Variations on the Bloch-Ogus Theorem

Transcription

1 Documenta Math. 51 Varatons on the Bloch-Ogus Theorem Ivan Pann, Krll Zanoullne Receved: March 24, 2003 Communcated by Ulf Rehmann Abstract. Let R be a sem-local regular rng of geometrc type over a feld k. Let U = SpecR be the sem-local scheme. Consder a smooth proper morphsm p : Y U. Let Y k(u) be the fber over the generc pont of a subvarety u of U. We prove that the Gersten-type complex for étale cohomology 0 H q ét (Y,C) Hq ét (Y k(u),c) u U (1) H q 1 ét (Y k(u),c( 1))... s exact, where C s a locally constant sheaf wth fnte stalks of Z/nZmodules on Y et and n s an nteger prme to char(k) Mathematcs Subject Classfcaton: 14F20, 16E05 Keywords and Phrases: étale cohomology, arthmetc resoluton 1 Introducton The hstory of the subject of the present paper starts wth the famous paper of D. Qullen [14] where he proves the geometrc case of the Gersten s conjecture for K-functor. One may ask whether the smlar result holds for étale cohomology. The frst answer on ths queston was gven by S. Bloch and A. Ogus n [2]. They proved the analog of Gersten s conjecture for étale cohomology wth coeffcents of n-th roots of unty. More precsely, let X be a smooth quas-projectve varety over a feld k and let x = {x 1,...,x m } X be a fnte subset of ponts. We denote by U = Spec O X,x the sem-local scheme at x. Consder the sheaf µ n of n-th roots of unty on the small étale ste X et, wth n prme to char(k). Then the man result of [2] (see Theorem 4.2 and n the twsted sheaf µ n

2 52 Ivan Pann, Krll Zanoullne Example 2.1) mples that the Gersten-type complex for étale cohomology wth supports 0 H q (U,µ n ) Hu(U,µ q n ) (U,µ n ) ( ) u U (0) Hu q+1 u U (1) s exact for all Z and q 0, where U (p) denotes the set of all ponts of codmenson p n U. The next step was done by O. Gabber n [8]. He proved that the complex ( ) s exact for cohomology wth coeffcents n any torson sheaf C on X et that comes from the base feld k,.e., C = p C for some sheaf C on (Speck) et and the structural morphsm p : X Spec k. It turned out that the proof of Gabber can be appled to any cohomology theory wth supports that satsfes the same formalsm as étale cohomology do. Ths dea was realzed n the paper [4] by J.-L. Collot-Thélène, R. Hoobler and B. Kahn. Namely, they proved that a cohomology theory wth support h whch satsfes some set of axoms [4, Secton 5.1] s effaceable [4, Defnton 2.1.1]. Then the exactness of ( ) follows mmedately by trval reasons [4, Proposton 2.1.2]. In partcular, one gets the exactness of ( ) for the case when U s replaced by the product U k T, where T s a smooth varety over k [4, Theorem 8.1.1]. It was also proven [4, Remark (3), Corollary B.3.3] the complex ( ) s exact for the case when dm U = 1 and the sheaf of coeffcents s replaced by a bounded below complex of sheaves, whose cohomology sheaves are locally constant constructble, torson prme to char(k). The goal of the present paper s to prove the latter case for any dmenson of the scheme U. Namely, we want to prove the followng µ n 1.1 Theorem. Let X be a smooth quas-projectve varety over a feld k. Let x = {x 1,...,x m } X be a fnte subset of ponts and U = Spec O X,x be the sem-local scheme at x. Let C be a bounded below complex of locally constant constructble sheaves of Z/nZ-modules on X et wth n prme to char(k), Then the E 1 -terms of the conveau spectral sequence yeld an exact complex 0 H q (U, C) Hu(U, q C) Hu q+1 (U, C) u U (0) u U (1) of étale hypercohomology wth supports. 1.2 Remark. For the defnton of a constructble sheaf we refer to [1, IX] or [9, V.1.8]. Observe that a locally constant sheaf wth fnte stalks provdes an example of a locally constant constructble sheaf (see [1, IX.2.13]). 1.3 Corollary. Let R be a sem-local regular rng of geometrc type over a feld k. We denote by U = SpecR the respectve sem-local affne scheme. Let C be a bounded below complex of locally constant constructble sheaves of Z/nZ-modules on U et wth n prme to char(k), Then the complex 0 H q ét (U, C) Hq ét (Spec k(u), C) H q 1 ét (Spec k(u), C( 1)) u U (1)

