Algebraic K-theory
 9780821808184, 0821808184

Table of contents :
Quaternionic exercises in K-theory Galois module structure / Ted Chinburg [und weitere] --
The Bloch-Ogus-Gabber theorem / Jean-Louis Colliot-Thélène, Raymond T. Hoobler and Bruno Kahn --
Milnor's conjecture and Galois theory I / Wenfeng Gao and Ján Mináč --
Ultraproducts and the discrete cohomology of algebraic groups / J.F. Jardine --
Lambda-operations, K-theory and motivic cohomology / Marc Levine --
Quasi-motives of curves / Stephen Lichtenbaum --
A chain complex for the spectrum homology of the algebraic K-theory of an exact category / Randy McCarthy --
Hypercohomology spectra and Thomason's descent theorem / Stephen A. Mitchell --
Kahler differentials of certain cusps / Leslie G. Roberts --
Local fundamental classes derived from higher-dimensional K-groups / Victor P. Snaith --
Local fundamental classes derived from higher-dimensional K-groups : II / Victor P. Snaith --
Weak approximation, R-equivalence and Whitehead groups / Nguyeñ Quoć Tȟańg.

Citation preview

Selected Title s i n Thi s Serie s 16 Victo r P . Snaith , Editor , Algebrai c K-Theory , 1 99 7 15 S t e p h e n P . B r a h a m , Jac k D . G e g e n b e r g , an d R o b e r t J . McKellar , Editors , Sixt h Canadian conferenc e o n genera l relativit y an d relativisti c astrophysics , 1 99 7 14 M o u r a d E . H . Ismail , D a v i d R . M a s s o n , an d M i z a n R a h m a n , Editors , Specia l functions, g-serie s an d relate d topics , 1 99 7 13 P e t e r A . Fillmor e an d J a m e s A . M i n g o , Editors , Operato r algebra s an d thei r applications, 1 99 7 12 Dan-Virgi l Voiculescu , Editor , Fre e probabilit y theory , 1 99 7 11 Collee n D . Cutle r an d Danie l T . Kaplan , Editors , Nonlinea r dynamic s an d tim e series: Buildin g a bridg e betwee n th e natura l an d statistica l sciences , 1 99 7 10 Jerrol d E . M a r s d e n , Georg e W . Patrick , an d W i l l i a m F . Shadwick , Editors , Integration algorithm s an d classica l mechanics , 1 99 6 9 W . H . K l i e m a n n , W . F . Langford , an d N . S . Namachchivaya , Editors , Nonlinea r dynamics an d stochasti c mechanics , 1 99 6 8 Larr y M . B a t e s an d D a v i d L . R o d , Editors , Conservativ e system s an d quantu m chaos, 1 99 6 7 W i l l i a m F . Shadwick , P e r i n k u l a m S a m b a m u r t h y Krishnaprasad , an d Tudo r Stefan R a t i u , Editors , Mechanic s day , 1 99 6 6 A n n a T . Lawnicza k an d R a y m o n d Kapral , Editors , Patter n formatio n an d lattic e gas automata , 1 99 6 5 J o h n C h a d a m , M a r t i n Golubitsky , W i l l i a m Langford , an d Bria n W e t t o n , Editors, Patter n formation : Symmetr y method s an d applications , 1 99 6 4 W i l l i a m F . Langfor d an d W a y n e N a g a t a , Editors , Norma l form s an d homoclini c chaos, 1 99 5 3 A n t h o n y B l o c h , Editor , Hamiltonia n an d gradien t flows , algorithm s an d control , 1 99 4 2 K . A . Morris , Editor , Contro l o f flexible structures , 1 99 3 1 Michae l J . Enos , Editor , Dynamic s an d contro l o f mechanica l systems : Th e fallin g ca t and relate d problems , 1 99 3

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FIELDS INSTITUT E COMMUNICATIONS T H E FIELD S INSTITUT E FO R RESEARC H I N MATHEMATICA L SCIENCE S

Algebraic K-Theor y Victor P . Snait h Editor

American Mathematica l Societ y Providence, Rhod e Islan d

The Field s Institut e for Researc h i n Mathematica l Science s T h e Field s I n s t i t u t e i s n a m e d i n h o n o u r o f t h e C a n a d i a n m a t h e m a t i c i a n J o h n Charle s Fields (1 863-1 932) . Field s wa s a visionar y w h o receive d m a n y h o n o u r s fo r hi s scientifi c work, includin g electio n t o t h e Roya l Societ y o f C a n a d a i n 1 90 9 a n d t o t h e Roya l Societ y o f L o n d o n i n 1 91 3 . A m o n g o t h e r accomplishment s i n t h e servic e o f t h e internationa l m a t h e m a t i c s community , Field s wa s responsibl e fo r establishin g t h e world' s m o s t prestigiou s prize fo r m a t h e m a t i c s r e s e a r c h — t h e Field s Medal . T h e Field s I n s t i t u t e fo r Researc h i n M a t h e m a t i c a l Science s i s s u p p o r t e d b y g r a n t s fro m t h e O n t a r i o Ministr y o f E d u c a t i o n a n d Trainin g a n d t h e N a t u r a l Science s a n d Engineerin g Research Counci l o f C a n a d a . T h e I n s t i t u t e i s sponsore d b y M c M a s t e r University , t h e University o f Toronto , t h e Universit y o f Waterloo , a n d Yor k Universit y a n d ha s affiliate d universities i n O n t a r i o a n d acros s C a n a d a .

1991 Mathematics Subject

Classification.

Primar

y 1 9-06 .

Library o f Congres s Cataloging-in-Publicatio n D a t a Algebraic K-Theor y / Victo r P . Snaith , editor . p. cm . — (Field s Institut e Communications , ISS N 1 069-526 5 ; v. 1 6 ) Papers fro m th e Secon d Grea t Lake s Conferenc e o n Algebrai c K-Theory , hel d Mar . 1 99 6 a t th e Fields Institut e fo r Researc h i n Mathematica l Science s i n memor y o f Rober t Wayn e Thomason . Includes bibliographica l references . ISBN 0-821 8-081 8- 4 (alk . paper ) 1. K-theory—Congresses . 2 . Geometry , Algebraic—Congresses . 3 . Topology—Congresses . I. Snaith , V . P . (Victo r Percy) , 1 944 - . II . Thomason , Rober t Wayne , 1 952-1 995 . III . Grea t Lakes Conferenc e o n Algebrai c K-Theor y (2n d : 1 99 6 : Field s Institut e fo r Researc h i n Mathe matical Sciences ) IV . Series . QA612.33.A3841 99 7 512 / .55—dc21 97-2024 3 CIP

C o p y i n g an d reprinting . Materia l i n this boo k ma y b e reproduce d b y any mean s fo r educationa l and scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y service s that collec t fee s fo r deliver y o f documents an d provide d tha t th e customar y acknowledgmen t o f th e source i s given. Thi s consen t doe s no t exten d t o othe r kind s o f copyin g fo r genera l distribution , fo r advertising o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercia l us e o f material shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematica l Society , P. O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o reprint-permissionOams.org. Excluded fro m thes e provision s i s materia l i n article s fo r whic h th e autho r hold s copyright . I n such cases , request s fo r permissio n t o us e o r reprin t shoul d b e addresse d directl y t o th e author(s) . (Copyright ownershi p i s indicate d i n th e notic e i n th e lowe r right-han d corne r o f th e first pag e o f each article. ) © 1 99 7 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . This publicatio n wa s prepare d b y th e Field s Institute . Visit th e AM S homepag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0

2 01 00 9 9 98 9 7

Contents Preface vi

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Robert W . Thomaso n (1 952-1 995 ) i

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Quaternionic Exercise s i n K-Theor y Galoi s Modul 1 e Structur e T E D CHINBURG , M A N F R E D K O L S T E R , G E O R G E P A P P A S and V I C T O R P . SNAIT H

The Bloch-Ogus-Gabbe r Theore m 3 1 JEAN-LOUIS C O L L I O T - T H E L E N E , RAYMON D T . H O O B L E R and BRUN O K A H N

Milnor's Conjectur e an d Galoi s Theor y I 9

5

WENFENG GA O an d JA N MlNA C

Ultraproducts an d th e Discret e Cohomolog y o f Algebrai1 c1 Group s 1 J. F . JARDIN E

Lambda-Operations, K-Theor y an d Motivi 1 c Cohomolog y 3 1 M A R C LEVIN E

Quasi-Motives o f Curve s 8

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S T E P H E N LICHTENBAU M

A Chai n Comple x fo r th e Spectru m Homolog y o f th1 e Algebrai c 9 K-Theory o f a n Exac t Categor y

9

RANDY MCCARTH Y

Hypercohomology Spectr a an d Thomason' s Descen t Theore m 22 1 S T E P H E N A . MITCHEL L

Kahler Differential s o f Certai n Cusp s 27 LESLIE G . R O B E R T S

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Local Fundamenta l Classe s Derive d fro m Higher-Dimensiona l K-Groups V I C T O R P . SNAIT H

Local Fundamenta l Classe s Derive d fro m Higher-Dimensiona l K-Groups: I I V I C T O R P . SNAIT H

Weak Approximation , R-Equivalenc e an d Whitehea d Group s NGUYEN Q U O C T H A N G

Appendix I Appendix I I

Preface The Second Great Lakes Conference on Algebraic K-Theory was hosted by The Fields Institut e fo r Researc h i n Mathematica l Science s i n Marc h 1 996 . I t prove d to b e a mathematicall y ver y excitin g occasio n wit h man y interestin g ne w result s being unveiled. Th e success of the meeting together with a desire to commemorat e Bob Thomason , on e o f K-theory' s mos t influentia l figures, le d t o thi s Proceeding s volume. As editor, I would lik e to expres s m y thanks an d those o f the organizer s of th e conference t o the staff o f The Fields Institute wh o made this meeting so rewardin g and enjoyable . I n particular , I wish to exten d specia l thank s t o th e Presiden t an d Scientific Director , Joh n Chadam , fo r invitin g th e Grea t Lake s K-Theor y Confer ence t o Th e Field s Institute , an d t o th e Publication s Assistant , Ern a Unrau , fo r making the editin g of this volum e s o easy fo r me.

Victor Snait h

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Robert Wayn e T H O M A S O N 1952-1995

In the mi d 1 970's , when Bo b Thomaso n wa s a sta r graduat e studen t a t Prince ton University , algebrai c K-theor y wa s i n a ver y tentativ e state . I t ha d begu n as Whitehead-ia n algebra-to-capture-geometry , showin g lot s o f low-dimensiona l promise. Then , wit h th e wor k o f Quillen , algebrai c K-theor y ha d suddenl y ex panded int o a grand cosmologica l entit y wit h it s own , extremel y fascinating , ontol ogy and epistomology . I t wa s clear that th e subjec t constitute d a major mathemat ical nexu s bu t th e evidenc e wa s i n th e for m o f hard-won hint s abou t ho w Quillen' s K-theory migh t unif y suc h diverse phenomena as homotopy types of diffeomorphis m groups of manifolds, specia l values of L-functions, polylogarithms , intersectio n the ory o f algebrai c varieties , Wei l cohomolog y theorie s an d s o on. A s a result , n o on e was sure whether K-theor y wa s the domai n of topologists, algebraist s or some othe r mathematical species . I n man y place s thi s hardl y mattered , th e subjec t coul d stil l easily be shrugged of f a s a new fad wit h fine ambitions, lot s of conjectured promise , but wit h n o "bi g theorem " t o mar k som e post-Quille n progress . Bob Thomaso n wa s th e on e wh o wa s t o provid e suc h a "bi g theorem" .

