ON THE GRAYSON SPECTRAL SEQUENCE. Andrei Suslin

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1 ON THE GRAYSON SPECTRAL SEQUENCE Andre Susln Introducton The man purpose of these notes s to show that Grayson s motvc cohomology concdes wth the usual defnton of motvc cohomology - see [V2, S-V] for example and hence Grayson s spectral sequence [Gr] for a smooth semlocal scheme X essentally of fnte type over a feld F takes the form E pq 2 = Hp q (X, Z( q)) = K p q (X). One can use next the globalzaton machnery developed n [S-F] to get a smlar lookng spectral sequence for any smooth scheme of fnte type over a feld. Moreover, t s not hard to see that the resultng spectral sequence concdes wth the one constructed n [F-S]. What s nce, however, wth ths approach s that t avods completely the use of the paper of Bloch and Lchtenbaum [B-L], whch many people stll fnd doubtful, stll s not publshed and possbly never wll be. The man result of the paper says that the canoncal homomorphsm of complexes of sheaves Z Gr (n) Z(n) s a quas-somorphsm. There are essentally three reasons behnd ths quas-somorphsm. Frst, cohomology sheaves of the complex Z Gr (n) are homotopy nvarant K0 -sheaves and hence homotopy nvarant pretheores - M. Walker [W]. Second, the complex Z Gr (n) s defned by a ratonally contractble presheaf, as s the complex Z(n); see Proposton 2.2 below, whch mples vanshng of certan polyrelatve cohomology groups n the semlocal case as n Theorem 2.7 below. Fnally, just as the usual motvc cohomology does, Grayson s motvc cohomology satsfes cohomology purty,.e., f Z Y s a smooth subscheme of pure codmenson d, then we have canoncal Gysn somorphsms H p Z (Y, ZGr (n)) = H p 2d (Z, Z Gr (n d)). Gven these three propertes the proof of the comparson theorem s mmedate. In vew of the frst pont and the usual propertes of homotopy nvarant pretheores (see [V1]) t suffces to establsh that for any feld extenson E/F the correspondng map n motvc cohomology (0.0) H p (E, Z Gr (n)) H p (E, Z(n)) 1 Typeset by AMS-TEX

2 2 ANDREI SUSLIN s an somorphsm. Makng a base change we may even assume that E = F. Followng the method developed n [S-V], we proceed to show that the map (0.0) s an somorphsm by nducton on the weght n. In case n = 0 there s nothng to prove. For any n both sdes are zero for p > n. It s easy to check by a drect computaton that the map n queston s an somorphsm n degree p = n, but we won t use ths computaton. Instead we note that for any p and any m we have a degree shft somorphsm H p (F, Z Gr (n)) = H m+p ( m, { m 1 } m =0 ; ZGr (n)) where on the rght we have polyrelatve cohomology of the m-smplex wth respect to all ts faces. Let Z be the famly of supports on m, consstng of all closed subschemes Z m contanng no vertces. Consder the long exact sequence for polyrelatve cohomology wth supports H m+p 1 ( ˆ m, { ˆ m 1 } m =0; Z Gr (n)) H m+p Z ( m, { m 1 } m =0; Z Gr (n)) H m+p ( m, { m 1 } m =0; Z Gr (n)) H m+p ( ˆ m m 1, { ˆ } m =0; Z Gr (n)) and a natural homomorphsm from ths exact sequence to a smlar sequence for polyrelatve cohomology wth supports wth coeffcents n Z(n). Vanshng of polyrelatve cohomology n the semlocal case mples that H ( ˆ m, { m 1 ˆ } m =0 ; ZGr (n)) = H ( ˆ m m 1, { ˆ } m =0 ; Z(n)) = 0 for > n. Thus takng m large enough we see that the natural map H m+p Z ( m, { m 1 } m =0; Z Gr (n)) H m+p ( m, { m 1 } m =0;Z Gr (n)) = = H p (F, Z Gr (n)) (and a smlar map for Z(n)-cohomology) s an somorphsm. Fnally the map n cohomology wth supports H m+p Z ( m, { m 1 } m =0 ; ZGr (n)) H m+p Z ( m, { m 1 } m =0 ; Z(n)) s an somorphsm n vew of the cohomology purty and the nductve assumpton. Of the three man propertes of Grayson s motvc cohomology the most subtle one undoubtedly s the cohomology purty. My orgnal plan was to use the fbratons produced by Grayson n [Gr] and frst prove cohomology purty for cohomology wth coeffcents n the spectra consttutng Grayson s tower. Ths can be done by frst comparng these cohomology theores to the cohomology theores provded by the tower constructed n [F-S] and then backtrackng the constructon used n [F-S] and returnng to the spectra correspondng to the topologcal fltraton on the K-theory of X. The latter cohomology theores were nvestgated by M. Levne n [Le], who proved, n partcular, a localzaton theorem for them that s equvalent to, or rather slghtly stronger than, the cohomologcal purty theorem,

3 ON THE GRAYSON SPECTRAL SEQUENCE 3 snce t does not requre Z to be smooth. Ths plan s clearly workable but rather lengthy and techncal. Fortunately the recent paper of Voevodsky [V3] makes ths plan unnecessary. In ths paper Voevodsky provdes an ncredbly short and absolutely ngenous proof of a cancellaton theorem for motvc cohomology that s easly adaptable to the Grayson motvc cohomology. Here are the contents of the paper by secton. In 1 we dscuss varous general facts about K0 -presheaves, K 0-presheaves and presheaves wth transfers, recallng, n partcular, some results from M. Walker s thess. The man result of 2, Theorem 2.7, gves vanshng of certan polyrelatve cohomology groups n the semlocal case. In 3 we recall the defnton of Grayson s cohomology and establsh some of ts basc elementary propertes. Secton 4 contans our adaptaton of Voevodsky s proof of the Cancellaton Theorem. In 5 we dscuss at some length Grayson s cohomology wth supports and n 6 we gve the proof of the comparson theorem. We borrowed the dea to use polyrelatve cohomology of the semlocal scheme ˆ n wth respect to ts faces from the paper of Gesser and Levne [G-L], [G-L1], who showed that these polyrelatve cohomology groups can be used as an effectve tool n dealng wth varous problems of the motvc cohomology theory. One of the man achevements of the paper [G-L] was the proof of the fact that Bloch-Kato Conjecture mples Belnson-Lchtenbaum Conjecture over a feld of any characterstc provded one replaces motvc cohomology wth hgher Chow groups. In 7 we show how Theorem 2.7 can be used to sgnfcantly smplfy (as we hope) the arguments used n [S-V] and get rd of the assumpton about resoluton of sngulartes. Ths new verson of the proof of the fact that Bloch-Kato Conjecture mples Belnson-Lchtenbaum Conjecture s shorter, clearer and more to the pont (n my opnon) than the versons presented n [S-V] and [G-L]. Throughout the paper we denote by Sm/F the category of smooth separated schemes of fnte type over the base feld F and by Sch/F the category of all separated schemes of fnte type over F. 1. K 0 -presheaves, K 0-presheaves and presheaves wth transfers. For any X, Y Sm/F set P(X, Y ) = The category of coherent O X Y -modules P such that Supp P s fnte over X and the coherent O X -module (p X ) (P ) s locally free. Note that the second condton mposed on the sheaf P above s equvalent to sayng that P s flat over X. To abbrevate the language we ll refer to sheaves P P(X, Y ) as sheaves fnte and flat over X. Note that the category P(X, Y ) s closed under extensons n the abelan category of all coherent O X Y -modules and hence has a natural structure of an exact category. Moreover, the functor (p X ) : P(X, Y ) Locally free O X -modules