3 Varatons on the Bloch-Ogus Theorem 53 s exact, where k(u) s the functon feld of U, k(u) s the resdue feld of u and C() = C µ n. Proof. Follows by purty for étale cohomology (see the proof of [4, 1.4]). 1.4 Corollary. Let R be a sem-local regular rng of geometrc type over a feld k. Let p : Y U be a smooth proper morphsm, where U = Spec R. Let C be a locally constant constructble sheaf of Z/nZ-modules on Y et wth n prme to char(k). Then the complex 0 H q ét (Y,C) Hq ét (Y k(u),c) H q 1 ét (Y k(u),c( 1)) u U (1) s exact, where Y k(u) = Speck(u) U Y. Proof. The cohomology of Y wth coeffcents n C concde wth the hypercohomology of U wth coeffcents n the total drect mage Rp C [9, VI.4.2]. Observe that the bounded complex Rp C has locally constant constructble cohomology sheaves. Now by the man result of paper [12], a bounded complex of sheaves on U et wth locally constant constructble cohomology sheaves s n the derved category somorphc to a bounded complex of locally constant constructble sheaves. Hence, there exsts a bounded complex C of locally constant constructble sheaves that s quas-somorphc to the complex Rp C. Replace Rp C by C and apply the prevous corollary. 1.5 Remark. The assumptons on the sheaf C are essental. As t was shown n [7] for the case k = C and C = Z there are examples of extensons Y/U for whch the map H q DR (Y ) Hq DR (Y k(u)) s not njectve. 1.6 Remark. The njectvty part of Theorem 1.1 (.e., the exactness at the frst term) has been proven recently n [15] by extendng the arguments of Voevodsky [16]. The structure of the proof of Theorem 1.1 s the followng. Frst, we gve some general formalsm (sectons 2, 3 and 4). Namely, we prove that any functor F that satsfes some set of axoms (homotopy nvarance, transfers, fnte monodromy) s effaceable (Theorem 4.7). Then we apply ths formalsm to étale cohomology (secton 5). More precsely, we check that the étale cohomology functor F(X,Z) = HZ (X, C) satsfes all the axoms and, hence, s effaceable. It mples Theorem 1.1 mmedately. We would lke to stress that our axoms for the functor F are dfferent from those n [4]. The key pont of the proof s that we use Geometrc Presentaton Lemma of Ojanguren and Pann (see Lemma 3.5) nstead of Gabber s. Ths fact together wth the noton of a functor wth fnte monodromy allows us to apply the technques developed n [10], [11] and [17]. Acknowledgments Ths paper s based on deas of an earler unpublshed manuscrpt of the frst author. Both authors want to thank INTAS project and TMR network ERB FMRX CT for fnancal support.

4 54 Ivan Pann, Krll Zanoullne 2 Defntons and Notatons 2.1 Notaton. In the present paper all schemes are assumed to be Noetheran and separated. By k we denote a fxed ground feld. A varety over k s an ntegral scheme of fnte type over k. To smplfy the notaton sometmes we wll wrte k nstead of the scheme Spec k. We wll wrte X 1 X 2 for the fbered product X 1 k X 2 of two k-schemes. By U we denote a regular sem-local scheme of geometrc type over k,.e., U = Spec O X,x for a smooth affne varety X over k and a fnte set of ponts x = {x 1,...,x n } of X. By X we denote a relatve curve over U (see 3.1.()). By Z and Y we denote closed subsets of X. By Z and Z we denote closed subsets of U. Observe that X and U are essentally smooth over k and all schemes X, Z, Y, Z, Z are of fnte type over U. 2.2 Notaton. Let U be a k-scheme. Denote by Cp(U) a category whose objects are couples (X,Z) consstng of an U-scheme X of fnte type over U and a closed subset Z of the scheme X (we assume the empty set s a closed subset of X). Morphsms from (X,Z) to (X,Z ) are those morphsms f : X X of U-schemes that satsfy the property f 1 (Z ) Z. The composte of f and g s g f. 2.3 Notaton. Denote by F : Cp(U) Ab a contravarant addtve functor from the category of couples Cp(U) to the category of (graded) abelan groups. Recall that F s addtve f one has an somorphsm F(X 1 X 2,Z 1 Z 2 ) = F(X 1,Z 1 ) F(X 2,Z 2 ). Sometmes we shall wrte F Z (X) for F(X,Z) havng n mnd the notaton used for cohomology wth supports. Now notons of a homotopy nvarant functor, a functor wth transfers and a functor that satsfes vanshng property wll be gven. 2.4 Defnton. A contravarant functor F : Cp(U) Ab s sad to be homotopy nvarant f for each U-scheme X smooth or essentally smooth over k and for each closed subset Z of X the map F Z (X) F Z A 1(X A 1 ) nduced by the projecton X A 1 X s an somorphsm. 2.5 Defnton. One says a contravarant functor F : Cp(U) Ab satsfes vanshng property f for each U-scheme X one has F(X, ) = Defnton. A contravarant functor F : Cp(U) Ab s sad to be endowed wth transfers f for each fnte flat morphsm π : X X of U-schemes and for each closed subset Z X t s gven a homomorphsm of abelan groups Tr X X : F π 1 (Z)(X ) F Z (X) and the famly {Tr X X } satsfes the followng propertes:

