson Soué Vanishing Conjecture) For n > 0 the compex Z(n) is acycic outside the interva [1, n], the n-th cohomoogy sheaf H n (Z(n)) coincides with the sheaf Kn M of Minor K n -groups. Be3 (Reationship to K-theory) For any X Sm/F there exists a natura spectra sequence p,q 2 = H p Zar (X, Z(q)) K 2q p(x)

21 BLOCH-KATO CONJCTUR AND MOTIVIC COHOMOLOGY 21 which rationay degenerates and defines isomorphisms of HZar (X, Z( )) Q with subsequent factors of the γ fitration on K (X) Q. Be4 (The Beiinson-Lichtenbaum Conjecture) For any prime to charf the compex Z/(n) = Z(n) L Z/ is quasiisomorphic to τ n Rπ (µ n ). Here π is the obvious morphism of sites (Sm/F) et (Sm/F) Zar and τ n denotes trancation at the eve n. In this section we define (foowing [V 2]) the motivic compexes Z(n) and estabish some of there eementary properties. Consider the sheaf Z tr (G n m ). According to definitions (see 0) Z tr (G n m ) coincides with Z tr (G n m )/D n (here G m stands for the mutipicative group scheme G m = A 1 \ {0}) and D n is the sum of images of homomorphisms given by the embeddings of the form Z tr (G n 1 m ) Z tr (G n m ) (x 1,..., x n 1 ) (x 1,..., 1,..., x n 1 ). The subsheaf D n is in fact a direct summand of Z tr (G n m ) (see 0). The corresponding projection p : Z tr (G n m ) D n is given by the formua p = I ( 1) card(i) 1 (p I ), where I runs through a non empty subsets of {1,..., n} and p I : G n m G n m is the coordinate morphism, repacing a I-entries by 1 G m. The sheaf Z tr (G n m ) coincides with the compementary direct summand and may be aso identified with the intersection of kernes of homomorphisms (p I ) over a non empty subsets I {1,..., n} (since (p I ) p = (p I ) for any I ). Furthermore, with these notations we have a canonica direct sum decomposition (3.0) Z tr (G n m ) = n m=0 I {1,...,n} card I=m Z tr (G m m ) = n m=0 Z tr (G m m ) (n m) Definition 3.1. The motivic compex Z(n) of weight n on Sm/F is the compex C (Z tr (G n m ))[ n] = C (Z tr (G n m )/D n)[ n]. For a smooth scheme X over F we define its motivic cohomoogy groups HM i (X, Z(n)) as Zariski hypercohomoogy (X, Z(n)). H i Zar Note that Z(n) is a compex of sheaves with transfers in Nisnevich topoogy (actuay even in the etae topoogy) with homotopy invariant cohomoogy presheaves. Thus Coroary 1.2 shows that HZar (X, Z(n)) = H Nis (X, Z(n)). Using further Theorem 1.5 we see that HM i (X, Z(n)) = Hom DM (F)(M(X), Z(n)). The motiv Z(1) is caed the Tate motiv. For any X DM (F) we denote by X(n) the tensor product X Z(n). In particuar for an abeian group A we denote

22 22 ANDRI SUSLIN AND VLADIMIR VOVODSKY by A(n) the tensor product compex A Z(n). Proposition 2.8 impies easiy, that A(n) may be identified with the compex C (Z tr (G n m ) A)[ n]. In particuar Z/(n) = C (Z tr (G n m )/)[ n]. Proposition 2.8 impies that for each m, n we have a canonica isomorphism Z(m) Z(n) Z(m + n) and the foowing diagrams commute for a m, n, p Z(m) Z(n) Z(p) Z(m + n) Z(p) = = Z(m) Z(n + p) Z(m + n + p). Lemma 3.2. (1) The compex Z(0) is canonicay quasisomorphic to the constant sheaf Z, positioned in degree 0, the compex Z(1) is canonicay quasiisomorphic to the sheaf O positioned in degree one. (2) The compex Z(n) is acycic in degrees > n. (3) For a n, m 0 we have a natura isomorphism Z tr (G n m ) tr Z tr (G m m ) = Z tr (G (n+m) m ) and hence a natura quasiisomorphism Z(n) Z(m) = Z(n+ m). Proof. Identification of the compex Z(0) is straightforward. The compex Z(1) is a canonica direct summand in C (Z tr (G m ))[ 1]. To identify Z(1) we start by computing the cohomoogy presheaves of the compex C (Z tr (G m ))[ 1]. Let U Sm/F be any smooth affine scheme. Denoting by H i the i-th cohomoogy presheaf of C (Z tr (G m ))[ 1] we have (using the notations introduced in [S-V]) the foowing identifications H i (U) = H 1 i (C (Z tr (G m ))(U)) = H sin 1 i(u G m /U). Computation of singuar homoogy of reative smooth curves - [S-V 3] shows that H i (U) =0 i 1 H 1 (U) =Pic(U P 1, U 0 U ). The exact sequence reating reative Picard group to the absoute ones shows immediatey that Pic(U P 1, U 0 U ) = O (U) Z. This computation impies easiy that a cohomoogy sheaves of Z(1) except for H 1 vanish and H 1 (Z(1)) = O. Acycicity of the compex Z(n) in degrees > n is obvious from the construction. The ast statement foows from the identification Z tr (G n m ) tr Z tr (G m Z tr (G (n+m) m ) and right exactness of the functor tr. m ) = Remark The proof of the Lemma 3.2 gives the foowing expicit way to identify the presheaf H 1 (Z(1)) with O. To each f O (U) = Hom Sm/F (U, G m ) we associate the cass of its graph Γ f Z tr (G m )(U) in H 1 (Z(1))(U) = Z tr (G 1 m )(U)/ / Im(Z tr (G 1 m )(U A1 )). Vice versa start with an irreducibe cyce Z U G m finite and surjective over U. This cyce defines an invertibe function p 2 O (Z), taking the norm of this function from Z to U gives us an invertibe function f Z = N Z/U (p 2 ) O (U). The two above maps define mutuay inverse isomorphisms of O (U) and H 1 (Z(1))(U) (at east for affine U s).