4 4 ANDREI SUSLIN s obvously exact. Consder the followng abelan groups K 0 (X, Y ) = K 0 (P(X, Y )), K 0(X, Y ) = K 0 (P(X, Y )), Cor(X, Y ) = Free abelan group generated by closed ntegral subschemes Z X Y fnte and surjectve over a component of X These groups are related by canoncal homomorphsms K 0 (X, Y ) K 0(X, Y ) Cor(X, Y ) Here the frst map s the obvous surjectve homomorphsm and the second one takes the class [P ] of the coherent sheaf P P(X, Y ) to Z l O X Y,z P z [Z], where the sum s taken over all closed ntegral subschemes Z X Y fnte and surjectve over a component of X and z denotes the generc pont of the correspondng scheme Z. Let X, Y, U Sm/F be three smooth schemes. In ths case we have a natural bfunctor P(X, Y ) P(Y, U) P(X, U) P Q (p X,U ) (p X,Y (P ) O X Y U p Y,U (Q)) One checks easly that the sheaf on the rght really belongs to P(X, U) and moreover the above bfunctor s bexact. Thus we get a natural composton law K 0 (X, Y ) K 0 (Y, U) K 0 (X, Y ) In a smlar way one defnes composton laws K0 (X, Y ) K 0 (Y, U) K 0 (X, U) Cor(X, Y ) Cor(Y, U) Cor(X, U) All these composton laws are assocatve, and that allows us to make Sm/F nto an addtve category n three dfferent ways: takng Hom(X, Y ) to be ether K 0 (X, Y ), K 0(X, Y ) or Cor(X, Y ) respectvely. We denote the resultng addtve categores K 0 (Sm/F ), K 0(Sm/F ) and SmCor/F respectvely. Defnton 1.1. A K0 -presheaf (resp. K 0-presheaf, presheaf wth transfers) on the category Sm/F s an addtve contravarant functor F : K0 (Sm/F ) Ab (resp. K 0 (Sm/F ) Ab, SmCor/F Ab). Remark An advanced theory of presheaves wth transfers was developed by V. Voevodsky [V1], [V2]. K 0 -presheaves and K 0-presheaves were ntroduced and studed by M. Walker [W]. Accordng to what was sad at the begnnng of ths secton we have canoncal functors K 0 (Sm/F ) K 0(Sm/F ) SmCor/F

5 ON THE GRAYSON SPECTRAL SEQUENCE 5 Ths remark mples that every presheaf wth transfers determnes a K 0 -presheaf and every K 0 -presheaf determnes a K0 -presheaf. Note also that we have a canoncal functor Sm/F K0 (Sm/F ) that assocates to every morphsm f : X Y of smooth schemes the class O Γf K0 (X, Y ) of ts graph Γ f X Y. Qute often we use the same notaton f for the K0 -morphsm X Y determned by a morphsm f : X Y of schemes and sometmes (when t can not lead to a confuson) even for the coherent sheaf O Γf that represents ths morphsm. There s another useful constructon present n all three categores K0 (Sm/F ), K 0 (Sm/F ) and SmCor/F. We ll dscuss t only n case of K0 (Sm/F ) leavng the obvous modfcatons needed for two other categores to the reader. Let X, Y, X, Y be four smooth schemes. Let further P P(X, Y ), P P(X, Y ) be sheaves fnte and flat over X and X respectvely. In ths case the external tensor product P P s obvously fnte and flat over X X. Thus we get a bfunctor P(X, Y ) P(X, Y ) P(X X, Y Y ) whch s obvously addtve and bexact. Ths gves a canoncal operaton external tensor product K 0 (X, Y ) K 0 (X, Y ) K 0 (X X, Y Y ) Accordng to what was sad above every K0 -presheaf determnes a presheaf Sm/F Ab n the usual sense. Ths remark allows us to consder presheaves wth transfers (resp. K 0 -presheaves, K0 -presheaves) as presheaves n the usual sense provded wth an approprate addtonal data. In ths secton we ll be mostly dealng wth the followng questons (1) What happens to a K0 -presheaf structure (resp. K 0-presheaf structure, presheaf wth transfers structure) on a presheaf F when we sheaffy t n Zarsk or Nsnevch topologes. (2) Fnd condtons that assure that the gven K0 -presheaf structure on F (resp. K 0 -presheaf structure) may be descended to a K 0 -presheaf structure (resp. presheaf wth transfers structure). Most part of the results presented below are taken from [V1, V2], [W], so unless the result s new we just gve the formulaton and ndcate the reference, sometmes gvng the sketch of the proof (f t s not too long). In the computatons we are about to perform the followng formula computng the composton of the usual morphsms wth ones gven by K0 (X, Y ) s often used. The verfcaton of ths formula s qute straghtforward and we leave t as an easy exercse to the reader. Lemma For any P P(X, Y ) and any morphsm f : X X, g : Y Y of smooth schemes we have the followng denttes [P ] f =[(f 1 Y ) (P )] K 0 (X, Y ) g [P ] =[(1 X g) (P )] K 0 (X, Y )

6 6 ANDREI SUSLIN Recall that a presheaf F : Sm/F Ab s sad to be homotopy nvarant provded that for any X Sm/F the canoncal homomorphsm (p X ) : F(X) F(X A 1 ) s an somorphsm. Lemma 1.2 (M.Walker [W]). Let F be a homotopy nvarant K0 -presheaf on the category Sm/F. Then the K0 -presheaf structure on F descends unquely to a K 0 -presheaf structure. Proof. Unqueness of the K 0 -presheaf structure on F compatble wth the orgnal K0 -presheaf structure s obvous. To prove ts exstence we have to verfy that for any X, Y Sm/F and any short exact sequence 0 P P P 0 n P(X, Y ) the homomorphsm F(Y ) [P ] [P ] [P ] F(X) gven by the K 0 -presheaf structure on F s trval. To do so we consder the followng commutatve dagram wth exact rows n P(X A 1, Y ) (1.2.1) 0 P [T ] P [T ] P [T ] 0 = T 0 P [T ] Q P [T ] 0 where the rght-hand square s a pull-back dagram. Set β = [Q] [P [T ]] [P [T ]] K 0 (X A1, Y ) and consder the followng dagram F(Y ) β F(X A 1 )) 0 F(X) 1 Snce F s homotopy nvarant the homomorphsms 0 and 1 are equal. On the other hand Lemma allows us to conclude that 1 β = (β 1 ) = ([P ] [P ] [P ]), whereas 0β = (β 0 ) = 0, snce the bottom lne of the dagram (1.2.1) splts beng specalzed at T = 0. For the future use we wrte down explctly the propertes of the sheaf Q constructed above. Lemma Startng wth any short exact sequence 0 P P P 0 n P(X, Y ) the above constructon gves us a sheaf Q P(X A 1, Y ) wth the followng propertes: (1) Q 0 = P P ; Q 1 = P (2) [Q 1 ] [Q 0 ] = [P ] [P ] [P ] K0 (X, Y ). Lemma 1.3 (V.Voevodsky). Let F be a homotopy nvarant presheaf wth transfers on the category Sm/F. Then the assocated Zarsk sheaf F Zar has a unque

