APPLICATION OF MOTIVIC COMPLEXES TO NEGLIGIBLE CLASSES

Emmanuel Peyre

Description

Niveau: Supérieur, Doctorat, Bac+8
APPLICATION OF MOTIVIC COMPLEXES TO NEGLIGIBLE CLASSES? Emmanuel Peyre Abstract. — Lichtenbaum's complex enables one to relate Galois cohomology to K -cohomology groups. In this paper, we consider the first terms of the Hochschild- Serre spectral sequence for the cohomology of these complexes, which was devel- oped by Kahn, in the case of quotients of “big” open sets in cellular varieties. In the particular case of a faithful representation W of a finite group G over an alge- braically closed field k, this yields that the group of negligible classes in the cohomol- ogy group H3(G,Q/Z(2)) is canonically isomorphic to the second equivariant Chow group of a point. It also implies that the unramified classes in the cohomology group H3(k(W )G, (Q/Z)?(2)) come from the cohomology of G, which had been proved by Saltman when k is the field of complex numbers. Using the motivic complexes of Voevodsky, we then prove similar results in degrees four and five. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• hochschild-serre spectral


• generalized flag variety


• module over


• rham-witt sheaf


• cellular varieties


• galois group


• sheaf corresponding

Sujets

• Generalized flag variety
• Galois group

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 6
• Langue: English

Extrait

APPLICATION OF MOTIVIC COMPLEXES TO NEGLIGIBLE CLASSES ∗

Emmanuel Peyre

Abstractone to relate Galois cohomology to complex enables . — Lichtenbaum’s K this paper, we consider the first terms of the Hochschild--cohomology groups. In Serre spectral sequence for the cohomology of these complexes, which was devel- oped by Kahn, in the case of quotients of “big” open sets in cellular varieties. In the particular case of a faithful representationWof a finite groupGover an alge- braically closed fieldk, this yields that the group of negligible classes in the cohomol- ogy groupH 3 (GQZ(2)) is canonically isomorphic to the second equivariant Chow group of a point. It also implies that the unramified classes in the cohomology group H 3 (k(W) G (QZ) ′ (2)) come from the cohomology ofG, which had been proved by Saltman whenkis the field of complex numbers. Using the motivic complexes of Voevodsky, we then prove similar results in degrees four and five.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Hochschild-Serre spectral sequence for Lichtenbaum’s complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Application to the case of finite groups . . . . . . . . . . . . . . . . . . . . . . . . 10 4. Application of Voevodsky’s motivic complexes . . . . . . . . . . . . . . . . 23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1. Introduction

TheunramifiedcohomologygroupswerefirstdevelopedbyColliot-The´le`neand Ojanguren as invariants for stable rationality which generalize the unramified Brauer

1991Mathematics Subject Classification 12G05;. — primary 14C25, 19D45, secondary 14E20. ∗ AlgebraicK Math., vol. 67,-theory (Seattle 1998), Proc. Sympos. Pure AMS, Providence, 1999, pp. 181–211