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It mus t hav e bee n clea r al l th e wa y u p th e educationa l ladde r tha t Bo b wa s exceptionally clever . Fo r example , a t eightee n h e ha d wo n firs t priz e i n th e Nort h America wid e competitio n entitle d "Grea t Book s o f th e Wester n World" . Conse quently, I was unawar e tha t Bo b wa s stil l a graduat e studen t whe n I firs t me t hi m around 1 976 . Andr e Wei l i s repute d t o hav e advise d tha t on e shoul d lear n fro m "the masters , no t th e students " - I ca n onl y conclud e tha t h e di d no t hav e Bo b Thomason i n mind ! My friendshi p wit h Bo b go t of f t o a rathe r shak y start . I n 1 97 5 topologists ' contributions t o algebrai c K-theor y ofte n too k th e for m o f result s abou t infinit e loopspaces. l b Madsen , J0rge n Tornehav e an d I ha d recentl y resolve d al l o f Pete r May's conjecture s relatin g t o th e infinit e loopspace s o f geometri c topology . Pran k Adams, a s a n editor , suggeste d sendin g th e pape r t o th e Annal s o f Mathematic s i n Princeton. Midwester n homotop y theor y wa s evidently no t s o popular i n Princeto n and th e typescrip t gathere d a lo t o f cobwebs , bu t a n edito r gav e a cop y t o Bo b t o read. Eventuall y Adam s too k th e pape r awa y i n disgust , publishin g i t a fe w week s later i n the Proceeding s o f the Cambridg e Philosophica l Society . Anyway , th e poin t is that I was feelin g anythin g bu t cheerfu l toward s Bo b a t ou r firs t meeting , whic h took plac e a t a conferenc e i n Evanston . However , withi n minute s o f meetin g Bo b it wa s clea r t o anyon e tha t her e wa s someon e wh o ha d n o patienc e fo r intr a Iv y League politics . Her e wa s someon e inten t o n doin g mathematic s fo r it s ow n sake ! During th e nex t fe w years , mostl y throug h listenin g an d no t understanding , I learne d a lo t fro m Bo b abou t th e prevailin g visio n fo r K-theory . Hi s intensit y notwithstanding, Bo b Thomaso n wa s alway s ver y approachabl e - focusse d o n re search goals , h e invariabl y ha d tim e patientl y t o explai n thing s t o lesse r mortal s such a s myself . Also , h e ha d a knac k fo r puttin g acros s th e conceptual picture . I n the lat e 1 970' s he, Bill Dwyer, Eri c Priedlande r an d I got involve d i n studyin g mo d p K-theory whic h ha s bee n inflicte d wit h Bot t periodicity . Whe n A i s a commu tative rin g ther e i s a ma p o f grade d rings , TT^(BA^) — • K*(A), fro m th e stabl e homotopy o f the classifyin g spac e o f the unit s o f A (wit h a disjoin t poin t attached ) to th e algebrai c K-theor y o f A. Thi s ma p come s fro m Quillen' s group-completio n theorem an d exist s als o fo r homotop y wit h coefficients . I n orde r t o sa y somethin g about thi s map I had manage d t o calculate the stable homotopy whe n inflicte d wit h Bott periodicit y an d ha d therefor e inflicte d algebrai c K-theor y wit h simila r period icities so as to be able to detect th e asymptoti c residue s of this map. I had obtaine d a fe w rudimentar y result s abou t ne w element s i n K-theory , bu t Bo b immediatel y saw tha t m y Bot t periodicitie s wer e reall y connecte d wit h th e periodicit y i n etal e cohomology. Fo r som e tim e h e ha d bee n tryin g t o establis h Quillen' s predictio n that K-theor y wa s relate d t o etal e cohomology , afte r som e lo w dimensiona l excep tions. Bo b alread y possesse d a numbe r o f ver y origina l construction s i n whic h on e did homologica l algebr a with spectr a an d stabl e homotop y theor y instea d o f group s and modules . Wit h thes e technique s h e ha d bee n tryin g t o prov e rathe r to o muc h about th e predicte d connection . However , wit h th e ai d o f ou r four-autho r paper , Bob soo n ha d th e require d theore m - tha t Bot t periodi c algebrai c K-theor y mo d pn o f an y reasonabl e schem e ha d a n Atiyah-Hirzebruc h spectra l sequenc e startin g with etal e cohomology . Thi s wa s exactl y a resul t o f the much-sought-afte r type . I t showed that , i n hig h dimensions , th e predicte d connectio n betwee n K-theor y an d cohomology existed . This connection , togethe r wit h it s concomitan t insights , serve d a s th e basi s fo r many o f Bob best theorems . No w he had a machine through whic h to funne l result s from topologica l K-theor y int o algebrai c K-theory , a sheet-ancho r givin g muc h o f

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the stabilit y o f purpos e t o hi s late r research . Hi s subsequen t strin g o f importan t results form s th e backbon e o f moder n K-theory , particularl y i n it s application s t o arithmetic an d algebrai c geometry . Hi s main theore m wa s a tour d e force. I t i s very hard goin g an d a prope r stud y take s a lon g time , lot s o f wor k durin g whic h on e passes throug h phase s whic h Bo b wryl y describe d a s th e progressio n fro m "thril l seeker" t o "reckles s cheat " t o "hones t man" . Personally , I doub t tha t I eve r mad e it pas t th e "reckles s cheat " phase . However , man y othe r author s hav e studie d th e ideas an d methods . I a m ver y gratefu l t o Stev e Mitchel l who , i n thi s volume , has writte n a n "essay whic h wil l smoot h th e wa y fo r others " throug h Thomason' s Theorem. I alway s fel t a certai n sadness , an d her e I a m no t referrin g t o hi s untimel y death, abou t Bob' s career . Befor e comin g t o Nort h Americ a i t ha d neve r occurre d to m e tha t on e mus t no t onl y prov e th e "right " theore m bu t on e mus t prov e i t i n the "right " place . Th e fictions , fad s an d faction s o f mathematica l societ y ar e quit e clear to me now, but i n those days I never understood wh y Bob's achievement s wen t so comparativel y unrewarded . T o hi s K-theoreti c friends , therefore , i t cam e a s n o great surpris e when , i n Octobe r 1 989 , Bob move d t o a positio n i n th e C.N.R.S . a t the Unversit e d e Pari s 7 . Wit h tha t move , Bo b wa s mathematicall y wher e h e fel t most a t home . Sadly, a t th e heigh t o f hi s mathematica l powers , Bo b died , tragically . H e would hav e bee n forty-thre e year s ol d o n Novembe r 5 , 1 995 . A longtim e diabete s sufferer, jus t befor e hi s birthda y h e wen t int o diabeti c shoc k an d die d i n hi s Pari s apartment. H e wil l b e misse d immensel y but , I lik e t o think , mathematicall y hi s name wil l survive a s lon g a s doe s the subjec t algebrai c K-theory , whic h h e loved s o much. An d tha t wil l b e a ver y lon g time !

Victor Snait h February 1 99 7

References Dwyer, W. , Priedlander , E. , Snaith , V . an d Thomaso n R . W . [1 982] , Algebraic K-theory eventually surjects onto topological K-theory, Invent . Math . 66 , 481 -491 . M R 84i : 14015 Gillet, H . an d Thomaso n R . W . [1 984] , The K-theory of strict hensel local rings and a theorem of Suslin, J . Pur e Appl . Alg . 34 , 241 -254 . M R 86e:1 801 4 Latch, D. , Thomason , R . W . an d Wilso n S . [1 979] , Simplicial sets from categories, Math . Zeit . 164, 1 95-21 4 . M R 80e:5501 2 May, J . P . an d Thomason , R . W . [1 978] , The uniqueness of infinite loop space machines, Topolog y 17, 205-224 . M R 80g:5501 5

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Publications o f Bo b Thomaso n [1974] A note on spaces with normal product with some compact space, Proc . AM S 44 , 509-51 0 . MR 51 #41 6 5 [1977] Homotopy colimits in Cat , with applications to algebraic K-theory and loop space theory, Ph.D. thesis , 1 2 4 pages , Princeto n University , availabl e fro m Universit y Microfilms , An n Arbor. M I 481 0 4 [1979] Homotopy colimits in the category of small categories, Math . Proc . Cambridg e Philos . Soc . 85, 91 -1 09 . M R 80b:1 801 5 [1979] Uniqueness of delooping machines, Duk e Math . J . 46 , 21 7-252 . M R 80e:5501 3 [1980] Ca t as a closed model category, Cahier s Top . Geom . Diff . 21 , 305-324 . M R 82b:1 800 5 [1979] First quadrant spectral sequences in algebraic K-theory, i n Lectur e Note s i n Math . 763 , Springer-Verlag, 332-355 . M R 81 c:1 801 8 [1982] First quadrant spectral sequences in algebraic K-theory via homotopy colimits, Comm . Algebra 1 0 , 1 589-1 668 . 83k : 18006 [1980] Beware the phony multiplication on Quillen's A~ 1 A, Proc . AM S 80 , 569-573 . MR 81 k:1 801 0 [1989] Algebraic K-theory and etale cohomology, Ann . Scient . Ec . Norm . Sup. , 437-552 . 1 8, , 4 e seri e erratu m 22 , 675-677 , errat a 91 j:1 401 3 , M R 87k:1 401 6 [1982] The Lichtenbaum-Quillen conjecture for K/ ^*[/3-1 ], i n Curren t Trend s i n Algebrai c Topol ogy CM S Conf . Proc . vol . 2 , Par t 1 , 1 1 7-1 39 . M R 84f:1 802 4 [1983] Riemann-Roch for algebraic versus topological K-theory, J . Pur e Appl . Al g 27 , 87-1 09 . MR 85c:1 401 3 [1986] Bott stability in algebraic K-theory, AM S Contemp . Math. , 55 , 389-406 . M R 87m:1 802 2 [1983] The homotopy limit problem, AM S Contemp . Math. , 1 9 , 407-41 9 . M R 84j:1 801 2 [1984] Absolute cohomological purity, Bull . Soc . Math . Franc e 1 1 2 , 397-406 . M R 87e:1 401 8 [1987] Algebraic K-theory of group scheme actions, Annal s o f Math . Stud y 1 1 3 , Princeto n U . Press, 539-563 . M R 89c:1 801 6 [1986] Lefschetz-Riemann-Roch theorem and coherent trace formula, Invent . Math . 85 , 51 5-543 . MR 87j:1 402 8 [1986] Comparison of equivariant algebraic and topological K-theory, Duk e Math . J . 5 3 , 795-825 . MR88h:18011 [1987] Equivariant resolution, linearization and HilberVs fourteenth problem over arbitrary base schemes, Adv . Math . 65 , 1 6-34 . M R 88g:1 404 5 [1988] Equivariant algebraic vs. topological K-homology Atiyah-Segal-style, Duk e Math . J . 56 , 589-636. M R 89f : 14015 [1987] The finite stable homotopy type of some topoi, J . Pur e Appl . Alg . 47, 89-1 04 . M R 88k : 14010 [1989] A finiteness condition equivalent to the Tate conjecture over ¥ q, i n AM S Contemp . Math . 83, 385-392 . M R 90b:1 402 9 [1989] Survey of algebraic vs. etale topological K-theory, i n AM S Contemp . Math . 8 3 , 393-443 . MR90h:14029 [1991] The local to global principle in algebraic K-theory, i n Proc . Internationa l Congres s o f Mathematicians, Kyot o 1 99 0 Springer-Verla g 1 , 381 -394 . M R 93e:1 901 0 [1992] Une formule de Lefschetz en K-theorie equivariante algebrique, Duk e Math . J . 68 , 447-462 . MR93m:19007 [1992] Le principe de scindage et Vinexistence de K-theorie de Milnor globale, Topolog y 3 1 , 571-588. M R 93j:1 900 5 [1993] Les K-groupes d'un schema eclate et une formule d'intersection excedentaire, Invent . Math . 112, 1 95-21 5 . M R 93k:1 900 5 [1993] Les K-groupes d'un fibre projectif i n Algebrai c K-theor y an d algebrai c topology , NAT O ASI Serie s C , vol.407 , Kluwer , 243-248 . [1995] The classification of triangulated subcategories, Compositi o Math. , preprint , t o appear . [1995] Symmetric monoidal categories model all connective spectra, Theor y Appl . Categorie s 1 , 78-118. (electroni c journa l http://www.tac.mta.ca/tac/ ) Thomason, R . W . an d Trobaugh , T . [1 990 ] Higher algebraic K-theory of schemes and of derived categories, i n Th e Grothendiec k Festschrif t III , Progres s i n Math . 88 , Birkhauser , 247-435 . MR92f:19001 Thomason, R . W . an d Trobaugh , T . [1 988 ] Le theoreme de localisation en K-theorie algebrique, C.R. Acad . Sci . (Paris ) 307 , 829-831 . M R 89m:1 801 5

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Thomason, R . W . an d Wilson , S . [1 980 ] Hopf rings in the bar spectral sequence, Quart . J . Mat h Oxford 31 , 507-51 1 . M R 82f:5503 0

P h . D . These s Supervise d b y Bo b Thomaso n Harada, M . [1 987 ] A proof of the Riemann-Roch theorem, Ph.D . thesis , John s Hopkin s University , Baltimore. Yao, D . [1 990 ] Higher algebraic K-theory of admissible abelian categories and localization theorems, Ph.D . thesis , John s Hopkin s University , Baltimore .