5 Varatons on the Bloch-Ogus Theorem 55 () for each fbered product dagram of U-schemes wth a fnte flat morphsm π X f X 1 π π1 f X X 1 and for each closed subset Z X the dagram F(f 1) F Z (X ) F Z 1 (X 1) Tr X X Tr X 1 X 1 F Z (X) F(f) F Z1 (X 1 ) s commutatve, where Z = π 1 (Z), Z 1 = f 1 (Z) and Z 1 = π 1 1 (Z 1); () f π : X 1 X 2 X s a fnte flat morphsm of U-schemes, then for each closed subset Z X the dagram + F Z (X 1 X 2) F Z 1 (X 1) F Z 2 (X 2) Tr X 1 X 2 X F Z (X) Tr X 1 X 1 +Tr X 2 X 2 s commutatve, where Z = π 1 (Z), Z 1 = X 1 Z and Z 2 = X 2 Z ; () f π : (X,Z ) (X,Z) s an somorphsm n Cp(U), then two maps Tr X X and F(π) are nverses of each other,.e. 3 The Specalzaton Lemma F(π) Tr X X = Tr X X F(π) = d. The followng defnton s nspred by the noton of a good trple used by Voevodsky n [16]. 3.1 Defnton. Let U be a regular sem-local scheme of geometrc type over the feld k. A trple (X,δ,f) consstng of an U-scheme p : X U, a secton δ : U X of the morphsm p and a regular functon f Γ(X, O X ) s called a perfect trple over U f X, δ and f satsfy the followng condtons: () the morphsm p can be factorzed as p : X π A 1 U pr U, where π s a fnte surjectve morphsm and pr s the canoncal projecton on the second factor; () the vanshng locus of the functon f s fnte over U;

6 56 Ivan Pann, Krll Zanoullne () the scheme X s essentally smooth over k and the morphsm p s smooth along δ(u); (v) the scheme X s rreducble. 3.2 Remark. The property () says that X s an affne curve over U. The property () mples that X s a regular scheme. Snce X and A 1 U are regular schemes by [6, 18.17] the morphsm π : X A 1 U from () s a fnte flat morphsm. The followng lemma wll be used n the proof of Theorems 4.2 and Lemma. Let U be a regular sem-local scheme of geometrc type over an nfnte feld k. Let (p : X U,δ : U X,f Γ(X, O X )) be a perfect trple over U. Let F : Cp(U) Ab be a homotopy nvarant functor endowed wth transfers whch satsfes vanshng property (see 2.4, 2.6 and 2.5). Then for each closed subset Z of the vanshng locus of f the followng composte vanshes F Z (X) F(δ) F δ 1 (Z)(U) F(dU) F p(z) (U) 3.4 Remark. The mentoned composte s the map nduced by the morphsm δ : (U,p(Z)) (X, Z) n the category Cp(U). Observe that we have δ 1 (Z) p(z), where p(z) s closed by () of 3.1. Proof. Consder the commutatve dagram n the category Cp(U) (X, Y) δ d X (X, Z) δ (U,Z ) d U (U,Z) where Z = δ 1 (Z), Z = p(z) and Y = p 1 (Z ). It gves the relaton F(d U ) F(δ) = F(δ) F(d X ). Thus to prove the theorem t suffces to check that the followng composte vanshes F Z (X) F(dX) F Y (X) F(δ) F Z (U) ( ) By Lemma 3.5 below appled to the perfect trple (X,δ,f) we can choose the fnte surjectve morphsm π : X A 1 U from () of 3.1 n such a way that t s fbers at the ponts 0 and 1 of A 1 look as follows: (a) π 1 ({0} U) = δ(u) D 0 (scheme-theoretcally) and D 0 X f ; (b) π 1 ({1} U) = D 1 and D 1 X f. Observe that Y = π 1 (A 1 Z ). Let Z 0 = π 1 ({0} Z ) D 0 and Z 1 = π 1 ({1} Z ) be the closed subsets of Y. By defnton Z 0, Z 1 are the closed subsets of D 0 and D 1 respectvely. Snce Z s contaned n the vanshng locus