7 ON THE GRAYSON SPECTRAL SEQUENCE 7 structure of a presheaf wth transfers for whch the canoncal homomorphsm F F Zar s compatble wth transfers. Moreover, F Zar s also homotopy nvarant. Proof. Every presheaf wth transfers s obvously a pretheory n the sense of Voevodsky [V1]. Furthermore, Voevodsky proved n [V1] that the Zarsk sheaf assocated to a homotopy nvarant pretheory s agan a homotopy nvarant pretheory. Ths mples mmedately that F Zar s homotopy nvarant. To show that F Zar s actually a presheaf wth transfers amounts to the verfcaton of the followng two ponts: (1) Let X, Y Sm/F be two smooth schemes, let further Z X Y be a closed ntegral subscheme fnte and surjectve over a component of X and let a F(Y ) be a secton that des n F Zar (Y ), then Z (a) F(X) des n F Zar (X). (2) In the same stuaton as above let a F Zar (Y ) be any secton of F Zar over Y. Then there exsts an open coverng X = X and for each an open Y Y and a secton a F(Y ) such that Z X Y X Y, a Y = φ(a ) φ In both cases t suffces to consder the case when the scheme X s local wth the closed pont x 0 X. Let z 0,..., z n be all ponts of Z over x 0 and let y 0,..., y n be ther mages n Y. Note that for any open U Y, contanng all y the scheme Z s contaned n X U and hence Z (a) = Z (a U ). Consder the semlocalzaton Y y of Y at the set of ponts y 0,..., y n. Accordng to another theorem of Voevodsky [V1], Proposton 4.24 for any pretheory F the sectons of F and F Zar over any smooth semlocal scheme Y y concde,.e., (1.3.1) lm F(U) = lm F Zar (U) U y 0,...,y n U y 0,...,y n Ths formula readly mples that there exsts U y 0,..., y n such that a U = 0 and thus Z (a) = Z (a U ) = 0. Second statement also follows from the formula (1.3.1). Lemma 1.4 (M. Walker [W]). Let F be a homotopy nvarant K 0 -presheaf. Then the assocated Zarsk sheaf F Zar has a unque structure of a K 0 -presheaf for whch the canoncal homomorphsm F φ F Zar s a homomorphsm of K 0 -presheaves. The presheaf F Zar s homotopy nvarant and moreover has a canoncal structure of a homotopy nvarant pretheory. Proof. Walker verfed n [W] that homotopy nvarant K 0 -presheaves satsfy many propertes enjoyed by homotopy nvarant pretheores. In partcular, they satsfy the formula (1.3.1). To show that F Zar has a unque structure of a K 0 -sheaf t suffces now to repeat the prevous argument replacng everywhere the closed subscheme Z by a coherent sheaf P P(X, Y ) ( fnte and flat over X). The remanng facts are not to hard to prove ether see [W] for detals.

8 8 ANDREI SUSLIN Remark Wth some more work one can show that n condtons of Lemma 1.4 the Zarsk sheaf F Zar actually has a canoncal structure of a Zarsk sheaf wth transfers. Snce for most practcal purposes pretheores are as good as sheaves wth transfers we don t try to gve (a rather lengthy) proof of ths fact here. Lemma 1.5. Let F be a K0 -presheaf (resp. K 0-presheaf, presheaf wth transfers). Then the assocated Nsnevch sheaf F Ns has a unque structure of K0 -presheaf (resp. K 0 -presheaf, presheaf wth transfers) for whch the canoncal homomorphsm F φ F Ns s a homomorphsm of K0 -presheaves (resp. of K 0-presheaves, presheaves wth transfers). Proof. In case of presheaves wth transfers ths fact s proved n [V2] Lemma In case of K 0 -presheaves and K 0-presheaves the proof s essentally the same as for presheaves wth transfers, so we just sketch t brefly. Once agan we have to verfy two thngs (1) Let X, Y Sm/F be two smooth schemes, let further P P(X, Y ) be a coherent sheaf fnte and flat over X. Let fnally a F(Y ) be a secton that des n F Ns (Y ). Then P (a) F(X) des n F Ns (X). (2) In the same stuaton as above let a F Ns (Y ) be any secton of F Ns over Y. Then there exsts a Nsnevch coverng {X X} and for each a g scheme Y Y over Y, a sheaf P P(X, Y ) and a secton a F(Y ) such that P f = g P, a Y = φ(a ) Ths tme, snce we are workng now n Nsnevch topology, we may assume X to be a Henselan local scheme. The closed subscheme Supp P = Spec O X Y / Ann P X Y beng fnte over X s a dsjont sum of local Henselan schemes Supp j. Ths decomposton of Supp P determnes a canoncal drect sum decomposton P = j P j wth support of P j beng equal to Supp j. It suffces clearly to treat the case P = P j,.e., we may assume that Supp P s a local Henselan scheme wth closed pont z Supp P. Let y be the mage of z n Y. By the assumpton there exsts an étale neghborhood (U, u) s (Y, y) such that a U = 0. The scheme (Supp P, z) beng Henselan we conclude from the defntons that the morphsm p Y : (Supp P, z) (Y, y) factors unquely through (U, u). The correspondng morphsm Supp P X U s obvously a closed embeddng. Ths allows us to defne a coherent O X U -module Q, whose support dentfes wth that of P under the above closed embeddng. Obvously Q P(X, U) and (1 X s) (Q) = P,.e., [P ] = s [Q] K 0 (X, Y ). Fnally P (a) = Q (s (a)) = 0. We end ths secton wth an applcaton of the prevous results to polyrelatve cohomology. For any presheaf F : Sm/F Ab we denote by C n (F) a new presheaf on the category Sm/F defned by the formula C n (F)(X) = F(X n ). Thus C (F) s a smplcal presheaf and we use the notaton C (F) for the correspondng non-negatve complex of sheaves wth dfferental (of degree 1) equal to the alternatng sum of face operatons. We use the notaton C (F) for the same complex rendexed cohomologcally (.e., C = C ). f

9 ON THE GRAYSON SPECTRAL SEQUENCE 9 Let X Sm/F be a smooth scheme and let {X } n =0 be a famly of subschemes n X such that all ntersectons X 0... X k are smooth. We denote by Z Zar (X; {X } n =0 ) the followng complex (wth dfferental of degree +1) of Zarsk sheaves:... 1 <...< k Z Zar (X 1... X k )... Z Zar (X ) Z Zar (X) Here Z Zar (X) stands n degree zero, 1 <...< k Z Zar (X 1... X k ) stands n degree k and the dfferental s the alternatng sum of maps nduced by nclusons. We use the notaton Z Ns (X; {X } n =0 ) for the obvous counterpart of the above complex wth Zarsk topology replaced by the Nsnevch topology. If F s any complex of Zarsk sheaves we defne polyrelatve cohomology of X wth respect to {X } n =0 as (hyper)ext-groups for complexes of sheaves: H p Zar (X, {X } n =0 ; F ) = Ext p (Z Zar (X; {X } n =0 ), F ) In a smlar way one defnes polyrelatve Nsnevch cohomology wth coeffcents n a complex of Nsnevch sheaves. The followng Lemma summarzes some of the standard (and obvous) propertes of polyrelatve cohomology. We do not specfy the topology n the formulaton of ths Lemma snce t works for both the Zarsk and the Nsnevch topologes. Lemma 1.6. a. We have a natural long exact sequence H p (X, {X } n =0 ; F ) H p (X, {X } n 1 =0 ; F ) H p (X n, {X X n } n 1 =0 ; F ) δ b. We have a natural spectral sequence δ H p+1 (X, {X } n =0; F ) E 1 pq = 1 <...< q H p (X 1... X q, F ) = H p+q (X, {X } n =0 ; F ) c. Lettng F I be an njectve (or flasque) resoluton of the complex F, the polyrelatve cohomology H (X, {X } n =0 ; F ) concdes wth cohomology of the bcomplex (1.6.1) I (X) I (X )... 1 <...< k I (X 1... I (X k )... For any complex of presheaves I and any smooth scheme X, provded wth a famly of closed subschemes {X } n =0 such that all ntersectons X 0... X k are smooth we use the notaton I (X, {X } n =0 ) for the bcomplex (1.6.1). Thus the last statement of Lemma 1.6 may be rephrased by sayng that for a flasque resoluton I of F we have a natural dentfcaton H (X, {X } n =0 ; F ) = H (I (X, {X } n =0 )). Ths remark shows, n partcular, that for any complex of sheaves F we have canoncal homomorphsms H (F (X, {X } n =0)) H (I (X, {X } n =0)) = H (X, {X } n =0; F ).