T h e M a t h e m a t i c s o f Bo b T h o m a s o n Weibel, C . A . [1 996 ] Robert W. Thomason 1( 952-1 995), Weibel, C . A . [1 997 ] The Mathematical Enterprises of 34(1), 1 -1 3 .

A . M . Soc . Notice s 4 3 , 860-862 . Robert W. Thomason, Bull . A . M . Soc .

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Fields Institut e Communication s Volume 1 6 , 1 99 7

Quaternionic Exercise s i n K-Theor y Galoi s Modul e Structure Ted Chinbur g Department o f Mathematic s University o f Pennsylvani a Philadelphia, P A US A 1 906 6 [email protected]

Manfred Kolste r Department o f Mathematic s McMaster Universit y Hamilton, Ontario , Canad a L8 S 4K 1 [email protected]

George Pappa s Department o f Mathematic s Princeton Universit y Princeton, N J US A 0854 4 gpappasOmath.princeton.edu

Victor P . Snait h Department o f Mathematic s McMaster Universit y Hamilton, Ontario , Canad a L8 S 4K 1 snaithQmcmaster.ca

A b s t r a c t . I n [5 ] invariant s o f th e Galoi s modul e structur e o f algebrai c Re groups o f ring s o f algebrai c integer s wer e introduced . Thes e invariant s liv e i n the class-grou p o f the grou p rin g of the Galoi s group s an d ar e conjectured t o b e related t o Ar t in roo t number s an d specia l value s o f L-functions . Quaternio n extensions ar e th e tes t cas e fo r thes e conjectures . A formul a i s given , i n term s of L-functions , fo r a n infinit e famil y o f quaternio n extensions .

1 Introductio n Let G b e a finit e grou p an d denot e b y CC(Z[G]) th e class-grou p o f finitel y generated Z[G]-module s o f finit e projectiv e dimension . Suppos e tha t X—> A —> B

—>Y

1991 Mathematics Subject Classification. Primar y 1 1 R70 ; Secondar y 1 9F27 . TC an d G P researc h partiall y supporte d b y a n NS F grant . MK an d VP S researc h partiall y supporte d b y a n NSER C grant . ©1997 America n Mathematica l Societ y 1

2

Ted Chinburg , Manfre d Kolster , Georg e Pappas , an d Victo r P . Snait h

is a 2-extensio n o f finitel y generate d Z[G]-module s representin g a n elemen t o f Extz\GAY,X) inducing , vi a cup-product , isomorphism s betwee n th e Tat e coho mology group s H l(G;Y) an d H l+2{G\X) fo r al l i. Unde r thes e circumstance s a representative 2-extensio n ma y b e chose n fo r whic h A an d B ar e finitel y generate d Z[G]-modules o f finit e projectiv e dimension . Th e Eule r characteristi c o f suc h a 2-extension i s define d t o b e th e elemen t [A] - [B] e CC(Z[G\). In th e cas e whe n G i s equa l t o G(L/K), th e Galoi s grou p o f a n extensio n o f global fields , L/K, interestin g example s o f suc h Eule r characteristic s ar e afforde d by th e invariants , Q(L/K,n) (n — 1,2,3), constructe d i n [4] . Furthe r examples , denoted b y Qi(L/K,3), wer e constructe d i n ([1 3] , Chapte r 7 ) fro m 2-extension s in whic h Y an d X ar e th e algebrai c K-group s o f th e rin g o f 5-integer s o f L i n dimensions 2 and 3 , respectively. Her e S i s any finite G(L/K)-invarian t se t o f finit e primes containin g thos e ove r prime s o f K whic h ramif y i n L/K, th e constructio n being independen t o f th e choic e o f S. I n [5 ] thi s constructio n i s generalise d usin g K-groups i n dimension s 2 s an d 2 s + 1 t o yiel d a famil y o f Eule r characteristics , denoted b y Q S(L/K,3) fo r s > 1 , t o complet e a patter n i n whic h th e origina l Chinburg invarian t rightfull y fit s a s Q,Q(L/K,3). O n th e othe r hand , associate d t o this situation , ther e i s the Cassou-Nogues-Frohlic h clas s WL/K e

CC(Z[G{L/K)])

which i s define d i n term s o f th e roo t number s o f th e L-function s o f irreducibl e symplectic representation s o f G{L/K) ([2] , [4] , [1 3] , [1 4]) . Surprisingly, i n th e cas e o f functio n fields i n characteristi c p , on e ca n sho w that W L/K = Q S(L/K,3) [5] . I n th e totall y rea l numbe r field case on e ca n sho w that a simila r relatio n hold s i n th e classgrou p o f a maxima l orde r o f th e rationa l group-ring o f G{L/K) ([5] , [1 3] , Chapter 7) . The purpos e o f this pape r i s to investigat e th e relationshi p betwee n W L/K an d Oi (L/K, 3) in the case of an infinite famil y o f totally rea l extensions of the rational s whose Galoi s group s ar e isomorphi c t o Q$, th e quaternio n grou p o f orde r eight . These extensions , denote d b y N/Q, wil l b e constructe d i n Sectio n 2 . Ther e wil l b e one suc h extensio n fo r eac h prime , p = 3{modulo 8) , which ma y b e characterised a s the uniqu e totall y rea l extensio n wit h G(N/Q) = Q s i n whic h onl y on e od d prim e (namely p) ramifies . Th e se t S ma y b e take n t o b e th e se t o f prime s o fT V ove r 2 and p. The example s wer e chosen becaus e the y exhibi t a numbe r o f simultaneous sim plifications. Firstly, bein g totall y real , th e invarian t Qi(N/Q,3) ma y b e define d b y th e method o f [1 3 ] a s th e Eule r characteristi c o f a canonica l 2-extensio n o f th e for m

Kfd(0NiS)

^ A ^ B

^K

2(ON,S)'

constructed i n [1 0] . Her e K l3nd(0N)S) = K l3nd(N) denote s th e indecomposabl e K 3ls group an d K2(ON,SY equa l t o th e kerne l o f th e natura l map , surjectiv e i n ou r examples, fro m ^ ( O ^ s ) t o th e direc t su m o f th e torsio n i n th e K^s o f eac h o f the rea l place s o f N. Th e group s A an d B i n thi s extensio n ma y b e take n t o b e finite an d cohomologicall y trivial . I n fact , i f (Q/Z)(2 ) i s th e Galoi s modul e give n by th e secon d Tat e twis t o f th e root s o f unit y ther e i s a canonica l isomorphis m o f the for m [1 0 ] ^x(iV;(Q/Z)(2))^K2(iV).

3

Quaternionic Exercise s i n K-Theor y Galoi s Modul e S t r u c t u r e

Let f^T v denot e th e absolut e Galoi s grou p o f TV . Ther e i s a shor t exac t sequenc e o f ^Q-modules 0 - + (Q/Z)(2 ) — J n c ^ ( ( Q / Z ) ( 2 ) ) - ^ (Q/Z)(2)

+

-^0

in whic h th e first ma p i s the diagona l embeddin g an d G^ i s the subgrou p o f orde r two generate d b y comple x conjugation , c . Th e associate d lon g exac t cohomolog y sequence give s ris e t o a 2-extensio n o f Z[G(TV/Q)]-module s o f th e for m (Q/Z)(2) n ~ — > ( / n d £ ( ( Q / Z ) ( 2 ) ) ) n * -^ ( ( Q / Z ) ( 2 )

+)

n

"— H

1

^ ; (Q/Z)(2)) '

in whic h th e las t grou p denote s th e kerne l o f th e ma p fro m iJ-^TV ; (Q/Z)(2)) = K2 (TV) t o th e direc t su m o f th e i ^ ' s o f eac h o f th e rea l place s o f 7V . Ther e i s a canonical isomorphis m o f th e for m [1 0 ] ( Q / Z ) ( 2 ) n " -^Ki

nd

{N)

and pullin g bac k thi s extensio n b y mean s o f th e canonica l injectio n o f K2(ON,S)' into K2(N) yield s the 2-extension whos e Euler characteristic is equal to Q\(TV/Q, 3). Secondly, ther e i s a n isomorphis m o f th e for m ([1 4] , Chapte r 5 , Sectio n 2 ) CC(Z[Q8)) ^ (Z/4) * = {±1 } . The uniqu e tw o dimensiona l irreducibl e comple x representation , z/ , o f G(TV/Q ) = Qs i s symplectic an d th e roo t numbe r clas s i s given b y th e formul a WN/Q =

W Q M € {±1 } * CC(Z[Q S])

where WQ(V) i s th e Arti n roo t numbe r o f v defin e b y th e functiona l equatio n o f the Arti n L-functio n [1 2] . In al l the quaternio n examples , TV/Q , of 2.8, the Arti n roo t number , WQ(U), i s trivial. Our mai n resul t i s th e followin g computatio n o f £7i(TV/Q , 3). Theorem 1 . 1 Let E denote the set of rational primes {2,p } with p = 3 (modulo 8 ) and let TV/Q be the associated totally real Qg-extension introduced above. Then

fti(JV/Q,3) = - L Q , E ( - 1 , . ) ( - 1 ) ^ ^ ^ G C £ ( Z [ 0 8 ] ) * (Z/4) * where z = L Q , E ( — 1, Ind(l)). Here LQ^(S,P) denotes the Artin L-function of a representation of Q Q with the Euler factors at 2 and p removed and C = G(TV/Q(y/2, y/p)), the centre ofQg. Also, when the L-value is a non-zero rational number, LQ^(S wher e c denote s comple x conjugation an d x, y exten d t o iV(z ) b y fixing i . Th e clas s grou p o f N(i) ha s od d order. Thi s follow s fro m ([6] , Sectio n 1 6.4 ) o r ma y b e prove d directl y fro m th e cohomology sequence s derive d fro m th e shor t exac t sequenc e involvin g divisor s an d principal divisor s whic h define s th e class-grou p o f N(i). B y ([6 ] Sectio n 1 6.4) , th e prime idea l (/? ) ramifie d i n N(i)/N. Therefor e ther e exist s z\ G 0^{i) generatin g the uniqu e idea l o f N(i) abov e (3 and s o z\c{z{) = /3u\ fo r som e u\ G £)*N. Sinc e x e Qs interchange s 1 ± \/ 2 i t mus t als o interchang e th e ideal s o f N(i) generate d by z\ an d c{z\). Similarly , y £ Qs fixes eac h o f thes e ideals . Therefor e th e primar y decompositions o f (x + l)(zi/c(zi)) an d (y— l)(zi/c(zi)) ar e bot h trivia l an d bein g units o f norm on e a t al l infinit e place s implie s tha t the y ar e root s o f unit y i n N(i). When p = 3 the root s o f unit y i n N(i) ar e o f orde r twenty-fou r an d otherwis e ar e of orde r eight . I n al l case s ther e exis t integers , a(x) an d a(y), suc h tha t * ( * l / c ( * l ) ) = e 2i4X\z1 /c(z1 ))-1 , y ( ^ / c ( 8