7 Varatons on the Bloch-Ogus Theorem 57 of f and D 0, D 1 X f we have Z D 0 = Z D 1 =. The latter means that there are two commutatve dagrams n the category Cp(U) (X, Y) d X (X, Z) I 0 I 0 (D 0, Z 0) d D0 (D0, ) and (X, Y) d X (X, Z) I 1 I 1 (D 1, Z 1) d D1 (D1, ) where I 0, I 1 are the closed embeddngs D 0 X and D 1 X respectvely. By vanshng property 2.5 we have F(D 0, ) = F(D 1, ) = 0. Then applyng F to the dagrams we mmedately get F(I 0 ) F(d X ) = 0 and F(I 1 ) F(d X ) = 0. (1) Let 0, 1 : U A 1 U be the closed embeddngs whch correspond to the ponts 0 and 1 of A 1 respectvely. The homotopy nvarance property 2.4 mples that F( 0 ) = F( 1 ) : F A1 Z (A1 U) F Z (U). (2) The base change property 2.6.() appled to the fbered product dagram (X, Y) π I 1 (D1, Z 1) π gves the relaton (A 1 U, A 1 Z ) 1 (U,Z ) F( 1 ) Tr X A 1 U = Tr D1 U F(I 1), (3) where Tr X A 1 U : F Y (X) F A1 Z (A1 U) and Tr D1 U : F Z 1 (D 1) F Z (U) are the transfer maps for the fnte flat morphsm π and π D1 respectvely. Consder the commutatve dagram F(d X) (F(δ),F(I 0)) F Z (X) F Y (X) F Z Z (U D 0 0) F Z (U) F Z 0 (D 0 ) Tr X A 1 U Tr d+tr D 0 U F A 1 Z (A1 U) F(0) F Z (U) (4) where the central square commutes by 2.6.() and the rght trangle commutes by 2.6.(). In the dagram we dentfy U wth δ(u) by means of the somorphsm δ : U δ(u) and use the property 2.6.() to dentfy Tr δ(u) U wth F(δ). The followng chan of relatons shows that the composte ( ) vanshes and we fnsh the proof of the lemma. F(δ) F(d X ) (1) = (d+tr D0 U ) (F(δ),F(I 0)) F(d X ) (4) = F( 0 ) Tr X A 1 U F(d X ) (2) = +

8 58 Ivan Pann, Krll Zanoullne F( 1 ) Tr X A 1 U F(d X ) (3) = Tr D1 U F(I 1) F(d X ) (1) = 0 The followng lemma s the sem-local verson of Geometrc Presentaton Lemma [10, 10.1] 3.5 Lemma. Let R be a sem-local essentally smooth algebra over an nfnte feld k and A an essentally smooth k-algebra, whch s fnte over the polynomal algebra R[t]. Suppose that e : A R s an R-augmentaton and let I = ker e. Assume that A s smooth over R at every prme contanng I. Gven f A such that A/Af s fnte over R we can fnd an s A such that 1. A s fnte over R[s]. 2. A/As = A/I A/J for some deal J of A. 3. J + Af = A. 4. A(s-1)+Af=A. Proof. In the proof of [10, 10.1] replace the reducton modulo maxmal deal by the reducton modulo radcal of the sem-local rng. 4 The Effacement Theorem We start wth the followng defnton whch s a slghtly modfed verson of [4, 2.1.1]. 4.1 Defnton. Let X be a smooth affne varety over a feld k. Let x = {x 1,...,x n } be a fnte set of ponts of X and let U = Spec O X,x be the semlocal scheme at x. A contravarant functor F : Cp(X) Ab s effaceable at x f the followng condton satsfed: Gven m 1, for any closed subset Z X of codmenson m, there exst a closed subset Z U such that (1) Z Z U and codm U (Z ) m 1; F(j) F(d U) (2) the composte F Z (X) F Z U (U) F Z (U) vanshes, where j : U X s the canoncal embeddng and Z U = j 1 (Z). 4.2 Theorem. Let X be a smooth affne varety over an nfnte feld k and x X be a fnte set of ponts. Let G : Cp(k) Ab be a homotopy nvarant functor endowed wth transfers whch satsfes vanshng property (see 2.4, 2.6 and 2.5). Let F = p G denote the restrcton of G to Cp(X) by means of the structural morphsms p : X Spec k. Then F s effaceable at x.