10 10 ANDREI SUSLIN Proposton 1.7. Assume that F s a K0 -presheaf. Assume further that X s a smooth semlocal scheme provded wth the famly {X } n =0 of closed subschemes such that all ntersectons X 0... X k are smooth (semlocal) schemes. Then the canoncal map H (C (F)(X, {X } n =0 )) H (C (F) Zar (X,{X } n =0 )) HZar(X, {X } n =0; C (F) Zar ) s an somorphsm. Proof. The standard spectral sequence argument shows that t suffces to deal wth the absolute case,.e., to show that canoncal homomorphsms C (F)(X) HZar (X, C (F) Zar ) s an somorphsm. Denote by H q the q-th cohomology presheaf of C (F). Then H q s a homotopy nvarant K0 -presheaf and hence a homotopy nvarant K 0 -presheaf (see Lemma 1.2). Consder next the hypercohomology spectral sequence E pq 2 = Hp Zar (X, Hq Zar ) = Hp+q Zar (X, C (F) Zar ) Accordng to Lemma 1.4 the sheaves H q Zar are homotopy nvarant pretheores and hence H p Zar (X, Hq Zar ) = 0 for p > 0 see [V1], Lemma Ths mples that the above hypercohomology spectral sequence degenerates and provdes somorphsms H q Zar (X, C (F) Zar ) = Γ(X, H q Zar ). Fnally, as was mentoned n the proof of Lemma 1.4, M. Walker has shown that for any homotopy nvarant K 0 -presheaf H q and any smooth semlocal scheme X the natural map Γ(X, H q ) Γ(X, H q Zar ) s an somorphsm. Remark For any K0 -presheaf F and any smooth scheme X, provded wth the famly {X } n =0 of closed subschemes such that all ntersectons X 0... X k are smooth, the canoncal map H Zar(X, {X } n =0; C (F) Zar ) H Ns(X, {X } n =0; C (F) Ns ) s an somorphsm. Thus one can replace Zarsk cohomology n 1.7 wth Nsnevch cohomology. Proof. Once agan t suffces to treat the absolute case. Snce cohomology sheaves of the complex C (F) Zar are homotopy nvarant pretheores our statement follows mmedately from general propertes of such sheaves see [V1]. 2. Ratonally contractble presheaves. In ths secton we prove one of the man new results of ths paper (Theorem 2.7), whch gves vanshng of certan polyrelatve cohomology groups n the semlocal case. It should be mentoned that ths Theorem s a motvc cohomology verson of a result proved by Gesser and Levne n [G-L1] for hgher Chow groups. The present formulaton s however much more general and the proof we hope s much more understandable.

11 ON THE GRAYSON SPECTRAL SEQUENCE 11 For any presheaf F : Sm/F Ab let C 1 F denote the followng presheaf C 1 F(X) = lm F(U) X {0,1} U X A 1 Note that there are two obvous presheaf homomorphsms (restrctons to X 0 and X 1 respectvely) 0, 1 : C 1 F F. Defnton 2.1 ([S-V] 9). A presheaf F s called ratonally (or genercally) contractble f there exsts a presheaf homomorphsm s : F C 1 F, such that 0 s = 0, 1 s = 1 F. The followng result, whch s a mnor generalzaton of [S-V] (9.6), serves as a man supply of ratonally contractble sheaves. Proposton 2.2. Let X Sm/F be a smooth connected scheme and let x 0 X be a ratonal pont of X. Assume that there exsts an open subscheme W X A 1, contanng X {0, 1} x 0 A 1 and a morphsm of schemes f : W X such that f X 0 = x 0, f X 1 = 1 X, f x0 A 1 = x 0. Then the followng presheaves are ratonally contractble: (1) Z(X)/Z(x 0 ) (2) Z tr (X)/Z tr (x 0 ) (3) U K 0 (U, X)/K 0 (U, x 0 ) (4) U K 0 (U, X)/K 0 (U, x 0) Proof. We start wth the easest case, that of the presheaf Z(X)/Z(x 0 ). Let g : Y X be a morphsm of schemes over F. Consder the morphsm g 1 A 1 : Y A 1 X A 1 and let U be the nverse mage of W under ths morphsm. Thus U Y A 1 s an open subscheme, contanng Y {0, 1}, and the composton f (g 1 A 1) defnes a morphsm s(g) : U X. Thus assocatng to g Z(X)(Y ) the secton s(g) Z(X)(U) we get a homomorphsm of presheaves s : Z(X) C 1 Z(X), such that 1 s = 1 Z(X), 0 s = x 0. Moreover, the prescrbed behavor of f wth respect to x 0 mples readly that the homomorphsm s takes Z(x 0 ) to C 1 Z(x 0 ) and hence defnes the ratonal homotopy operator s : Z(X)/Z(x 0 ) C 1 (Z(X)/Z(x 0 )) wth the requred propertes. In the remanng cases the proof s essentally the same; one should nterpret the sectons of the above presheaves as morphsms n an approprate sense and make sure that the constructons performed above make sense for these morphsms. We gve detals concernng the last two presheaves and leave the second presheaf (whch was treated n [S-V], (9.6) ) to the reader. Start wth a coherent sheaf P on Y X fnte and flat over Y and denote by Z Y X ts support. Note that the sheaf P 1 A 1 s a coherent O Y A 1 X A 1-module fnte and flat over Y A1. Moreover, the support of ths sheaf concdes wth Z A 1. Let further T X A 1 be the closed subscheme complement of W. The ntersecton (Z A 1) [(Y A 1 ) T ] (Y A 1 ) (X A 1 ) does not contan ponts where the second A 1 -coordnate s

12 12 ANDREI SUSLIN equal to 0 or 1, on the other hand both A 1 -coordnates on ths ntersecton concde and hence t does not contan ponts where the frst A 1 coordnate s equal to 0 or 1. Moreover, ths ntersecton s obvously fnte over Y A 1. The above remarks mply that p Y A 1((Z A 1) [(Y A 1 ) T ]) Y A 1 s a closed subscheme not ntersectng Y {0, 1}. Denote by U = U P Y A 1 the complementary open subscheme. By the very constructon we have the followng ncluson: [U (X A 1 )] (Z A 1) U W The restrcton of the sheaf P 1 A 1 to U (X A 1 ) s fnte and flat over U and ts support s contaned n U W, whch mples readly that the restrcton (P 1 A 1) U W s stll fnte and flat over U. Fnally we set Q = (1 U f) ((P 1 A 1) U W ). The sheaf Q s a coherent O U X -module, fnte and flat over U. One checks easly that assocatng to P the class of Q n lm Y {0,1} U Y A 1 K 0 (U, X) (resp. n lm Y {0,1} U Y A K 1 0 (U, X) defnes a functon on the category P(Y, X) whch s addtve wth respect to all short exact sequences (resp. to splt short exact sequences only) and hence defnes a homomorphsm (resp. a homomorphsm s : K 0 (Y, X) lm K 0 (U, X) Y {0,1} U Y A 1 s : K0 (Y, X) lm K0 (U, X). Y {0,1} U Y A 1 Fnally one checks easly that s s a homomorphsm of presheaves and has all the requred propertes. Remark 2.3. Note that a drect summand n a ratonally contractble presheaf s obvously also ratonally contractble. Thus Grayson s presheaf U K0 (U, G n m ) beng a drect summand n U K0 (U, G n m )/K0 (U, ) s ratonally contractble. Lemma 2.4. Let F be a ratonally contractble presheaf. Then the presheaf C n (F) s equally ratonally contractble. Proof. The choce of a vertex v n defnes a splttng of the presheaf C n (F) nto a drect sum of F and the kernel Cn 1 (F) of the restrcton map defned by v: v : C n (F) F. Moreover, the sheaf Cn(F) 1 s contractble (see [S-V] Lemma 1.9) and hence ratonally contractble. For any n 0 denote by ˆ n the semlocalzaton of n at the set of ts vertces v 0,..., v n n. Snce all the structure maps of the cosmplcal scheme take vertces to vertces we conclude that ˆ s a cosmplcal semlocal scheme. Applyng a presheaf F to ths cosmplcal scheme we get a smplcal abelan group,.e., a complex. The followng elementary property of ratonally contractble sheaves s the bass for all further applcatons.