2l

) ) = e2

{ V)

4 Zl/c(Zl).

If UJ satisfie s LJ = z\jc{z\) le t L = 7V(£ 64,o;) an d M — -/V(£64). L/Q i s a Galoi s extension whos e grou p i s generate d b y element s X,Y, c and a G G(L/M) wher e c

Quaternionic Exercise s i n K-Theor y Galoi s Modul e S t r u c t u r e 7

is comple x conjugatio n an d th e actio n i s give n b y th e followin g formula e i n whic h the integers , a{X) an d ct(Y), ar e multiple s o f 3 when p > 3 : X(w)

- ?1 9 2 w

Y{w)

- S1 9 2 w >

X(^)

_

Y(fa) = ^ 6 4

-l 5

,

X((3)

= 0 / (^ + 1),

Y{(3)

= P{{u + Vy' /p)/y/2),

Xv(u) = Xv{—u) = —Xv(§v) — —Xv{—5v) for all local unit s u. By Lemma 3.2, S — {l,e,e2 ,e 3 } i s a set of representatives for ( 0 | C u / ( l - h / ) ) / { ± l , ± 5 } , where e = 1 + y/2. Thu s w e can write (1 4 ) a s ^(x,) = 2 -

5

/

2

x,(c)X:xJ

S

)(^(^+^(^)-^„(f)-^(^))).

(16) For u £ 0* K (Z/1 6) 9 — • A f — > B' — • (Z/8) 3 — > 0 where A ' an d i? ' ar e finite, cohomologicall y trivia l Z[Q8]- m °dules. In thi s sectio n w e will develo p a yog a fo r evaluatin g th e Eule r characteristi c [A'] - [B'\ G CC(Z[Qs)) * (Z/4)* . Define fre e Z[Q8]-modules , Do an d D\, b y D 0 - Z[Q 8] < a o > , 6 1 = Z[Q 8] < &o , &o,&o > •

)

Quaternionic Exercise s i n K-Theor y Galoi s Modul e S t r u c t u r e

15

We hav e th e star t o f a fre e resolutio n o f th e for m

...-^b2-^bx-^b^ (z/8)

3 —>o

where v(l) = 1 (modulo 8) , d(bo) = (x — 3)ao , d(b' 0) = (y — l)& o an d d(&o ) = 8ao The clas s o f th e 2-primar y 2-extensio n i s represente d b y a 2-cocycle , j : £> 2 — > (Z/16)g, whic h factorise s t o giv e a Z[(5s]-niodul e homomorphism , F , well-define d up t o th e additio n o f map s whic h exten d ove r D\ an d fittin g int o a commutativ e diagram o f th e for m Ker(d) -

D

d

>

x

£>

0

"

» (Z/8)

3

(Z/8)

3

\F

(Z/16) 9 ^

A '*

B

'^

Lemma 4. 1 j4n y choice of Z[Qs]-module homomorphism, F : Ker(d) — > (Z/16)g, is surjective and P — Ker(F) is a finitely generated, projective Z[Qg] module such that [A!) - [B'\ = [P] G CC(Z[Q 6]) = (Z/4)* . Proof Th e resul t i s easil y prove d (c.f . [3] ) onc e w e hav e show n tha t F i s surjective. T o se e thi s le t ... ^ D

2

- ^D

x

- ^ D 0 — > Z —+ 0

be a fre e Z[Qs]-niodul e resolutio n o f the trivia l module , Z . Le t z G H4(Q$; (Z/8) 3 ) = Z/ 2 b e a generator. Th e cup-produc t wit h th e clas s of the 2-primar y 2-extensio n i / 4 ( Q 8 ; ( Z / 8 ) 3 ) — # 6 (Q 8 ;(Z/1 6) 9 ) is an isomorphism . W e may represent z by a cocycle of the form D 4 —> Ker(d2) — > (Z/8) 3 . Startin g wit h 0 w e ma y construc t a chai n ma p o f th e fre e resolutio n . . . - ^ DQ

-^ D

5

^ > £ 4 — • #er(d 2) — 0

into th e resolutio n o f (Z/8) 3 whic h result s i n a homomorphism , £(z ) : DQ —> D2 — > Ker(d). Th e imag e o f z unde r th e cup-produc t isomorphis m i s represente d by th e cocycl e F6(z) : DQ —> (Z/1 6)g . Therefor e / mus t b e surjectiv e i f ever y cocycle representin g th e generato r o f iJ 6 (Q 8 ; (Z/16)g) i s surjective . From ([1 5] , p . 1 9 ) th e standar d fre e ZfQgl-resolutio n o f Z ha s D7 = Z[Qs] < e >, DQ = Z[Q 8] < c , c' > , D 5 = Z[Q 8] < 6,6 ' > with d 5(c) = ( 1 + z) 6 - ( 1 + y)b' , d b(d) = (xy + 1 ) 6 + (x - 1 )6 ' an d d 6(e) = (x — l) c — (a; y — l)c' . Fro m thes e formula e on e find s easil y tha t th e generato r o f H6(Qs] (Z/1 6)g ) i s represente d b y an y homomorphis m fro m DQ t o (Z/1 6) g o f th e form (c , c') H ^ (2a + 1 ,2 6 + 1 ) , al l o f whic h ar e surjective . 4.2 I n D\ — Z[Qs] < 6 0, 6Q.6Q > w e may define som e elements by the followin g formulae:

Ted Chinburg , Manfre d Kolster , Georg e P a p p a s , an d Victo r P . Snait h

16

A = (x - 3)f # - 86 0, B = ( y - 1 )6 £ - 8*/ 0, C = (1 + y + y 2 + y 3)b'0, Xi = ( 1 - y 3)b0 + (3y 3 - xy 3 - xy - xy 2)b'0, )bo - (3y + 3y 2 + 1 1 / + z2/ 2)&o,

3

X2=(y- V

X3=(y2-y3)b0-(3y2+xy)b'0, 3

X4=(x- xy

)b0 + (3xy 3 + y 2)b'0,

X5 = (xy - xy 3)b0 + (y + y 2 - 3xy - 3xy 2)b'0, X6 = (xy 2 - xy 3)b0 - (y 3 + 3xy 2)b'0, X7 = (3y 3 + xy 3)b0 - (8y

3

- 3xy - 3xy 2)b'Q + b%.

The followin g result s ar e straightforwar d t o verify . Lemma 4. 3 In terms of the elements of 4..2 a Z-basis for Ker(d) is given by {ylA,0 (Z/1 6) g of the following form in which s,z G Z.

z A B C

xl x2 x3 xA x5 x6 x7

F{Z) (modulo 1 6 ) -9(2s + l ) -6(2s + l ) (-3-4z)(2s + l ) (2 + 4*)(2 s + l ) 2(2s + l ) 9(2s + l ) -(2s+1 ) (-1 - 4z)(2s + 1) (2s + 1 ) -2z(2s + 1 )

The Euler characteristic corresponding to such a cocycle is non-trivial if and only if z = 0,1 (modulo 4) . Proof Usin g Propositio n 4. 4 an d th e fac t tha t F mus t b e surjective , i t i s straightforward t o sho w tha t F ha s th e for m describe d i n th e table . O f course , this doe s no t guarante e tha t al l value s o f s an d z eve n giv e ris e t o a ZfQg]-modul e homomorphism. Wit h hindsigh t w e shal l se e tha t the y al l do , bu t w e d o no t nee d this fac t sinc e w e ar e onl y concerne d wit h th e (non-empty ) se t o f F' s representin g 2-extensions whic h hav e Eule r characteristic s i n th e class-group . A Z-basi s fo r Ker(J ) i s give n b y y l(A + 9X 6) ( 0 < i < 3) , y J (B + 6X 6) ( 0 < j < 2) , X 7 - f(X 7)X6, C + ( 3 + 4z)X 6, xC + (1 1 + 4z)X 6, X 1 - ( 2 + 4z)X 6 , X2 - 2X 6, X 3 - 9X 6 , X 4 + X 6, X b + ( 1 + 4z)X 6 an d X 6. Al l bu t th e las t elemen t are i n P = ker(F : Ker(d) — > (Z/1 6)g ) s o tha t a Z-basi s fo r P i s give n b y y*(A + 9X6) ( 0 < i < 3), ^ (B + 6X 6 ) ( 0 < j < 2), X 7-f(X7)X6j C+ ( 3 + 4z)X 6 , xC+{ll + 4z)X6, X 1 -(2 + 4z)X6, X 2-2X6, X 3-9X6, X 4 + X6, X 5 + (l+4z)X 6 and 1 6X6 Now conside r th e surjectiv e map , TT : P — > Z[Qs ] < ^ o > , give n b y projectin g each element t o its 6Q-coordinate. Th e kernel of this map, Q = ker(n), ha s a Z-basi s given b y C+ ( 3 + 4z)X 6 , xC + (1 1 + 4z)X 6 , X x - ( 2 + 4z)X 6 , X 2 - 2X 6 , X 3 - 9 X 6 , X4 + X 6 , X 5 + ( 1 + 4Z)XQ an d 1 6X 6 . I f P i s projective the n Q i s projective o f ran k one. Let Q ± = {zGQ|x 2 (z) = ± z } . We ar e goin g t o manufactur e a projectiv e submodule , 5 , o f Q suc h tha t Q/S is a finite grou p o f od d orde r an d fo r whic h w e ca n evaluat e th e clas s o f S i n th e classgroup. If Q i s projectiv e the n Q + = ( 1 + x 2)Q i s a fre e Z[Qg 6]-module an d a basi s element ma y be found b y the followin g process . Le t a = ( l + x + x2 + x 3 ) ( l + y), A = (l+x+x 2 +:r 3 )(l-2/), / 1 = (l+y+y 2+y3){l-x) an d p = {l+xy+(xy) 2 + {xy) 3)(l-y) in Z[Q —

>R

—>0

in whic h S i s projectiv e an d 5 + = Z[Qf] < ( 1 + x 2)(C + ( 3 + 4£)X 6 ) > an d S - = Q_ . If x + : (Z/8) * — > {±1 } ha s kerne l equa l t o {±1 } the n th e clas s o f R i n C£(Z[Q 8 ]) = {±1 } i s equa l t o x+(|i?| ) = x+(( l + 2z)( 5 + 4z)) , b y th e formul a o f ([14] Theorem 5.2.23 ; [3] , Section 4.3.6, p. 44 ; [1 3] , Lemma 6.3.7, [1 6] , p. 88 ) whic h is restate d fo r th e reader' s convenienc e i n 5.4 . Henc e x+(|^| ) £ (Z/4) * depend s only o n z (modul o 4 ) an d i s non-trivia l precisel y whe n z = 0, 1 (modul o 4) . According t o ([1 4] , Chapte r 5 , Sectio n 2 ) th e clas s o f S i s determined b y com paring a basi s elemen t o f S+ wit h on e fo r 5 _ i n th e followin g manner . I f w e writ e Ui = ( 1 — x 2)Xi the n 5 _ = Q _ = ( 1 — x 2)Q an d i s generate d ove r th e integra l quaternions, H z , b y ( 3 + 4z)*7 6, ( l l + 8*)[/ 6 (sinc e ( l - x 2 ) C = 0) , U X - ( 2 + 4z)[/ 6 , f/2 - 2{7 6, f/ 3 - 9f/ 6, ^ 4 + C/e , ^ 5 + ( 1 + 4z)E/ 6 an d 1 6f7 6. However , b y 4.4(v ) an d (vii), U$ = U4 + UQ and 4. 4 easil y implie s tha t UQ is a basi s elemen t fo r 5 _ ove r Hz- T o compar e base s w e tak e th e basi s o f S+ an d lif t i t fro m S/S- = S+ t o C + ( 3 -+- 4z)X 6 G 5. Nex t w e multiply thi s liftin g b y ( 1 - x 2) t o giv e (l-x2)(C+{3 +

4z)X

6)eS„.