9 Varatons on the Bloch-Ogus Theorem 59 Proof. We may assume x Z n non-empty. Indeed, f x Z = then the theorem follows by the vanshng property of F (see 2.5). Let f 0 be a regular functon on X such that Z s a closed subset of the vanshng locus of f. By Qullen s trck [14, 5.12], [13, 1.2] we can fnd a morphsm q : X A n 1, where n = dmx, such that (a) q f=0 : {f = 0} A n 1 s a fnte morphsm; (b) q s smooth at the ponts x; (c) q can be factorzed as q = pr Π, where Π : X A n s a fnte surjectve morphsm and pr : A n A n 1 s a lnear projecton. Consder the base change dagram for the morphsm q by means of the composte r : U = Spec O X,x j X q A n 1. X r X X p U q r A n 1 So we have X = U A n 1 X and p, r X denote the canoncal projectons on U, X respectvely. Let δ : U X = U A X be the dagonal embeddng. Clearly δ s a secton of p. Set f = rx (f). Take nstead of X t s rreducble component contanng δ(u) and nstead of f t s restrcton to ths rreducble component (snce x Z s non-empty the vanshng locus of f on the component contanng δ(u) s non-empty as well). Now assumng the trple (p : X U,δ,f) s a perfect trple over U (see 3.1 for the defnton) we complete the proof as follows: Let Z = r 1 X (Z) be the closed subset of the vanshng locus of f. Let Z = p(z) be a closed subset of U. Snce r X δ = j we have δ 1 (Z) = j 1 (Z) = Z U. By Specalzaton Lemma 3.3 appled to the perfect trple (X,δ,f) and the functor j F : Cp(U) Ab the composte vanshes. In partcular, the composte F Z (X) F(δ) F Z U (U) F(dU) F Z (U) F Z (X) F(j) F Z U (U) F(dU) F Z (U) ( ) vanshes as well. Clearly Z Z U. By 3.1.() we have dm X = dm U+1. On the other hand the morphsm r X : X X s flat (even essentally smooth) and, thus, codm X (Z) = codm X Z. Therefore, we have codm U (Z ) = codm X (Z) 1 = m 1. Hence, t remans to prove the followng:

10 60 Ivan Pann, Krll Zanoullne 4.3 Lemma. The trple (p : X U, δ, f) s perfect over U. Proof. By the property (c) one has q = pr Π wth a fnte surjectve morphsm Π : X A n and a lnear projecton pr : A n A n 1. Takng the base change of Π by means of r : U A n 1 one gets a fnte surjectve U-morphsm π : X A 1 U. Ths checks () of 3.1. Snce the closed subset {f = 0} of X s fnte over A n 1 the closed subset {f = 0} of X s fnte over U and we get 3.1.(). Snce q s smooth at x the morphsm r : U A n 1 s essentally smooth. Thus the morphsm r X : X X s essentally smooth as the base change of the morphsm r. The varety X s smooth over k mples that X s essentally smooth over k as well. Snce q s smooth at x the morphsm p s smooth at each pont y X wth r X (y) x. In partcular p s smooth at the ponts δ(x ) (x U). Snce U s sem-local δ(u) s sem-local and p s smooth along δ(u). Ths checks () of 3.1. Snce X s rreducble 3.1.(v) holds. And we have proved the lemma and the theorem. To prove Theorem 4.2 n the case when F s defned over some smooth affne varety we have to put an addtonal condton on F. In order to formulate ths condton we ntroduce some notatons. 4.4 Notaton. Let ρ : Y X X be a fnte étale morphsm together wth a secton s : X Y over the dagonal embeddng : X X X,.e., ρ s =. Let pr 1, pr 2 : X X X be the canoncal projectons. We denote p 1, p 2 : Y X to be the composte pr 1 ρ, pr 2 ρ respectvely. For a contravarant functor F : Cp(X) Ab consder t s pull-backs p 1F and p 2F : Cp(Y ) Ab by means of p 1 and p 2 respectvely. From ths pont on we denote F 1 = p 1F and F 2 = p 2F. By defnton we have F (Y Y,Z) = F(Y Y p X,Z). 4.5 Remark. In general case the functors F 1 and F 2 are not equvalent. Moreover, the functors pr 1F and pr 2F are dfferent. But n the case when F comes from the base feld k,.e., F = p G where p : X Spec k s the structural morphsm and G : Cp(k) Ab s a contravarant functor, these functors concde wth each other. 4.6 Defnton. We say a contravarant functor F : Cp(X) Ab has a fnte monodromy of the type (ρ : Y X X,s : X Y ), where ρ s a fnte étale morphsm and s s a secton of ρ over the dagonal, f there exsts an somorphsm Φ : F 1 F 2 of functors on Cp(Y ). A functor F : Cp(X) Ab s sad to be a functor wth fnte monodromy f F has a fnte monodromy of some type. 4.7 Theorem. Let X be a smooth affne varety over an nfnte feld k and x X be a fnte set of ponts. Let F : Cp(X) Ab be a homotopy nvarant functor endowed wth transfers whch satsfes vanshng property (see 2.4, 2.6 and 2.5). If F s a functor wth fnte monodromy then F s effaceable at x.