13 ON THE GRAYSON SPECTRAL SEQUENCE 13 Proposton 2.5. Assume that the presheaf F s ratonally contractble. Then the complex F( ˆ ) s contractble and hence acyclc. Proof. For each n 0 the homomorphsm s defnes a map s ˆ : F( ˆ n ) C n 1 F( ˆ n ) = F( n ˆ A 1 ) where the latter hat ndcates that the scheme n A 1 s semlocalzed wth respect to the set of ts vertces,.e., ponts v 0, v 1. Consder fnally the usual trangularzaton of n A 1, defned by the famly of maps ψ : n+1 n A 1 (0 n), where ψ s the lnear somorphsm takng the vertex v j to v j 0 f j and to v j 1 1 f j >. Once agan ψ take vertces to vertces and hence defne maps on the correspondng semlocalzatons. The contractng homotopy operator for the complex F( ˆ ) s gven by the formula σ(u) = n ( 1) (ψ ) (s(u)) =0 Consder a smooth semlocal scheme ˆ n and the famly of ts closed subschemes n 1 { ˆ } n =0. Snce all ntersectons are obvously smooth we may consder (for any presheaf F) the correspondng non-negatve complex F( ˆ n n 1, { ˆ } n =0 ). Corollary 2.6. Assume that the presheaf F s ratonally contractble. Then the non-negatve complex F( ˆ n n 1, { ˆ } n =0 ) s acyclc n postve degrees, and ts zerodmensonal homology group concdes wth the ntersecton of the kernels of all the face maps : F( ˆ n ) F( ˆ n 1 ): H 0 (F( ˆ n, { n 1 ˆ } n =0 ) = F( ˆ n ker def ) = n =0 Ker( : F( ˆ n ) F( ˆ n 1 ). ˆ n 1 Proof. Rendex the complex F( ˆ n, { } n =0 ) homologcally and then shft t so that F( ˆ n ) stands n degree n. It was proved n [F-S] Theorem 1.2, that the resultng homologcal complex s canoncally quas-somorphc to (σ n M(F( ˆ )). Here σ n denotes the operaton of stupd truncaton at level n and M(F( ˆ )) s the Moore complex correspondng to the smplcal abelan group F( ˆ ). Proposton 2.5 shows that the complex F( ˆ ) s acyclc and hence σ n M(F( ˆ ) s acyclc n degrees n. Returnng to our orgnal cohomologcal complex we conclude that the complex F( ˆ n n 1, { ˆ } n =0 s acyclc n degrees 0. Fnally the statement concernng ts zero-dmensonal homology s obvous. Theorem 2.7. Let F be a ratonally contractble K0 -presheaf. Then the canoncal embeddng C (F)( ˆ n ) ker C (F)( ˆ n n 1, { ˆ } n =0 ) s a quas-somorphsm. In partcular, the complex C (F)( ˆ n n 1, { ˆ } n =0 ) s acyclc n postve degrees and hence H p Zar ( ˆ n, { n 1 ˆ } n =0; C (F) Zar ) = H p Ns ( ˆ n n 1, { ˆ } n =0; C (F) Ns ) = 0

14 14 ANDREI SUSLIN for p > 0. Proof. The frst statement follows mmedately from Corollary 2.6 snce presheaves C n (F) are ratonally contractble accordng to Lemma 2.4. The second statement follows from the frst one Proposton 1.7 and Remark Grayson s Cohomology. For any X Sm/F the category P(X, G n m ) may be dentfed wth the category of vector bundles P over X equpped wth an n-tuple (α 1,..., α n ) of commutng automorphsms. We denote by K0 (Gn m ) the presheaf of abelan groups on Sm/F gven by the formula K0 (Gn m )(X) = K 0 (P(X, Gn m )). Canoncal embeddngs G n 1 m k G n m (α 1,..., α n 1 ) (α 1,..., 1 k,..., α n 1 ) k = 1,..., n defne embeddngs of presheaves K0 (Gn 1 m ) ( k) K 0 (G n m). Moreover, one checks easly that the sum of mages of these embeddngs s a canoncal drect summand n K0 (Gn m) see for example [S-V], 0. We defne the presheaf K0 (G n m ) as the quotent of K 0 (Gn m ) modulo n k=0 ( k) (K 0 (Gn 1 m )). We defne the complex of Zarsk sheaves Z Gr (n) va the formula Z Gr (n) = C (K 0 (G n m )) Zar[ n] Fnally we defne Grayson s motvc cohomology of X as the Zarsk hypercohomology of X wth coeffcents n the complex of sheaves Z Gr (n). Proposton 3.1. a) The complex Z Gr (0) concdes wth Z. b) For any n the cohomology sheaves of Z Gr (n) are homotopy nvarant pretheores and, n partcular, are strctly homotopy nvarant. Hence for any X Sm/F we have canoncal somorphsms H Zar (X, ZGr (n)) =H Zar (X, ZGr (n) Ns ) = H Ns (X, ZGr (n) Ns ) H Zar(X, Z Gr (n)) =H Zar(X A 1, Z Gr (n)) Proof. The frst statement s obvous. The second statement follows from Lemma 1.2, Lemma 1.4 and the standard propertes of homotopy nvarant pretheores see [V1]. To smplfy notaton we keep the same notaton Z Gr (n) for the Nsnevch sheaffcaton Z Gr (n) Ns of the complex Z Gr (n). Proposton 3.1 shows that ths does not lead to any knd of confuson. For any schemes X, Y Sm/F and any ntegers m, n 0 we have canoncal parngs see 1 K 0 (G n m )(X) K0 (G m m )(Y ) K0 (G (n+m) m )(X Y )