This i s a n integra l quaternio n time s th e basi s element , UQ. I n fact , i t i s ( 3 + 4z)U&. Th e clas s o f S i n CC(Z[Q$\) = (Z/4) * i s equa l t o th e reduce d nor m o f the quaternioni c multiplie r - i n thi s cas e ( 3 + 4z) 2 - whic h i s trivia l i n (Z/4)* . This complete s th e proo f sinc e th e exac t sequenc e definin g S implie s th e relatio n [Q] = [R] + [S] e CC(Z[Q

8}).

The followin g resul t give s a yog a whereb y t o extrac t th e Eule r characteristi c directly fro m th e 2-extensio n sequenc e i n Theore m 4.5 . Corollary 4. 6 Suppose that we have a 2-extension of Z[Q$]-modules of the form (Z/16) 9 —>A'—+B'—+ (Z/8)

3

which possesses an Euler characteristic, as in Theorem 4-5. Choose any element, 7 £ B', which maps to a generator of (Z/S)^. Hence there exist elements 6 , b' G A' mapping to (x — 3) 7 and (y — 1 ) 7 respectively. Form xb = ( Xy - xy 3)b + {y + y 2 - 3xy - 3xy 2)b' G A' and x6 = {xy 2 - xy 3)b - {y 3 + 3xy 2)b' G A'. In fact £5 , £6 li> e i n (Z/1 6) g and are units (modulo 1 6). Furthermore, there exists an integer, z, such that x5 = x 6(-l-4z) G(Z/ 1 6)

9.

The value of z (modulo 4) is independent of the choices and depends only on the equivalence class of the 2-extension. Euler characteristic, [A 1 ] — [B'\ of Theorem 4.5, is non-trivial if and only if z = 0,1 (modulo 4).

20

Ted Chinburg , Manfre d Kolster , Georg e P a p p a s , an d Victo r P . Snait h

Proof I t i s eas y t o se e tha t £ 5 an d XQ lie i n ( Z / 1 6 ) Q . Fro m th e discussio n in Sectio n 3 , o f th e homomorphism , F : Ker(d) — > (Z/1 6)g , w e se e tha t w e ma y perform th e yog a i n th e resolutio n fo r (Z/8) 3 t o obtai n X$,XQ £ Ker(d) an d se t Xi= = F(Xi) fo r i = 5,6 . Thes e choice s ar e unit s (modul o 1 6 ) b y Theore m 4.5 , from whic h i t i s easily see n tha t th e sam e i s true fo r al l choices. Similarly , fo r thes e elements x$ = XQ(— 1 — 4z) (modul o 1 6) . Therefor e th e resul t follow s fro m th e straightforward verificatio n tha t al l choice s yiel d th e sam e ratio , X$/XQ £ (Z/1 6)* . 5 Evaluatio n o f Sli(N/Q, 3 ) Let p = 3 (modul o 8 ) b e a prim e an d le t N/Q b e th e associate d Qg-extensio n constructed i n Sectio n 2 . Le t z an d UJ b e a s i n 2.9 . Therefor e UJ satisfie s UJ 8 = z/c(z) an d w e hav e a Galoi s extension , L = N(!;Q4,UJ) ove r Q , wit h G(L/Q) =< X,Y,a,c>. A s explained i n Sectio n 3 , the 2-Sylo w subgroup o f K2(ON,S)' i s cyclic of orde r eight . I t i s a subgrou p o f K2(N)' whic h i s i n tur n a subgrou p o f Hl(N;(Q/Z)(2))*K2{N). In order to apply the 2-primar y yog a of Corollary 4.6 we need to describe a n elemen t of order eigh t i n H 1 ^; (Q/Z)(2) ) whic h correspond s t o a generator o f the 2-Sylo w subgroup o f K2(ON,S)'- Observ e tha t th e absolut e Galoi s group , fl^? i s generate d by fijv(i) an d c , s o tha t i t act s triviall y o n fig ^ 8 = M s (2) insid e (Q/Z)(2) . Therefore th e homomorphis m o f 2.11, / : G(L/N(i)) —>

Z/ 8

given by the formul a $ 9) = g{oj)/oJ, extend s to G(L/N) ^ by settin g /(c ) = 0 and define s a cohomolog y clas s

G(L/N(i)) x

Z/2 < c >

[/]€i? 1 (AT; M8 (2)). Proposition 5. 1 The image of[f] in H 1 (N; (Q/Z)(2) ) corresponds under the isomorphism with K2{N) to a generator of the 2-Sylow subgroup of K2(ON,SY• Proof Th e Steinber g symbol , {2 , £s} ^ ^2{N(i)) ha s orde r eigh t becaus e it' s tame symbo l (th e imag e unde r th e boundar y ma p i n th e localisatio n sequence ) i s zero excep t i n multiplicativ e grou p o f th e residu e field correspondin g t o z , {ON(i)I{Z)Y- Thi s residu e field ha s orde r p 2 wher e th e tam e symbo l i s equa l t o £s- Th e transfer , Tr N^iyN({z1 ^s}) € ^(N) als o ha s orde r eigh t sinc e it s imag e i n K2(N(i)) i s ( 1 + c)({z^s}) whos e tam e symbo l i s concentrate d i n (0 N(i)/(z))* 0 (ON(i)/(z))* an d i s give n ther e b y (C8 5c(^s))- O n th e othe r hand , sinc e transfe r commutes wit h th e ma p t o th e su m o f th e K^s o f th e infinit e place s an d sinc e K2 of th e comple x number s i s trivial , th e elemen t Tr N^yN({z^s}) lie s i n ^(AT)' . Therefore thi s elemen t i s a generato r o f the 2-Sylo w subgrou p o f K2{ON,S)''• Now le t u s conside r th e cohomologica l side . Let F b e a field of characteristic differen t fro m 2 and le t F s denot e a separabl e closure o f F. Followin g Tate , conside r th e exac t sequenc e

0 —+ ^ — * O^T - ^ (F ST — 0 and th e resultin g coboundar y homomorphis m dF : H°{F; (F s )*) ^ F * — > H\F; /x

2n).

This coboundar y i s define d b y sendin g y £ F * t o th e 1 -cocycle , dp : Sip = G(FS/F) — > p2 n give n b y dp(y){g) = g{w)/w wher e w e (F s )* satisfie s iv 2" = y. Therefore, i f n = 3 and F = iV(i) , w e have dF(y){g) = g(w)/w wher e w 8 = y.

Quaternionic Exercise s i n K-Theor y Galoi s Modul e S t r u c t u r e 2 1

Tate constructe d a ma p o f th e for m H1(F;Q2/Z2(2))-^K2(F) which i s a n isomorphis m ont o th e 2-primar y torsio n subgroup . Thi s ma p i s char acterised i n th e followin g manner . Ther e i s a ma p h:K2(F)-^H2(F;Z2(2)) defined o n Steinber g symbol s b y /i({a , b}) = dp (a) U di?(6). Th e othe r ma p i s characterised b y th e fac t tha t th e compositio n H1 (F; (Q 2/Z2)(2)) ^

K

2(F)

2

^ H

(F; Z

2(2))

is equal t o th e coboundary , (Q 2 /Z 2 )(2) — 0 . Here Z 2 (2) = Z 2 0 Z 2 an d Z 2 i s the Galoi s modul e give n b y the invers e limi t o f th e groups o f two-primar y root s o f unit y i n F s. From th e followin g commutativ e diagram , wit h F — N(i), H\F; Z

2 (l))

0 H°(F; p s) >

H

l

(F; (Q

2 /Z 2 )(2))

10 M s ( 2 )

given b y t(g) = (g(w)/w) 0 £ s wher e w 8 — z. Th e cohomologica l transfe r an d the K 2 transfe r homomorphism s coincid e unde r Tate' s isomorphism . Therefor e th e cohomology clas s w e want i s given b y TrNil)/N([t}) eH\N;/i

8(2)).

The transfe r i n grou p cohomolog y fo r a subgrou p o f inde x tw o i s describe d explicitly i n ([1 5] , Chapte r 4 ) fro m whic h on e finds tha t a 1 -cocycl e representin g Trpj(iy]sr([i\) i s give n b y th e followin g formula , fo r g £ ^N(y/^i)^ TrN{iyN([t])(g) =

c(c- 1 gc(w)/w)®c(Z8)"+" (g{w)/w)®£ = Tr

8

N{i)/N{[t})(gc)

= c{c- 1 gc(w)/w) 0

c(&) " +" (gc 2{w)/w) 0 £ 8

where "+ " refer s t o "addition " i n M s 0 Ms - However , i n M s 0 Ms , c(c _1 #c(w)/w) 0 c(& ) = {gc{w)/c(w)) 0 Cs" 1 = (c{w)/gc{w)) 0 £ 8 and (c(^)/^c(^)) 0 & " + " (g{w)/w) 0 $ 8 =

((^(^/cH))/(^/cH)) 0 $8 - (^(W)/^) 0 &

22 T e

d Chinburg , Manfre d Kolster , Georg e Pappas , a n d Victo r P . Snait h

where u; 8 = z/c(z), a s required. 5.2 W e have to apply th e 2-primary yog a of 4.6 to the cohomology sequenc e

(Q2/Z2)(2)n" — , 7 n ^ ? > ( ( Q 2 / Z 2 ) ( 2 ) ) n " — (Q 2 /Z 2 )(2)°~ — H\N; (Q

2 /Z 2 )(2))

—*....

Recall that , b y 2.9, we have a Galoi s extension , L / Q , containing N/Q wit h grou p G(L/Q) =< X,Y,a,c > = < X,Y,a > x < c >. Defin e £ 7 = L(Ci2s ) the n E/Q is als o Galoi s wit h G(E/Q) ^ G(E/Q(i))x < c > an d G(E/Q(i)) ^ < X , y , a > where the action of X, Y, a is given by the same formulae a s for L / Q. I n particular, Y an d a fix £i28 and X(£i2s) = £1 2 8 • We shall perfor m th e yoga in the cohomology sequence obtaine d b y replacin g ^ Q , £l N an d (Q 2 /Z 2 )(2) b y G(E/Q), G(E/N) and /ii28(2) , respectively . Thi s i s permissible sinc e th e generator o f order eigh t in ^2(ON,SY originate s i n iJ 1 (A r ;/i8(2)) an d is represented b y the homomorphism c >— G{L/N{%)) -U ^ /x 8 = M8 (2).

G(L/N) * G(L/N{i))x
Z/128 — > Ind< {cl/Q)(Z/128) —

+ (Z/1 28) + — • 0

we for m th e cohomology sequenc e (Z/128)G(a) - ^ i i f ® a + ^ vXY

1

9a.