11 Varatons on the Bloch-Ogus Theorem 61 Proof. Smlar to the proof of Theorem 4.2 let f 0 be a regular functon on X such that Z s a closed subset of the vanshng locus of f. We may assume x Z s non-empty. Consder the fbered product dagram from the proof of Theorem 4.2 X r X X p U q r A n 1 We have the projecton p : X = U A n 1 X U, the secton δ : U X of p and the regular functon f = rx (f). Snce F s the functor wth fnte monodromy there s a fnte étale morphsm ρ : Y X X, a secton s : X Y of ρ over the dagonal embeddng and a functor somorphsm Φ : F 1 F 2 as n 4.4 and 4.6. Consder the base change dagram for the morphsm ρ : Y X X by means of the composte g : X (p,rx) U X (j,d) X X X g Y ρ X ρ g X X Then ρ s a fnte étale morphsm and there s the secton δ : U X of the composte p = p ρ : X U such that ρ δ = δ ( δ s the base change of the morphsm s : X Y by means of g : X Y ). Set f = ρ 1 (f). As n the proof of 4.2 we replace X by t s rreducble component contanng δ(u) and f by t s restrcton to ths component. By Lemma 4.8 below the trple ( p : X U, δ, f) s perfect. Let Z = r 1 X (Z) and Z = ρ 1 (Z). Set Z = p(z). Observe that p( Z) = Z. The commutatve dagram F(r X) F(j) F Z (X) F Z U (U) F(dU) F Z (U) F(δ) F( δ) F Z (X) F( ρ) F Z( X) shows that to prove the relaton F(d U ) F(j) = 0 (compare wth ( ) of the proof of 4.2) t suffces to check the relaton F(d U ) F( δ) = 0. Consder the pull-backs of the functors F 1, F 2 and the functor somorphsm Φ by means of the morphsm g : X Y. We shall use the same notaton F1, F 2 and Φ for these pull-backs tll the end of ths proof. So we have F 1 = g (p 1F) and F 2 = g (p 2F). The somorphsm Φ : F 1 = F 2 of functors over Cp( X)

12 62 Ivan Pann, Krll Zanoullne provdes us wth the followng commutatve dagram (F 1 ) Z( X) Φ = (F 2 ) Z( X) F 1( δ) (F 1 ) Z U (U) F1(dU) (F 1 ) Z (U) Φ = = Φ F 2( δ) (F 2 ) Z U (U) F2(dU) (F 2 ) Z (U) where the structure of an X-scheme on U s gven by δ. Snce r X ρ = p 2 g we have ρ (r X F) = F 2. Thus to check the relaton F(d U ) F( δ) = 0 for the functor F we have to verfy the same relaton F 2 (d U ) F 2 ( δ) = 0 for the functor F 2. Then by commutatvty of the dagram t suffces to prove the relaton F 1 (d U ) F 1 ( δ) = 0 for the functor F 1. Snce j p = p 1 g we have p (j F) = F 1 : Cp( X) Ab. Thereby t suffces to prove the relaton G(d U ) G( δ) = 0 for the functor G = j F : Cp(U) Ab. Ths relaton follows mmedately from Theorem 3.3 appled to the functor G, the trple ( p : X U, δ, f) and the closed subset Z X. 4.8 Lemma. The trple ( X, δ, f) s perfect over U. Proof. Observe that the trple (p : X U,δ,f) s perfect by Lemma 4.3, the morphsm ρ : X X s fnte étale and ρ δ = δ for the secton δ : U X of the morphsm p : X U. For the fnte surjectve morphsm of U-schemes π : X A 1 U the composte X ρ X π A 1 U s a fnte surjectve morphsm of U-schemes as well. Ths proves 3.1.(). Snce ρ s fnte and the vanshng locus of f s fnte over U the vanshng locus of the functon f s fnte over U as well. Ths proves 3.1.(). Snce ρ δ = δ, ρ s étale and p s smooth along δ(u) the morphsm p s smooth along δ(u). Snce the scheme X s essentally smooth over k and ρ s étale the scheme X s essentally smooth over k. Ths proves 3.1.(). Snce X s rreducble we have 3.1.(v). And the lemma s proven. 5 Applcatons to Étale Cohomology 5.1 Defnton. Let X be a smooth affne varety over a feld k, n an nteger prme to char(k) and C a bounded below complex of locally constant constructble sheaves of Z/nZ-modules on X et. Snce étale cohomology commute wth nductve lmts, we may suppose the complex C s also bounded from above. We consder the functor F : Cp(X) Ab whch s gven by F(Y f X,Z) = H Z(Y,f C), where the object on the rght hand sde are étale hypercohomology. Snce a locally constant constructble sheaf of Z/nZ-modules on X et s representable by a fnte étale scheme over X, we may assume that the hypercohomology are taken on the bg étale ste of X [9, V.1]. Hence, we have a well-defned functor.