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EMMANUEL PEYRE

group. It has been used in [CTO] and [Pe1] to give new examples of unirational varieties which are not stably rational. Unirational fields of special interest are given by Noether’s problem: ifGis a finite group andWa faithful representation ofGover a fieldk, then the field of invariant functionsk(W) G does not depend, up to stable equivalence, onW. The problem is to determine for which fieldskand groupsGthe fieldk(W) G  firstis stably rational. The counter-example overCwas constructed by Saltman in [Sa1] using the unramified Brauer group. Bogomolov [Bo] gave a complete description of the unramified Brauer group of the fieldC(W) G in terms of the cohomology of the groupG. The study of the higher unramified cohomology groups for these fields is made more complicated by the existence of negligible classes in the cohomology of finite groups which vanish when lifted to Galois groups. The first interesting results about the third unramified cohomology group for such fields have been obtained by Saltman in [Sa2]. More precisely, he proved that this cohomology group fork=Cis contained in the image of the inflation map H 3 (G,QZ)→H 3 (k(W) G ,QZ) and that, ifH 3 (G,QZ) n is the kernel of this map and ifGis ap-group, then there is a natural isomorphism H 3 (G,QZ) n H 3 (G,QZ) p +H 3 (G,QZ) c − e →N 3 (G) where H 3 (G,QZ) p = Ker  H 3 (G,QZ)→H 3 (G,C(W) ∗ )  which may be computed in terms of the cohomology ofG, H 3 (G,QZ) c = X Cores G H H 3 (H,QZ) n , H
G andN 3 Gis a kind of equivariant Chow group. The connection between Chow groups of codimension 2 and restriction maps in degree 3 appears also in [Pe2], [Pe3] and [Pe4], where we describe for any generalized flag varietyVan exact sequence H r 1 aZ (V,K 2 ) j →(PicV k s ⊗k s ∗ ) G →Ker  H 3 (G,QZ(2))→H 3 (k(V),QZ(2))  →CH 2 (V) tors →0 wherek s is a separable closure ofk,G= Gal(k s k), andK i is the sheaf associated to the presheaf of Quillen’sK-groupsU7→K i (U exact sequence was obtained). This usingtheworkofColliot-Th´ele`neandRaskindontheK-cohomology (see [CTR]) and a result of Bruno Kahn based on Lichtenbaum’s complexes (see [Li1], [Li2], [Li3] and [Kah1 sequence was also considered by Merkur]). This ′ ev who proved in [Me1] that the mapjis injective. More recently, Kahn gave in [Kah2] a direct proof of this exact sequence and a description of the unramified cohomology group of degree three of these twisted generalized flag varieties using the Hochschild-Serre spectral sequence for the hyper- cohomology of Lichtenbaum’s complexes.

NEGLIGIBLE CLASSES

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One of the purposes of this text is to show that an easy generalization of the results of Kahn enables one to state the results for generalized flag varieties and for finite groups in a uniform way. In fact we prove that ifGis a finite group,Wa faithful representation ofGover an algebraically closed fieldkof exponential characteristicpsuch that the complement of the open setUon whichGacts freely inWhas a codimension bigger than 4, then there is an exact sequence 0→CH 2 G (k)→H 3 (G,QZ(2)) →H raZ 0  UG,H e´ 3 t (QZ(2))  →H raZ 0  W,H et 3 ´ (Q p Z p (2))  whereUGis the quotient ofUbyG, CH 2 G (k) is the equivariant Chow group of Speck andH te´ 3 (QZ(2)) is the sheaf corresponding to the presheaf V7→H et´ 3 (V,QZ(2)) The connection with the results of Saltman becomes clear if one takes into account the inclusions H n 3 rk (k(W) G ,QZ(2))⊂H r 0 Za (UG,H 3 te´ (QZ(2))) ⊂H 3 (k(W) G ,QZ(2)) The second section of this paper contains a partial description of theK-cohomo- logy groups of big open sets in cellular varieties followed by an easy generalization of the results of Kahn, the third applies the previous computations to the case of finite groups and makes explicit the connection with Saltman’s work and the fourth extends the results to higher degrees using the work of Voevodsky.

2. Hochschild-Serre spectral sequence for Lichtenbaum’s complex

2.1. Notations. —In the sequel we use the following notations: Notation2.1.1. — For any fieldL, letLbe an algebraic closure ofLandL s be the separable closure ofLinL any variety. ForVoverLwe denote byL(V) the function field ofVand for any extensionL ′ ofLbyV L ′ the productV× SpecL SpecL ′ . We putV s =V Ls  denotes by. OneV (i) the set of points of codimensioniinV, and, for anyx∈V, byκ(x) its residue field. The Chow groups of cycles of codimensionion Vmodulo rational equivalence are denoted by CH i (X). IfLis a field, letpbe the exponential characteristic ofL, that is 1 ifLis of characteristic 0 and the usual characteristic otherwise. Ifnis prime topandVa variety overL, let n batteelehe´offhsaen-th roots of unity and for anyrandiin Z >0 , letW r Ω V i log be the logarithmic part of the corresponding De Rham-Witt sheaf W r Ω i V (see [Il,§ [I.5.7]). ByBK, corollary 2.8], forV= SpecLone has W r Ω L j log (L)← e −K jM (L)p r K jM (L) Ifn=n ′ p r with (n ′ , p) = 1, then one puts ZnZ(j) = n ⊗j ⊕W r Ω V j log [−j]