-u,i v,i

Therefore, i n orde r t o evaluat e th e invers e image s unde r 0 o f x$ an d XQ it suffice s to comput e th e multiple s o f Yl v v ® & whic h occur . Furthermore , th e presenc e o f the facto r ^2 v mean s tha t w e may manipulat e th e X' s an d 7' s a s i f the y wer e i n the group-rin g o f Qg. W e find easil y tha t th e ter m i n £ 5 i s — (8/169)a(F)/2 an d in XQ is (8/1 69)a(F)/2 . Sinc e thes e ar e 2-adi c unit s w e se e tha t a(Y)/2 i s a n od d integer an d x5 = ( - l ) x 6 G Z/16. This correspond s t o z = 0 (modul o 4 ) i n Corollar y 4. 6 an d w e have established th e following result . P r o p o s i t i o n 5. 3 Let N/Q be 2.8 and let

one of the quaternion extensions of Theorem

0 — + (Z/1 6) 9 —+A'-^B'-* (Z/8) be a 2-extension representing fii(iV/Q,3). Then

3

—+ 0

the 2-primary part of the 2-extension defining

[A'} - [B'] = - 1 G {±1} * C£{Z[G(N/Q)]). In orde r t o evaluat e th e Eule r characteristi c o f th e od d primar y par t o f th e 2-extension definin g fii(iV/Q,3) w e shall us e th e followin g result . P r o p o s i t i o n 5. 4 ([1 4], Theorem 5.2.23, [3], Section 4.3.6, p. 44; [1 3 ] Lemma 6.3.7, [1 6] , p. 88) Let X be a finite group of odd order with a Q 8-action then [X] e CC(Z[Q 8}) * (Z/4) * 25 given by

[x] = x+(\x+\)(V\X-\) =

W\X-\)(-i)W4)l093UX+n e

(z/4)*

2

where X± — {z G X\x (z) = ±z } and 0 < (^/|J^C_| ) is the positive square root of the cardinality of X-. Alternatively, if z G Z, (-i)(i/4)^ 2 (|x+|) e (z/4) * is equal to x+i z) where x+ • (Z/8)* — > (Z/4) * is the surjective homomorphism with kernel equal to {±1 } . Proof o f Theore m 1 . 1 Th e class , fii(JV/Q,3), i s define d a s th e Eule r characteristic o f a 2-extensio n o f th e for m 0 -^ Ki

nd

(0NtS) —

A — * B — > K 2(ON,SY —

0

where S i s the se t o f primes o f N ove r {2,p } an d A, B ar e finit e groups . Th e Eule r characteristic i s the su m o f that o f th e 2-primar y par t - non-trivia l b y Propositio n 5.3 - an d tha t o f th e od d primar y part , whic h w e shal l denot e b y 0 __ > k? d(0N,s) —

A"

- ^ B" — k 2(0N^sy -^

0 .

Since al l thes e group s ar e finit e an d o f od d orde r w e hav e [A"} - [B"] = [kl nd(0N.s)} -

[k 2(0N,sY} e

CC(Z[Q 8}) £ * (Z/4)*.

Now we shal l appl y th e analysi s o f ([1 3] , Sectio n 7.2 ) t o calculat e [A"]-[B"]eCC(Z[Qs])^{Z/A)*.

Quaternionic Exercise s i n K-Theor y Galoi s Modul e S t r u c t u r e 2

7

Let xi,X 2 : Qs — > {±1 } b e give n b y x i O ) = - 1 = X2 E p+l'q i s th e composit e K - z

v

^

X

~

Zt+uA)

- ^ H%£\X,A)

-U

p q

H%£l

Zp+2(X

-

Z^A).

1.2 Passin g t o th e limit . W e no w assum e tha t X i s equidimensiona l an d noetherian o f dimensio n d an d tha t fo r al l p , codim ^ Z p > p . Orde r th e se t o f (d + 1 )-tuple s Z b y Z < Z' i f Z v C Z' v fo r al l p. Th e constructio n o f th e exac t couple C - i s covarian t wit h respec t t o thi s ordering . Passin g t o th e limi t no w defines a ne w exac t coupl e C wit h D™ - l i m 5 # ^ ( X , A) : = ff&Z^A), E^ =

an

d

+ plZp+i{X-Zp+uA)

\^Hl

where X^ denote s th e se t o f point s o f codimensio n p i n X. Th e followin g lemm a describes th e secon d direc t limi t mor e concretely : Lemma 1 .2. 1 a) / / T i , . . . ,T r are pairwise disjoint closed subsets of X, then ($H^(X,A)-^H{jTz(X,A). b) We have E?'i~ 1 1 1 1 H2 +*(X,A) ( . where, for x £ X^, H£+«(X,A)

)

: = limH^iU, A). U3x

Proof a ) B y inductio n o n r w e ma y assum e r = 2 . W e hav e a commutativ e diagram H^(X,A)

H*Ti{X,A) H*

TiUT2(X,A)

H^(X-T

UA)

H^(X-T2,A) in which the row and colum n ar e exact an d th e two diagonal map s are isomorphism s by excision . Th e clai m follows . b) Note that, i f the irreducible components of codimension p of Z p are Y\ ,... ,Y r, then Z p\Zp+i — Y[{Yi\Z p+\) (disjoin t union ) a s soo n a s Z p+\ contain s th e Y tr\Y3 and th e highe r codimensiona l component s o f Z p. Th e isomorphis m no w follow s from a) . •

36

Jean-Louis Colliot-Thelene , Raymon d T . Hoobler , an d B r u n o Kah n

The spectra l sequenc e associate d t o th e exac t coupl e C stil l converge s t o H*(X,A). I t i s calle d th e coniveau spectral sequence (compar e [2 , Remar k 3.1 0] , [15, p . 239]) : Ep'q = \[

p+q

1 {X,A)=>Hp+q(X,A). ( .2

H

)

The associate d filtration NpHn(X, A)

= Im(tf£ (p) (X, A) - > H n(X, A))

is called th e coniveau filtration o r filtration by codimension of support. It s E\ -terms yield Cousin complexes: (%>*

0 - J ] H*{X,A)-^ I

] H^(X^A)-^(1.3)

In th e nex t section , w e shall need : Lemma 1 .2. 2 For all n,p, the presheaf

U^ ] J H2(U,A) is a sheaf for the Zariski topology of X. This sheaf is flasque and can be identified with

U i x*H£(X,A) xex(p) where i x is the immersion x c -^ X and the abelian group H X(X, A) is considered as a (constant) sheaf on x for the Zariski topology. Proof Fo r x 6 X^ p\ defin e a preshea f F x o n th e categor y o f Zarisk i ope n subsets o f X b y F

f H2(U,A) i

*(U>-{ 0

i

f Usx

f Ujx.

By definitio n o f H^{X,A) (se e Lemm a 1 .2. 1 b)) , F X(U) = F X{X) i f U 3 x, hence F x i s the shea f i x*Hx(X, A) , whic h i s obviously flasque. • Suppose no w tha t X i s a smooth , irreducibl e variet y ove r a field fc, A i s a locally constant , constructibl e shea f an d th e stalk s o f A ar e m-torsion , wit h m prime t o th e characteristi c o f k. W e shal l us e cohomologica l purit y t o transfor m these complexe s int o one s whic h involv e onl y etal e cohomolog y withou t supports . For i £ Z , w e writ e A(i) = A ® M ^ where fi m i s the shea f o f m-t h root s o f unity . Le t Z b e a smoot h irreducibl e close d subvariety o f X o f codimensio n p. Cohomologica l purit y ([36 , ch . VI , Section s 5 and 6] , [SGA 4 1 /2 , p . 63 , Theorem V.3.4] ) the n give s canonica l isomorphisms :

HZ(X,A)^Hn-2?(Z,A(-p)).

37

T h e Bloch-Ogus—Gabbe r Theore m

Noting that , fo r a n arbitrar y irreducibl e close d subvariet y Z o f X , th e inter section Z D U i s smoot h fo r smal l enoug h ope n subset s U, henc e define s a smoot h pair Z n U C U, thi s yield s isomorphism s

H^(X,A)^H^(k(x),A(-p))1 for x G 1 ^ (wher e k(x) i s th e residu e field o f x) , s o tha t th e comple x (1 .3 ) take s the following , perhap s mor e familia r for m (compar e [2 , Proposition 3.9]) :

0-> #«(*(*), A )- J ] H«- 1 (fc(a;),^(-l))^--• • • - I I H"-"(k{x),A(-p))^..-

.

(1.4)

Here, k(X) i s th e functio n field o f X. Not e tha t # 9 - p (fc(:i;),yl(-p)) i s simpl y Galois cohomology of the residue field of x. S o the E\-terms o f the coniveau spectra l sequence hav e take n a n especiall y simpl e form . Not e als o tha t Ep{q = 0 for p>q.

2 Th e Effacemen t Theore m an d th e Bloch-Ogu s Theore m 2.1 Effaceabl e sheaves . I n this paper, w e are interested i n a special propert y of th e shea f A: Definition 2.1 . 1 Le t X b e a variet y ove r k. Le t t i , . . . , t r G X b e a finite number o f points containe d i n som e affin e ope n subse t o f X. A n etal e shea f A ove r X i s effaceable atti,...,t r i f th e followin g conditio n i s satisfied : Given p > 0 , fo r an y smal l enoug h affin e ope n neighbourhoo d W o f t i , . . . , t r and an y close d subse t Z C W o f codimensio n > p + 1 , ther e exist s a smalle r ope n neighbourhood o f t\,..., t r, U C W , an d a close d subse t Z' CU containin g Z n ( 7 such tha t (1) codimu(Z') > p ; (2) th e composit e H%{W, A) - > H% nU(U, A) - > H%, nU(U, A) i s 0 for al l n > 0 . The shea f A is effaceable if i t i s effaceable a t £ i , . . . , t r fo r a l H i , . . . , t r a s above . This conditio n look s ver y technical , bu t i t ha s far-reachin g consequences : Proposition 2.1 . 2 Let R = Ox,(t 1 ,...,tr) be the semi-local ring of X at ( £ i , . . . , t r) and Y = Spe c R. Suppose the sheaf A is effaceable atti,...,t r. Then, in the exact couple defining the coniveau spectral sequence for (Y,A), the map i PiQ is identically 0 for all p > 0. In particular, p , g _ f H"(Y,A) i ^2 ~ \ 0 i

1

fp = 0 fp > 0.

Strictly speaking , thi s argumen t i s onl y vali d whe n th e groun d field k i s perfect . Otherwis e a close d poin t o f X whos e residu e fiel d i s inseparabl e ove r k wil l produc e a counterexampl e t o the statemen t jus t befor e thi s equation . However , i f k i s imperfect , th e isomorphis m wil l hol d after passin g t o it s perfec t closure . Sinc e etal e cohomolog y i s invarian t unde r purel y inseparabl e extensions [36 , Chapte r II , p . 77 , Remar k 3.1 7] , th e isomorphis m hold s i n general . Compar e [2 . Remark 4.7] .

38 J e a n - L o u i

s Colliot-Thelene , Raymon d T . Hoobler , an d B r u n o K a h n

The Cousin complex (1 .3 ) yields an exact sequence:

Proof Conside r th e diagra +mH% Hnz{W,A)

H^lP+l)(W,A) .

H^

• H%,

nu(U,A)

nU(U,A)

H?uP)(U,A)

1 (P+ )(U,A)

H^(P)(Y,A).

Hy{p+1)(Y,A) •

The compositio n o f arrow s i n th e firs t ro w i s identicall y 0 fo r an y n. Therefore , the composition s H%(W,A) - • H£ {P+1 )(Y,A) - > H^ {p)(Y,A) ar e 0 . Passin g t o th e limit ove r Z , thi s give s tha t th e composition s H^ v(p+1 ) (W , A)— » Hy ip+1 )(Y,A)— > Hy{p) (Y , A) ar e 0 . Passin g t o th e limi t ove r W, w e get tha t th e ma p # £ ( p + 1 ) (y, . 4 ) i ^ ^ ' /T?