13 Varatons on the Bloch-Ogus Theorem Lemma. The étale cohomology functor F(X,Z) = HZ (X, C) s a functor endowed wth transfers (2.6). Proof. Let π : Y X be a fnte flat morphsm of schemes. For an X-scheme X set Y = X X Y and denote the projecton Y X by π. If X g X s an X-scheme morphsm then set Y = X X Y and denote by π : Y X the projecton on X and by g Y : Y Y the morphsm g d Y. If Z X s a closed subset then we set S = π 1 (Z), Z = X X Z, Z = X X Z, S = (π ) 1 (Z ), S = (π ) 1 (Z ). If Y = Y 1 Y 2 (dsjont unon) then set π = π Y, Y = g 1 Y (Y ), S = Y S, S = Y S and defne g Y, : Y Y to be the restrcton of g Y. Let C be a sheaf on the bg étale ste Et/X. If Z X s a closed subset then for an X-scheme X we denote Γ Z (X,C) = ker(γ(x,c) Γ(X Z,C)) and f Y = Y 1 Y 2 we denote Γ S (Y,C) = ker(γ(y,c) Γ(Y S,C)). Delgne n [5] constructed trace maps for fnte flat morphsms. In partcular, for a X-scheme X and for every presentaton of the scheme Y n the form Y = Y 1 Y 2 there are certan trace maps Tr π : Γ S (Y,C) Γ Z (X,C). These maps satsfy the followng propertes () (base change) the dagram Γ S Tr π (Y,C) g Y, Γ S (Y,C) Tr π Γ Z (X,C) g Y Γ Z (X,C) commutes; () (addtvty) the dagram commutes; Γ S (Y,C) Tr π Γ Z (X,C) + Γ S 1 (Y d 1,C) Γ S 2 (Y 2,C) Tr π 1 +Tr π 2 Γ Z (X,C) () (normalzaton) f π 1 : Y 1 X s an somorphsm then the composte map s the dentty; Γ Z (X,C) (π 1 ) Γ S 1 (Y 1,C) Tr π 1 ΓZ (X,C) (v) maps Tr π are functoral wth respect to sheaves C on Et/X.

14 64 Ivan Pann, Krll Zanoullne Now let 0 C I be an njectve resoluton of the sheaf C on Et/X. Then for a closed subset Z X and for a presentaton Y = Y 1 Y 2 one has H p S (Y,C) := H p (Γ S (Y,I )), H p Z (X,C) := H p (Γ Z (X,I )). Thereby the property (v) shows that the trace maps Tr π : Γ S (Y,Ir ) Γ Z (X,I r ) determne a morphsm of complexes Γ S (Y,I ) Γ Z (X,I ). Thus one gets the nduced map whch we wll denote by H p (Tr π ) : H p S (Y,C) Hp Z (X,C). And these trace maps satsfy the followng propertes (the same as n Defnton 2.6): () the base changng property; () the addtvty property; () the normalzaton property; (v) the functoralty wth respect to sheaves on Et/X. 5.3 Lemma. The étale cohomology functor F(X,Z) = HZ (X, C) s a functor wth fnte monodromy (4.6). Proof. Accordng to Defnton 4.6 we have to show that there exst a fnte étale morphsm ρ : Y X X together wth a secton s : X Y of ρ over the dagonal and a functor somorphsm Φ : F 1 F 2 on Cp(Y ), where F 1 = ρ pr 1F and F 2 = ρ pr 2F. To produce Y we use the followng explct constructon suggested by H. Esnault (we follow [15]): Let X be a fnte Galos coverng wth Galos group G such that the pullback of C to X s a complex of constant sheaves. Consder the étale coverng X k X X k X wth Galos group G G. Let X k X = ( X k X)G be the unque ntermedate coverng assocated wth the dagonal subgroup G = (g,g) G G. The dagonal map X X k X nduces a map s : X X k X whch s a secton to the projecton X k X X k X over the dagonal X = X X k X. Let Y be the connected component of X = m(s) n X k X. Then Y ρ X k X s a connected Galos coverng havng a secton s over X. To check that there s the functor somorphsm Φ we refer to the end of secton 4 of [15]. We also need the followng techncal lemma that s a slghtly modfed verson of Proposton 2.1.2, [4] 5.4 Lemma. Let U be a sem-local regular scheme of geometrc type over a feld k,.e., U = Spec O X,x for some smooth affne varety X and a fnte set of ponts x = {x 1,...,x n } of X. Suppose the étale cohomology functor