15 ON THE GRAYSON SPECTRAL SEQUENCE 15 These parngs defne further a canoncal homomorphsm of complexes C (K0 (G n m ))(X) C (K0 (G m))(y ) Tot C, (K0 (G (n+m) ))(X Y ) m m C (K 0 (G (n+m) m ))(X Y ) Here C p,q (K0 (G (n+m) m ))(X Y ) = K0 (G (n+m) m )(X Y p q ) and the second arrow s the shuffle map see e.g., [S-V] 0. Specalzng now to the case X = Y and composng the resultng homomorphsm of complexes wth the ( X ) map C (K0 (G (n+m) m ))(X X) C (K0 (G (n+m) m ))(X) we get a homomorphsm of complexes of abelan presheaves, whch upon sheaffcaton and shft defnes canoncal homomorphsms of complexes of sheaves Z Gr (n) Z Gr (m) µ n,m Z Gr (n + m). It s well-known (and easy to verfy) that the product maps µ n,m are strctly assocatve and commutatve. Ths observaton mples readly the followng result. Proposton 3.2. For any X Sm/F H (X, Z Gr (n)) n=0 s a bgraded assocatve rng wth dentty, whch s graded commutatve wth respect to the cohomologcal degree. Proposton 3.3. The presheaf K 0 (G n m ) s ratonally contractble and hence n, m 0 H p ( ˆ m m 1, { ˆ } m =0 ; ZGr (n)) = 0 for p > n Proof. The frst statement follows from Proposton 2.2 and Remark 2.3 and the second statement follows from Theorem 2.7. Recall that motvc complexes Z(n) are defned n a way very smlar to the one we used above to defne complexes Z Gr (n) see [V2, S-V]. Denote by Z tr (G n m ) the presheaf of abelan groups on Sm/F gven by the formula Z tr (G n m )(X) = Cor(X, G n m). It s easy to see that ths presheaf s actually a sheaf n the Nsnevch topology so that ths tme no sheaffcaton s needed. A straghtforward verfcaton shows that the sum of the mages of the homomorphsms Z tr (Gm n 1 ) Z tr (G n m) nduced by the canoncal embeddngs k : Gm n 1 Gn m (1 k n) s a drect summand n Z tr (G n m) (see [S-V] 0). One defnes the Nsnevch sheaf wth transfers Z tr (G n m ) as the quotent of Z tr(g n m ) modulo n k=0 ( k) (Z tr (Gm n 1 )). Fnally one defnes the complex of Nsnevch sheaves wth transfers Z(n) va the formula Z(n) = C (Z tr (G n m ))[ n] As was explaned at the begnnng of 1 we have canoncal homomorphsms of presheaves K0 (Gn m ) Z tr(g n m ). These homomorphsms nduce obvously homomorphsms K0 (G n m ) Z tr (G n m ), from whch we derve n an obvous way homomorphsms of complexes of Nsnevch sheaves f n : Z Gr (n) Z(n).

16 16 ANDREI SUSLIN Lemma 3.4. The homomorphsms f n : Z Gr (n) Z(n) are compatble wth products. Moreover the homomorphsm f 0 s an somorphsm of complexes. Proof. The second statement s trval snce both presheaves K0 (Spec F ) and Z tr (Spec F ) concde wth the constant sheaf Z. The frst statement s also clear snce product maps Z(m) Z(n) Z(m + n) may be defned n exactly the same way as product maps Z Gr (m) Z Gr (n) Z Gr (m + n) were defned above cf. [S-V] 2, [F-S] The Cancellaton Theorem for Grayson s Cohomology. In ths secton we repeat, essentally verbatm, Voevodsky s proof of the Cancellaton Theorem for motvc cohomology, replacng everywhere motvc cohomology by Grayson s cohomology. The only dfference s that we have to modfy slghtly Voevodsky s argument to make sure that t works n K0 -context. We also replace the fnal part of the Voevodsky proof wth an explct constructon of a homotopy operator. For any scheme X Sm/F and any abelan presheaf F : Sm/F Ab the group F(X) dentfes canoncally as a drect summand n F(X G m ). We use the notaton F(X G m ) for the complementary drect summand,.e., F(X G m ) = Ker(F(X G m ) res F(X e) = F(X)), where e G m s the dentty element. Fx smooth schemes X and Y. For our applcatons only the affne case s of nterest, so we ll be assumng that both X and Y are affne. Ths assumpton makes many constructons more transparent, however, as the reader wll see, t s rrelevant for the valdty of results proved below. For ths reason we ll formulate results n complete generalty but wll make verfcatons n the affne case only. Let P be a coherent sheaf on X G m Y G m fnte and flat over X G m. Denote by f 1 and f 2 the two nvertble functons on X G m Y G m correspondng to the two projectons onto G m. Proposton 4.1. a) For any n 0 the sheaf P/(f1 n+1 1)P s fnte and flat over X. b) There exsts an nteger N such that for any n N the sheaf P/(f1 n+1 f 2 )P s fnte and flat over X. Proof. Let A = F [X] be the coordnate rng of the affne scheme X. The frst part of our statement s clear: we know that P s a fntely generated projectve module over the Laurent polynomal rng A[f 1, f1 1 n+1 ] and clam that n ths case P/(f1 1)P s a fntely generated projectve A-module, whch s perfectly obvous. To prove the second part we note that multplcaton by f 2 determnes an automorphsm of the A[f 1, f 1 1 ]-module P, whch we denote by α. Our clam concerns the propertes of the module Coker(f n+1 1 α). To show that ths module s fntely generated over A we note that, consdered as an A[f 1, f1 1 ]-module, t s klled by det(f1 n+1 α). Denote by χ = χ α the characterstc polynomal of α. Thus χ = χ(t ) s a monc polynomal n T of degree d = rank P : χ(t ) = T d + b 1 T d b d.

17 ON THE GRAYSON SPECTRAL SEQUENCE 17 Here b A[f 1, f1 1 ] are Laurent polynomals n f 1. The constant term b d concdes wth det(α) and hence s nvertble n A[f 1, f1 1 ]. Thus t s of the form b d = a f1 s for some s Z and some a A. Note that det(f n+1 1 α) = χ(f n+1 1 ) = f d(n+1) 1 + b 1 (f 1 ) (d 1)(n+1) b d 1 f n a f s 1 A straghtforward examnaton shows that f we take n large enough then det(f1 n+1 α) f1 s s a monc polynomal n f 1 of degree d(n+1) s and wth nvertble constant term (equal to a). Ths readly mples that Coker(f1 n+1 α) s a fntely generated module over A[f 1, f1 1 n+1 ]/ det(f1 α) f1 s = A[f 1 ]/ det(f1 n+1 α) f1 s and hence s a fntely generated A-module. Moreover, under the same condtons on n, we see that the homomorphsm f1 n+1 α : P P s njectve and hence Coker(f1 n+1 α) has a two-term flat resoluton over A, whch may be used to compute the Tor-groups. To show that the fntely generated A-module Coker(f n+1 1 α) s projectve t suffces to show that for any prme deal µ A the group Tor A 1 (P/(f1 n+1 α)p, k(µ)) s trval. However, ths Tor 1 -group concdes wth the kernel of the homomorphsm f n+1 1 α(µ) : P A k(µ) P A k(µ) and the kernel of ths endomorphsm s trval snce ths kernel s a submodule n a fntely generated projectve k(µ)[f 1, f1 1 ]-module and s klled by the monc polynomal det(f1 n+1 α(µ)) f1 s. For any coherent sheaf P on X G m Y G m that s fnte and flat over X G m we denote by ρ + n (P ) the drect mage of P/(f1 n+1 1)P under the projecton X G m Y G m X Y. Accordng to Proposton 4.1 the sheaf ρ + n (P ) s fnte and flat over X. We ll say that ρ n (P ) s defned provded that the sheaf P/(f n+1 1 f 2 )P s fnte and flat over X, n whch case we defne ρ n (P ) to be the drect mage of P/(f1 n+1 f 2 )P under the projecton X G m Y G m X Y. Proposton 4.1 shows that ρ n (P ) s defned for all n large enough, and when defned ρ n (P ) s a coherent sheaf on X Y fnte and flat over X. Fnally we ll say that ρ n (P ) s defned provded that ρ n (P ) s defned, n whch case we set ρ n (P ) = [ρ + n (P )] [ρ n (P )] K0 (X, Y ) Lemma 4.2. a) For any P 1, P 2 and any n, there s an somorphsm ρ + n (P 1 P 2 ) = ρ + n (P 1) ρ + n (P 2). b) Assume that ρ n (P 1 ) and ρ n (P 2 ) are defned. Then ρ n (P 1 P 2 ) s defned as well and ρ n (P 1 P 2 ) = ρ n (P 1 ) ρ n (P 2 ). In ths case we also have the dentty ρ n (P 1 P 2 ) = ρ n (P 1 ) + ρ n (P 2 ) K 0 (X, Y ) Lemma 4.3. Let Q be a coherent sheaf on X Y fnte and flat over X. (1) The sheaf ρ n (Q 1 G m ) s defned for all n 0. Moreover, ρ + n (Q 1 G m ) = Q n+1 ρ n (Q 1 Gm ) = Q n