(P)

(F, A)



is itsel f 0 . Corollary 2.1 . 3 (Th e Bloch-Ogu s theorem ) Let A be effaceahle on X. Then, the E2-term of the coniveau spectral sequence for {X,A) is E™ = H?

ar(X,H0(A))

q

where 7i (A) is the Zariski sheaf associated to the presheaf U i-» H

q

(U,A).

Proof Conside r th e comple x o f flasqu e Zarisk i sheave s associate d t o th e Cousin complexe s (1 .3 ) (compar e Lemm a 1 .2.2) :

0 - ] J i x.H«(X,A)^ U wH2 +«(A-,A)^...-> U i x,H*+*(X,A) xex xexw

xex(p~) (2.1)

Proposition 2.1 . 2 implie s tha t (2.1 ) i s a resolutio n o f H q(A), wit h globa l Sec tions (1 .3) . Th e conclusio n follows . • 2.2 Th e effacemen t theorem . Th e mai n resul t o f [1 4 ] is: Theorem 2.2. 1 (Gabber) For X smooth over k, any torsion sheaf (on the small etale site of X) of the form p* A$ is effaceahle, where p : X— » Spec/ c is the structural morphism. Specializing t o twiste d root s o f unit y an d usin g sequenc e (1 .4) , w e ge t a mor e familiar case : Corollary 2.2. 2 (Bloch-Ogus, [2]) Let X be smooth, irreducible overk, R and Y as in Proposition 2.1 .2, and let m be an integer prime to the characteristic of k.

39

T h e Bloch-Ogus—Gabbe r Theore m

Then, for all i G Z and q > 0, we have an exact sequence:

R e m a r k s 2.2. 3 (1) I n Appendix A we shall remove the hypothesi s that Ao is torsion i n Theore m 2.2.1. (2) Effaceabl e sheave s hav e th e followin g trivia l stabilit y properties : • A sheaf A i s effaceable a t £ i , . . . , tr i f and onl y i f A\w i s effaceable fo r a smal l enoug h ope n neighbourhoo d W oit\,... ,t r (effaceabilit y i s a local condition) . • Le t A, B b e tw o etale sheave s ove r X. The n A ® B i s effaceable i f an d only i f A an d B are . (3) Ove r a smooth k- variety X , i t is not tru e tha t al l etale sheaves are effaceable . As a n example , tak e k — R an d fo r X th e affin e lin e A ^ . Le t / : X' — > X be th e two-fol d coverin g give n b y th e equatio n x 2 + y 2 = 0 , wher e x i s th e parameter o f A ^ , an d le t A = /*Z/2 . Le t Y b e th e localizatio n o f X a t 0 and Y' — X' XxY. Sinc e / i s finite, ther e i s a n isomorphis m H*(Y,A) - ^ f P ( Y ' , Z/2) . On th e othe r hand , th e structura l morphis m Y' — » Spec R i s spli t by th e inclusio n o f th e close d poin t o f Y' ', henc e if * (Y 7, Z/2) contain s i/*(R, Z/2) 7 ^ 0 a s a direc t summand . However , th e tw o generi c point s ?7^, ?72 o f Y' ar e isomorphi c t o Spe c C(x), henc e th e Kumme r theor y clas s of - 1 i n H l(Y\Z/2) goe s t o 0 i n bot h i J ^ Z ^ ) an d if 1 (T^ , Z/2). Corre spondingly, i f r ; denotes th e generi c poin t o f Y , th e ma p

is no t injective . On e ca n giv e a simila r exampl e wit h X = A 2^ an d X' defined b y th e equatio n x 2 + y 2 + z 2 = 0 (fro m i / 2 onwards) , etc . Se e als o [9, p . 1 73] . (4) On e ma y howeve r produc e effaceabl e sheave s whic h ar e mor e genera l tha n those o f Theore m 2.2.1 : Proposition 2.2. 4 Let X — > X be a finite map between schemes of pure dimension d with X smooth, and let B be an etale sheaf over X. If B is effaceable a t / - 1 ( { ^ i 5 • • • ,tr}), then f*B is effaceable at t\,... ,t r. Proof Le t T = {*i,...,* r }, Z b e a s i n Definitio n 2.1 .1 , f = f~ l{T\ an d Z — f~ 1 (Z). Appl y th e effacemen t theore m t o (X,T , Z, B) an d ge t a pai r ([/ , Z') such tha t T C U an d th e compositio n

iq{x,B) - iq

n0(u,B)

- iq,

n0(u,B)

is 0 for al l n > 0 . Le t U = X-f(X-U) Z' = f(Z') so tha t T C U, Z c Z', codim x Z' > p an d f~ l(U) CU, get a commutativ e diagra m

Z

1

C f~ l{Z'). W

e the n

40

Jean-Louis Colliot-Thelene , Raymon d T . Hoobler , an d B r u n o K a h n

H%(X,f.B) »H%

nU{UJ,B)

.

n

H

z,nU(U,UB)

I

H%(X,B) .



2n0(U,B)

-

^ H*

ln0(U,B)

*

H^_

1 (z,)n0(U,B)

where the left vertica l map and top right vertical map are isomorphisms b y Shapiro' s lemma fo r etal e cohomolog y (exactnes s o f / * fo r a finit e morphism) . Propositio n 2.2.4 follows . • Note tha t i n th e proo f o f Propositio n 2.2.4 , i t i s no t necessar y t o assum e X smooth. S o i t provide s non-smoot h case s i n whic h th e effacemen t theore m holds . f

Corollary 2.2. 5 (Shapiro' s lemm a fo r Zarisk i cohomology ) Let X' — > X be a finite flat map between two smooth varieties over k, and let T be a sheaf of abelian groups over X' for the Zariski topology. Suppose that T is of the form H q(A) for some effaceable etale sheaf A over X'. Then the natural homomorphism

is an isomorphism for any p > 0. Proof Shapiro' s lemm a fo r etal e cohomolog y yield s isomorphism s o f coho mology group s wit h supports , fo r Z C X a close d subse t an d Z' — f~ 1 (Z):

HZ(XJ*A)^HZ,(X\A). Localizing, w e ge t isomorphism s o f th e E\ -terms o f th e conivea u spectra l se quences fo r A (ove r X') an d f*A (ove r X):

J] H^(XJ tA)^ ]

J H^(X',A).

This isomorphis m o f Cousin complexe s induce s a n isomorphis m o f their homol ogy groups : HpZar(X,Hq(f*A)) -

^ H^

q

v{X\H

{A))

by th e Bloch-Ogu s theore m ove r semi-loca l ring s (not e tha t f*A i s effaceabl e b y Proposition 2.2A). Finally , ther e i s a n isomorphis m o f Zarisk i sheave s

Hq(f*A)^f*Hq{A) which i s merely Shapiro' s lemm a fo r etal e cohomolog y applie d a t th e loca l ring s of

X. • Remarks 2.2. 6 (1) A n alternativ e argumen t woul d b e t o sho w tha t R qf*Jr = 0 for q > 0 . Thi s is wha t w e shall d o i n subsectio n 8.4 , removin g th e flatness hypothesis .

The Bloch-Ogus—Gabbe r Theore m

41

(2) "Shapiro' s lemm a fo r Zarisk i cohomology " i s fals e fo r arbitrar y Zarisk i sheaves! Fo r example , tak e X = A\ an d X' som e 2-fol d coverin g o f X split a t 0 (e.g . X' = A£ , f(x) = x 2 - 1 if cha rA; ^ 2) . Le t Y b e th e local ization o f X a t 0 an d Y' — f~l(Y). Le t T = j\A (extensio n b y 0) , wher e j : rj—>• X' i s th e inclusio n o f th e generi c poin t an d A i s a constan t sheaf . Then w e have : Hl(Y, f*T) = 0 for i > 0 (sinc e Y i s local) ; ff1(y,,^)=A The latte r i s easily see n b y a Cec h cohomolog y computation . The proo f o f Theorem 2.2. 1 i s given i n Section s 3 and 4 . W e shall i n fac t prov e something slightl y stronge r (an d simple r t o state) : Theorem 2.2. 7 (EfFacemen t Theorem ) Let X be a smooth, affine variety over k, £ i , . . . , £ r G X a finite number of points, p > 0 an integer and Z a closed subvariety of codimension > p + 1 . Let A be a sheaf of torsion abelian groups over the (small) etale site of X. Assume that A — p*A 0, where p : X— > Spec/c is the structural morphism and AQ is a Gal(ks/k)-module. If k is infinite, then there exist an open subset U of X containing all U and a closed subvariety Z' C X containing Z such that (1) cotim x{Z')>p; (2) the map H% nU(U,A) - > H%, nU(U,A) is

0 for all n > 0 .

If k is finite, then there exists (U,Z r) as above such that (at least) the composite HZ(X,A) -

H

n

znU(U,A)

-+

Hnz,nu(U,A) (2.2

)

is 0 for all n > 0 . Remark 2.2. 8 W e woul d lik e t o poin t ou t tha t (contrar y t o th e definitio n o f effaceability) th e statemen t i n Theore m 2.2. 7 i s no t local : th e proo f b y n o mean s implies that th e map of Theorem 2.2. 7 (2) remains 0 when U is replaced by a smaller open set. Therefore , i f in Theorem 2.2. 7 one replaces X b y Y a s in Proposition 2.1 .2 , it i s no t a t al l clea r tha t th e conclusio n stil l holds . I n othe r words , give n a close d subset Z c F o f codimensio n > p + 1 , although th e map s H^(Y, A) — > Hy{p) (Y , A) are al l 0 b y th e proo f o f Propositio n 2.1 .2 , i t i s no t clea r whethe r on e ca n find a single Z 1 C Y a s i n Theore m 2.2. 7 suc h tha t th e map s H%(Y,A) - > H%,(Y,A) ar e 0. Thi s show s th e subtlet y o f th e situatio n an d probabl y wh y Gersten' s conjectur e is s o difficult fo r genera l regula r loca l ring s o f dimensio n > 2 . 3 Som e Geometr y 3.1 Th e geometri c presentatio n theorem . Th e ke y t o th e proo f o f Theo rem 2.2.7 is a geometric presentation theore m whic h follows fro m lemma s of Gabbe r [14] supplemente d wit h som e remark s o f Gro s an d Suw a [22] . Thi s sectio n i s de voted t o a detaile d proo f o f thi s theorem . Fo r simplicity , w e writ e A n fo r A £ an d S x S ' rathe r tha n S x ^ S f fo r th e produc t o f tw o fc-schemes S an d S f. Theorem 3.1 . 1 (Geometri c Presentatio n Theorem ) LetX be a smooth, affine, irreducible variety of dimension d over an infinite field k; let t\,. .., t r G X be a finite set of points and Z a closed subvariety of codimension > 0 . Then there exists a map A d~1 x A 1 , an open set V C A. d~l, and an open set U C V ;_1 (1 ^) containing £ i , . . . , t r such that

(i) znu = znii>- 1 (V);

42

Jean-Louis Colliot-Thelene , R a y m o n d T . Hoobler , an d B r u n o K a h n

(2) ip\z is finite; (3) \z '• Z — • A d~l is finite. Proof Le t B = k[Z] b e th e affin e algebr a o f Z , an d x % the imag e o f Xi i n B. For i/j — (wi,... ,Ud-\) G Ed~l, ip\z i s finite i f an d onl y if , fo r al l z , x % is integra l over k\u\,... ,Ud_i] , wher e Uj is th e imag e o f Uj in B.