15 Varatons on the Bloch-Ogus Theorem 65 F(Y,Z) = HZ (Y, C) from Theorem 1.1 s effaceable at x. Then, n the exact couple [4, 1.1] defnng the conveau spectral sequence for (U, C), the map p,q s dentcally 0 for all p > 0. In partcular, we have E p,q 2 = H q (U, C) f p = 0 and E p,q 2 = 0 f p > 0. And the Cousn complex [4, 1.3] yelds the exact complex from Theorem 1.1. Proof. Consder the commutatve dagram HZ n (X, C) HZ U n (U, C) HZ n (U, C) H n X (m) (X, C) H n U (m) (U, C) H n U (m 1) (U, C) The composton of arrows n the frst row s dentcally 0 for any n. Therefore the compostons H n Z (X, C) Hn U (m) (U, C) H n U (m 1) (U, C) are 0. Passng to the lmt over Z, ths gves that the compostons H n X (m) (X, C) H n U (m) (U, C) H n U (m 1) (U, C) are 0. Passng to the lmt over open neghborhoods of x, we get that the map m,n m : H n U (m) (U, C) H n U (m 1) (U, C) s tself 0 for any m 1. Now we are ready to prove the man result of ths paper stated n the Introducton. Proof of Theorem 1.1. Assume the ground feld k s nfnte. In ths case, observe that the étale cohomology functor F(X,Z) = HZ (X, C) satsfes all the hypotheses of Theorem 4.7. Indeed, t s homotopy nvarant accordng to [4, 7.3.(1)]. It satsfes vanshng property by the very defnton. It has transfer maps by Lemma 5.2 and t s a functor wth fnte monodromy by Lemma 5.3. So that by Theorem 4.7 the functor F s effaceable. Now Theorem 1.1 follows mmedately from Lemma 5.4. To fnsh the proof,.e., to treat the case of a fnte ground feld, we apply the standard arguments wth transfers for fnte feld extensons (see the proof of [4, 6.2.5]). 5.5 Remark. If the complex C comes from the base feld k,.e., each sheaf n C can be represented as p C for some sheaf C on (Speck) et, where p : X Spec k s the structural morphsm. Then the étale cohomology functor F satsfes all the hypotheses of Theorem 4.2. Hence, Theorem 1.1 holds wthout assumng that F s a functor wth fnte monodromy.

16 66 Ivan Pann, Krll Zanoullne References [1] M. Artn, A. Grothendeck et J. L. Verder, Théore des topos et cohomologe étale des schémas, SGA 4, Tome 3, Lecture Notes n Math. 305, Sprnger-Verlag, v+640 pp. [2] S. Bloch, A. Ogus, Gersten s Conjecture and the homology of schemes, Ann. Sc. Ec. Norm. Super., 4(7): , [3] J.-L. Collot-Thélène, Bratonal nvarants, purty and the Gersten conjecture, n K-theory and algebrac geometry: connectons wth quadratc forms and dvson algebras (Santa Barbara, CA, 1992), 1-64, Proc. Sympos. Pure Math. 58, Part 1, Amer. Math. Soc., Provdence, RI, [4] J.-L. Collot-Thélène, R.T. Hoobler, B. Kahn, The Bloch-Ogus-Gabber Theorem, n Algebrac K-theory, 31-94, Felds Inst. Commun. 16, Amer. Math. Soc., Provdence, RI, [5] P. Delgne, SGA 4 1/2, Cohomologe étale, Lecture Notes n Math. 569, Sprnger-Verlag, 1977 [6] D. Esenbud, Commutatve Algebra, Graduate Texts n Mathematcs 150, Sprnger, [7] H. Esnault, E. Vehweg, Germs of de Rham cohomology classes whch vansh at the generc pont, C. R. Acad. Sc. Pars Sér. I Math., 326(7): , [8] O. Gabber, Gersten s conjecture for some complexes of vanshng cycles. Manuscrpta Math., 85(3-4): , [9] J.S. Mlne, Étale Cohomology, Prnceton Mathematcal Seres 33, Prnceton, New Jersey: Prnceton Unversty Press. XIII, [10] M. Ojanguren, I. Pann, Ratonally trval hermtan spaces are locally trval, Math. Z., 237: , [11] M. Ojanguren, I. Pann, A purty theorem for the Wtt group, Ann. Sc. Ec. Norm. Super., 32(1): 71-86, [12] I. Pann, A. Smrnov, On a theorem of Grothendeck. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 272(7): , [13] I. Pann, A. Susln, On a conjecture of Grothendeck concernng Azumaya algebras, St.Petersburg Math. J., 9(4): , [14] D. Qullen, Hgher algebrac K-theory I, n Algebrac K-theory I, , Lecture Notes n Math. Vol Sprnger-Verlag, Berln, 1973.

17 Varatons on the Bloch-Ogus Theorem 67 [15] A. Schmdt, K. Zanoullne, Generc Injectvty for Etale Cohomology and Pretheores, to appear n J. of Algebra, (also avalable on K-Theory preprnt server N 556) [16] V. Voevodsky, Cohomologcal theory of presheaves wth transfers, n Cycles, Transfers and Motvc Homology Theores, An. Math. Studes 143, Prnceton Unversty Press, [17] K. Zanoullne, The Purty Problem for Functors wth Transfers, K- Theory J., 22(4): , Ivan Pann, Laboratory of Algebra and Number Theory, Steklov Mathematcal Insttute at St.Petersburg RAN, 27, Fontanka St.Petersburg Russa Krll Zanoullne Laboratory of Algebra and Number Theory, Steklov Mathematcal Insttute at St.Petersburg RAN, 27, Fontanka St.Petersburg Russa

18 68 Documenta Mathematca 8 (2003)