18 18 ANDREI SUSLIN and hence ρ n (Q 1 Gm ) = [Q]. (2) The sheaf ρ n (Q e G m ) s defned for all n 0 (here e : G m G m s the trval morphsm that takes G m to the unt e G m ). Moreover, and hence ρ n (Q e Gm ) = 0 ρ + n (Q e Gm ) = Q n+1 ρ n (Q e G m ) = Q n+1 Proof. Let A = F [X], B = F [Y ] be the correspondng coordnate rngs. The A B[f 1, f 1 1, f 2, f 1 2 ]-module Q 1 G m /f n+1 1 f 2 concdes wth Q F F [f 1, f 1 1, f 2, f 1 2 ] /(f 1 f 2, f1 n+1 f 2 ) = Q F F [f 1, f1 1 n+1 ]/(f1 f 1 ). Consdered as an A-module ths module s obvously fntely generated projectve and moreover s somorphc to Q n. Smlarly Q e Gm /f1 n+1 f 2 concdes wth Q F F [f 1, f1 1, f 2, f2 1 ]/(f 2 1, f1 n+1 f 2 ) = Q F F [f 1, f1 1 n+1 ]/(f1 1). Ths module s fntely generated and projectve over A and s somorphc to Q n+1. The computatons concernng ρ + n are performed smlarly. Lemma 4.4. Let P be a coherent sheaf on X G m Y G m fnte and flat over X G m. Assume that ρ n (P ) s defned. (1) For any morphsm g : X X of smooth schemes the element ρ n ((g 1 Gm Y G m ) P ) s also defned and moreover we have the followng natural somorphsms ρ + n ((g 1 G m Y G m ) P ) = (g 1 Y ) (ρ + n (P )) ρ n ((g 1 G m Y G m ) P ) = (g 1 Y ) (ρ n (P )) (2) For any morphsm g : Y Y of smooth schemes the element ρ n ((1 X Gm g 1 Gm ) P ) s also defned and moreover we have the followng natural somorphsms ρ + n ((1 X G m g 1 Gm ) P ) = (1 X g) (ρ + n (P )) ρ n ((1 X G m g 1 Gm ) P ) = (1 X g) (ρ n (P )) Proof. Set B = F [Y ], A = F [X], A = F [X ]. The (A B)[f 1, f1 1, f 2, f2 1 ]-module (g 1 Gm Y G m ) P )/(f1 n+1 f 2 ) concdes wth A A P/(f1 n+1 f 2 ). Ths module s obvously fntely generated and projectve over A and moreover t concdes also wth (g 1 Y ) (ρ n (P )). The part concernng ρ + n s establshed smlarly. The second part s even easer snce n ths case the correspondng module does not change at all. Here s one more result of the same character establshed n the same absolutely straghtforward fashon. We skp the obvous detals.

19 ON THE GRAYSON SPECTRAL SEQUENCE 19 Lemma 4.5. Let P be a coherent sheaf on X G m Y G m fnte and flat over X G m. Assume that ρ n (P ) s defned. Then for any scheme X Sm/F ρ n (P 1 X ) s also defned. Moreover, we have the followng dentfcatons: ρ + n (P 1 X ) = ρ+ n (P ) 1 X ρ n (P 1 X ) = ρ n (P ) 1 X Next we have to construct, followng [S-V] 3 certan standard homotopes. Proposton 4.6. There exsts a coherent sheaf H on (G m G m A 1 ) (G m G m ) fnte and flat over (G m G m A 1 ) and such that [H 0 ] [H 1 ] = [σ] [1 Gm ] + [(a, e)] [(b, e)] [(e, a)] [(e, b)] + 2 [(e, e)] where σ : G m G m G m G m s the permutaton of coordnates morphsm, : G m G m s the nverson morphsm (x) = x 1, a and b are the frst and the second coordnate functons on G m G m respectvely and e = 1 G m s the dentty element. We start wth a few auxlary homotopes. Lemma Let a and b be two nvertble functons on a smooth scheme S. Then there exsts a coherent sheaf P on S A 1 G m, fnte and flat over S A 1 and such that [P 0 ] [P 1 ] = [ab] [a] [b] + [1] K 0 (S, G m) Proof. Let X denote the G m -coordnate functon and T denote the A 1 -coordnate functon. Consder the closed subscheme Y S A 1 G m gven by the equaton X 2 (T (a + b) + (1 T )(1 + ab))x + ab = 0 An mmedate verfcaton shows that ths scheme s fnte and flat over S A 1. The correspondng subschemes Y 0, Y 1 S G m are gven by the equatons (X 1)(X ab) = 0 and (X a)(x b) = 0 respectvely. Denotng the coordnate rng of S by A we note further that we have short exact sequences of A[X, X 1 ]-modules 0 A[X] X ab 0 A[X] X a X 1 X b A[X] (X ab)(x 1) A[X] (X a)(x b) A[X] X 1 0 A[X] X b 0 From these short exact sequences of A[X, X 1 ]-modules we derve (see (1.2.2)) coherent sheaves P 1 and P 2 on S A 1 G m fnte and flat over S A 1 for whch P0 1 = A[X] X ab A[X] X 1, P 1 1 = A[X] (X ab)(x 1) P0 2 = A[X] (X a)(x b), P 1 2 = A[X] X a A[X] X b Now t suffces to take P = O Y P 1 P 2.

20 20 ANDREI SUSLIN Corollary Let a, b and c be three nvertble functons on a smooth scheme S. Then there exsts a coherent sheaf Q on S A 1 G m G m, fnte and flat over S A 1 and such that [Q 0 ] [Q 1 ] = [(ab, c)] [(a, c)] [(b, c)] + [(1, c)]. Proof. Consder the closed embeddng c : S A 1 G m S A 1 G m G m, gven by the formula c (s, t, x) = (s, t, x, c(s)) and take Q = ( c ) (P ). Corollary Let a and b be two nvertble functons on a smooth scheme S. Then there exsts a coherent sheaf R on S A 1 G m G m, fnte and flat over S A 1 and such that [R 0 ] [R 1 ] = [(ab, ab)] [(a, a)] [(b, b)] + [(1, 1)] Proof. Consder the closed embeddng : S A 1 G m S A 1 G m G m, gven by the formula (s, t, x) = (s, t, x, x) and take R = (P ). Proof of Proposton 4.6 Take S = G m G m and denote by a and b the frst and the second coordnate functons respectvely. Accordng to Corollares and we have the followng homotopes between K 0 -morphsms from G m G m to tself: [(ab, ab)] [(a, a)] + [(b, b)] [(1, 1)] [(ab, ab)] [(a, ab)] + [(b, ab)] [(1, ab)] [(a, a)] + [(a, b)] [(a, 1)]+ + [(b, b)] + [(b, a)] [(b, 1)] [(1, ab)] These relatons combne to gve the followng homotopy: [(a, b)] + [(b, a)] [(a, 1)] + [(b, 1)] + [(1, ab)] [(1, 1)] Furthermore, [(a, b)] + [(a, b 1 )] 2[(a, 1)] and hence [σ] = [(b, a)] [(a, b 1 )] [(a, 1)] + [(b, 1)] + [(1, ab)] [(1, 1)]. Fnally Corollary mples that we have the followng relaton: [(1, ab)] [(1, 1)] [(1, ab)] [(1, 1)] [(1, a)] + [(1, b)] 2 [(1, 1)]. Start wth an element v K 0 (X G m, Y G m ) and consder the element v (1 Gm e) K 0 (X G m G m, Y G m G m ). Denote further by v σ (1 Gm e) the element (1 Y σ) (v (1 Gm e)) (1 X σ) obtaned from v (1 Gm e) by permutng two copes of G m both n X G m G m and n Y G m G m. Consder fnally the followng homotopy φ = φ v = (1 Y H) (((v (1 Gm e)) (1 X Gm )) 1 A 1) + (1 Y σ) (v (1 Gm e)) (1 X H) K 0 (X G m G m A 1, Y G m G m ) Lemma Assume that (1 Y e) v = v (1 X e) = 0. Then φ 0 φ 1 = v σ (1 Gm e) v (1 Gm e). Proof. Ths follows mmedately from the propertes of the homotopy H and the fact that v (1 Gm e) s klled by 1 Y (a, e); 1 Y (b, e); 1 Y (e, a); 1 Y (e, b); 1 Y (e, e) on the left and by smlar elements wth 1 Y replaced by 1 X on the rght.

21 ON THE GRAYSON SPECTRAL SEQUENCE 21 Theorem 4.7. For any X, Y Sm/F the natural homomorphsm C K 0 (Y )(X) C K 0 (Y G m)(x G m ) : u u (1 Gm e Gm ) s a quas-somorphsm of complexes. Proof. Note that C n K0 (Y G m)(x G m ) s a drect summand n K0 (X n G m, Y G m ) consstng of all those elements v K0 (X n G m, Y G m ) for whch v (1 X n e) = (1 Y e) v = 0. A straghtforward verfcaton shows that the elements u (1 Gm e Gm ) have both requred propertes and hence the map n queston s well-defned. Denote by L(X, Y ) the free abelan group generated by somorphsm classes of coherent sheaves P on X Y fnte and flat over X. Thus we have a canoncal surjectve homomorphsm L(X, Y ) K 0 (X, Y ). For an element V L(X G m, Y G m ) we ll say that the operaton ρ n (V ) s defned f ρ n (P ) s defned for every P appearng n V wth 0 coeffcent. Proposton 4.1 shows that for any V L(X G m, Y G m ) the operaton ρ n (V ) s defned for all n large enough. Moreover, Lemma 4.2 mples mmedately the followng fact. (4.7.1) If the mage of V n K0 (X G m, Y G m ) s trval then ρ n (V ) = 0 for all n large enough. We frst show that the homomorphsm C (K0 (Y ))(X) C (K0 (Y G m))(x G m ) nduces njectve maps on homology. Snce both sdes are smplcal abelan groups we may work wth the correspondng Moore complexes. Thus assume that u K0 (X n, Y ) s a Moore cycle for whch u (1 Gm e Gm ) s a boundary,.e., there exsts v K0 (X n+1 G m, Y G m ) such that n+1 (v) = u (1 Gm e Gm ), (v) = 0 for 0 n. Represent u and v by approprate elements U L(X n, Y ), V L(X n+1 G m, Y G m ) and fnd N such that ρ N (V ) s defned. Lemma 4.4 shows that ρ N ( (V )) s also defned and moreover (ρ N (V )) = ρ N ( (V )). Fact (4.7.1) shows next that upon ncreasng N we wll get the followng denttes: (ρ N (V )) = ρ N ( (V )) = 0 for 0 n n+1 (ρ N (V )) = ρ N ( n+1 (V )) = ρ N (U (1 Gm e Gm )) = u where the latter dentty follows from Lemma 4.3. Ths mples that u tself s a boundary and concludes the proof of the njectvty statement. We next prove the surjectvty of the nduced map n homology. Let v C n (K0 (Y G m))(x G m ) be a Moore cycle. Thus v K0 (X n G m, Y G m ) has the followng propertes: v (1 X n e) = (1 Y e) v = 0, (v) = 0 for 0 n Choose a representatve V L(X n G m, Y G m ) for v, consder the homotopy φ = φ v K 0 (X n G m G m A 1, Y G m G m ) and choose a representatve Φ L(X n G m G m A 1, Y G m G m ) for φ. Accordng to Lemma we have (4.7.2) [Φ 0 ] [Φ 1 ] = [V σ (1 Gm e)] [V (1 Gm e)]

22 22 ANDREI SUSLIN In the computatons we are about to perform we ll be usng the operatons ρ N wth respect to the newly adjoned copy of G m. Note that, accordng to (4.3), (4.5) and (4.4.2) we have the followng formulae. ρ N (V (1 Gm e)) = [V ] = v for all N 0 ρ N (V σ (1 Gm e)) = [ρ N (V )] (1 Gm e) for all N large enough Consder fnally the element ψ N = ρ N (Φ) K 0 (X n G m A 1, Y G m ) defned for all suffcently large N. Accordng to (4.7.1) and (4.7.2) for all N large enough we have the followng denttes: (ψ N ) 0 (ψ N ) 1 = ρ N (Φ 0 Φ 1 ) = v [ρ N (V )] (1 Gm e) Moreover, accordng to (4.4.1) and (4.7.1) for all N large enough we get: (ψ N ) = ρ N ( Φ) = 0, snce [ Φ] = φ v = 0. The latter relatons show that for N large enough ψ N s a Moore cycle of the smplcal abelan group C (K 0 (Y G m))(x G m A 1 ). Snce the two restrctons C (K 0 (Y G m))(x G m A 1 ) C (K 0 (Y G m ))(X G m ) defned by ratonal ponts 0, 1 A 1 defne the same map n homology we conclude that (ψ N ) 0 (ψ N ) 1 = v [ρ N (V )] (1 Gm e) s a boundary n C (K 0 (Y G m))(x G m ). Fnally the complex C (K 0 (Y G m))(x G m ) s a drect summand n C (K 0 (Y G m))(x G m ) and hence v [ρ N (V )] (1 Gm e) s a boundary n C (K 0 (Y G m))(x G m ) as well. Theorem 4.8. For any schemes X, Y Sm/F the natural homomorphsm H Zar (X, C (K 0 (Y )) Zar) H Zar (X G m, C (K 0 (Y G m)) Zar ) defned as the external multplcaton by 1 Gm e H 0 (G 1 m, C (K 0 (G 1 m )) Zar ) s an somorphsm. Proof. We frst compute H Zar (X G m, C (K 0 (Y G m)) Zar ) usng Leray spectral sequence E pq 2 = Hp (X, R q π (C (K 0 (Y G m)) Zar X Gm )) = = H p+q (X G m, C (K 0 (Y G m)) Zar ) where π : X G M X s the obvous projecton and we use the notaton X to denote the restrcton of a sheaf or a complex of sheaves onto the small Zarsk ste of X. Denote by H q the q-th cohomology presheaf of the complex C (K 0 (Y G m)) and by H q = H q Zar denote the q-th cohomology sheaf of the complex C (K 0 (Y G m )) Zar. To compute the complex Rπ (C (K 0 (Y G m)) Zar X Gm ) we use the hypercohomology spectral sequence E pq 2 = Rp π (H q X Gm ) = H p+q (Rπ (C (K 0 (Y G m)) Zar X Gm )) The stalks (R p π (H q X Gm )) x = H p (X x G m, H q ) are trval for p > 0 snce H q s a homotopy nvarant pretheory - see [V1] Lemma Thus E pq 2 = 0 for p 0 and