44 J e a n - L o u i

s Colliot-Thelene , R a y m o n d T . Hoobler , an d Brun o K a h n

Let K — k(£ d~l) b e th e functio n field o f £ d~1 an d7 7 th e generi c poin t o f £ d~l. We vie w7 7 as a rationa l poin t o f £ d~l ove r if , s o tha t7 7 = (771 ,.. . ,77^-1 ) wit h 77^ G £(K). Fo r simplicity , w e stil l writ e rjj fo r th e imag e o f rjj i n i f ® & B. Sinc e dim Z < d — 1 , there i s fo r al l i a non-zer o algebrai c relatio n i n K 0k B: fiiVi,-- ,r)d-i,Xi)

= 0

with fi G if [Xi,... ,Td] . W e claim that fi ca n b e chose n s o that i t give s an integra l dependence relatio n o n X{. To see this, w e argue a s i n [34 , p. 262 , proof o f Lemm a 2]: le t n — degfi an d f\ n* b e th e homogeneou s par t o f degre e n o f fi. Sinc e k is infinite , w e ca n find (£1 ,.. . , £d-i) G kd~1 suc h tha t f\ (t\, ... , £d-i> 1 ) 7 ^ 0 i n if. Lettin g g % = /^(T i + £iTd,.. . ,Td_ i + t d-iTd,Td), th e coefficien t o f T d i n g { i s f\n\tu... ,td_i,l ) an d w e ge t 9i(v[^-' ,r}'d-i>Xi) = 0 with 77 ' = 77 j — tj^i . Substitutin g bac k77 ^ instead o f 77 ' give s th e desire d integra l dependence relation . Therefor e ther e exist s a polynomia l g\ G k[£d~1][Ti,... , TJ such tha t • ^ ( r / ! , . . . ,r7 d _1 ,x 2 ) = O G / / c [ ^ - 1 ] ^ B ; • th e coefficien t ai o f Tdeg9i i n ^ i s / 0 . Then, w e ma y tak e fti = {(wi,.. . , u d - i ) e ^ "

1

|a,(^,.. . ,u d_i)^0for l < i < i V } . D

3.4 Securin g (p. Lemma 3.4. 1 Assume k is algebraically closed. With notation as in Theorem 3.1.1, there exists a non-empty open set f2 2 £ £ d such that, for

£ d _ 1 ,7 7 = Speci f the generi c poin t o f £ d~1 an d M v th e generi c fibre o f q. The n p induce s a ma p Mv— > £ x whic h i s dominant i f p i s dominant . Let K b e a n algebrai c closur e o f if . Thi s give s ris e t o th e geometri c poin t rj : Speci f—- > Specif— » £ r f _ 1 , an d w e hav e a n induce d morphis m Mjj — * £ ^ (where M ^ = M v ® K if) , whic h i s stil l dominan t i f p i s dominant . Now th e poin t rj defines a if-morphis m ipo : Z^— » A ^ 1 whic h i s finite a s a consequence o f 3.3.1 . Vie w 5 G X(k) a s a poin t o f X(K). Le t 2/Q , ( a = 1 , . . . ,n ) be th e finitely man y (if-rational ) point s o f Z-^ suc h tha t ipo(y a) = V>o(s) . Le t z$ (P = 1 ,.. . , m) b e th e finitely man y (if-rational ) point s o f Z ^ suc h tha t ipo(zp) =

46 J e a n - L o u i

s Colliot-Thelene , R a y m o n d T . Hoobler , an d Brun o K a h n

ipo{s')- W e have : A% = { (v,t,t')e (£x

{Z xZ\A z)) x

kK

= U V^ where Va,p = {v G % | v{ya) = v{zp)} x {(y a,Z(3)}. Thus th e projectio n p(Mjf) C £j^ decompose s a s P{Mrf)= (

J {ve£^\v{y a) =

v{z

p)}.

This i s a finite unio n o f prope r linea r subspace s o f &^. Henc e th e projectio n map M^j —> £-ft is no t dominant , so p : M— > £ d~l x £ i s no t dominan t an d th e Zariski closur e o f th e constructibl e se t p(M) i s a prope r close d subse t o f £ d. Now le t O 3 be th e complemen t o f \J { -p(M(t t, tj)) i n £ d: thi s i s a prope r ope n set tha t satisfie s conditio n 1 ) o f Lemma 3.4.2 . The proo f o f conditio n 2 ) i s entirely similar , usin g th e set s N(s) = {(lV)A) -JL+ H«

l -* F(P v,A)1

~ H^_

l F,(P v-F',A)

where th e botto m ro w i s par t o f a n exac t sequenc e fo r cohomolog y wit h supports . Since k* o (3 — 0 , i t obviousl y follow s tha t a — 0 . • Proof o f Theore m 2.2.7 . W e ma y assum e X irreducible . Suppos e first k infinite. Le t ip,(p,V,U b e a s i n Theore m 3.1 .1 , an d se t Z' = ^(^(Z)). W e appl y Theorem 4.2. 1 wit h V = V, F = ^(Z) an d F' = y(Z H U).

The Bloch-Ogus—Gabbe r T h e o r e m

51

In th e commutativ e diagra m H%nU(U,A)

h%(X)-^ h%(X') is a n isomorphis m fo r al l q. S U B l (Etal e Mayer-Vietoris) . C i s additiv e an d fo r Z , X',X, f commutative squar e

a s i n a) , th e

C(X' - Z)

C(X')

+ C(X

C(X)

- Z)

is homotopy cartesian . Recall tha t a commutativ e squar e o f complexe s A•

B

C

D

is homotopy cartesian i f th e natura l ma p fro m th e mappin g con e o f [C —> A © D] to B i s a homotop y equivalence . Lemma 5.1 . 2 The square of axiom SUBl is if the induced map f CZ{X)-^CZ(X') is a homotopy equivalence.

homotopy cartesian if and only

54 J e a n - L o u i

s Colliot-Thelene , Raymon d T . Hoobler , a n d B r u n o K a h n

Proof Thi s follows from th e triangulated categor y version of the nine diagram . More precisely , conside r th e map o f exact triangle s C(X')(BC(X-Z) d m g (

" - / ? C(X'-Z)®C(X-Z)

-

C

z(X')[l]

f C(X)

C(X - Z)

Cz{X){\]

in th e homotopy categor y K(A). B y [29, Proposition 5. 6 of chapter XI] , we ca n complete thi s diagram , u p to isomorphism, int o D'

D

C{X')®C{X

„s diag(v,Id)

(/, -«)

piy

i

z) e c{x - z)

Cz{X')[\]

f

(f,-Id) C{X - Z)

C(X)

D"

Cz(X)[l]

in whic h al l rows an d columns ar e exact triangles . I t is clear that : • th e middle to p vertical ma p induces a n isomorphism C(X f — Z) . D" ~ 0 ^ C Z(X) ^ > C Z{X'). • S U B 1 hold s ^ = ^ D - ^ D' ^=> D" ~ 0.

D'.

D

The clai m follows . Remarks 5.1.3

(1) W e say tha t a cohomolog y theor y (resp . a substratum ) satisfie s Zariski excision (resp . Zariski Mayer-Viet oris) i f axio m COH 1 (resp. SUB1 ) holds whe n w e let / ru n through ope n immersions . Th e obvious analogu e of Lemm a 5.1 . 2 (ordinar y excision ) holds . (2) Not e that i n SUB1 on e can replace the condition U C is additive" b y "C(0) = 0" (tak e th e case X' — Z i n the commutative square) . (3) Definitio n 5.1 . 1 allows u s to se t u p a conivea u exac t coupl e an d spectra l sequence as in Section 1 . Zarisk i excision allows us to recognize the E\ -terms of th e coniveau spectra l sequenc e i n the form o f Equation (1 .1 ) , producin g Cousin complexes in the sens e of [24] by Zariski sheafification . I n particular, if h* satisfies Zarisk i excision, we get a convergent coniveau spectral sequence, for al l X eS k E™

]J h X£X(P)

p +q p+q {X) x {X)^h

T h e Bloch-Ogus—Gabbe r T h e o r e m

55

analogous t o (1 .2) , wher e h%(X) = lim/i£

nC/ ([/)

(not e tha t E v{q = 0 fo r

U3x

p(£ [0,dimA]) . As we shall se e i n Sectio n 7 , if h* is defined b y a substratum , Zarisk i excisio n give s rise t o a n ( a prior i unrelated ) Brown-Gersten spectra l sequenc e a s well . Lemma 5.1 . 4 For a substratum C and the associated cohomology theory with supports h*: a) Axiom SUB1 implies axiom COH1 . b) The converse is true if, for all X, C(X) is fibrant in the sense of Definition C.l.l b) of Appendix C. Proof Par t a ) follow s triviall y fro m Lemm a 5.1 .2 . Par t b ) als o follow s fro m this lemma , Lemm a C.1 . 4 b ) an d Corollar y C.2.7 . • Remarks 5.1 . 5 (1) Obviously , Lemm a 5.1 . 4 hold s whe n etal e Mayer-Vietori s i s replace d b y Zariski Mayer-Vietoris . (2) B y Theore m C.3.1 , i f A verifie s axio m AB 5 and ha s a generator i n the sens e of [23 , 1 . 5 an d 1 .6 ] an d i f moreove r countabl e product s ar e exac t i n A (fo r example, A satisfie s AB4*) , the n ther e exist s anothe r substratu m F an d a natural transformatio n C > F suc h that , fo r al l A , (a) F(X) i s fibrant ; (b) ipx i s a monomorphis m an d a quasi-isomorphism . This applie s t o the cas e where A i s the categor y o f left /^-module s ove r som e ring [23 , Section 1 ] ,

s k.

Let u s no w introduc e ou r secon d axiom . T o do this, w e need a n assumptio n o n Assumption 5.1 . 6 a) Spe cA: G S k. b) I f A eS k, t h e n P ^ e S k.

Lemma 5.1 . 7 If S k satisfies Assumption 5.1 .6, then a) XeS k = > A ^ eS k. b) For any n > 1 , the open subsets of A£ are in S k. • In axiom s COH 2 an d SUB2 , w e assum e tha t S k satisfie s Assumptio n 5.1 .6 . COH2 ("Ke y lemma " fo r cohomology ) Le t V b e a n ope n subse t o f A £ (fo r some n) an d A'v -L+

, ^ 2

Pi

^V

V be the diagram representin g th e inclusion o f Ay an d th e section a t infinit y int o P v. Let F b e a closed subset o f V. The n th e diagram o f the key lemma of subsection 4. 1

56 J e a n - L o u i

s Colliot-Thelene , R a y m o n d T . Hoobler , an d Brun o K a h n

7T

A r

hUV)

F

is commutative . S U B 2 ("Ke y lemma " fo r substrata ) Le t V,F b e a s i n axio m COH2 . The n the diagra m

is homotop y commutative . We giv e a las t definition . Definition 5.1 . 8 Le t X £ Sk b e affin e an d le t t\,.. ., t r G l b e a finite se t o f points. A cohomolog y theor y wit h support s h* (resp . a substratu m C) ove r k i s strictly effaceable at (£1 ,... , t r) if , give n p > 0, for an y open neighbourhood W C X of t i , . . . , tr an d fo r an y close d subse t Z C W o f codimensio n > p + 1 , ther e exis t an ope n neighbourhoo d U C W o f £ i , . . ., t r an d a closed subse t Z f C W containin g Z suc h tha t coding(.Z 7 ) > p an d th e ma p h qznU(U)— > h qz,nU(U) i s 0 for al l g G Z (resp. th e ma p Cznu(U) — > Cz>nu{U) i s nullhomotopic) . I t i s strictly effaceable if this conditio n i s satisfie d fo r an y (X , £ i , . . ., £ r) a s above , wit h X smooth . Example 5.1 . 9 Suppos e k infinit e an d le t A b e a shea f o f torsio n abelia n groups ove r th e smal l etal e sit e o f Spe c k. B y Theore m 2.2.7 , th e cohomolog y theory wit h support s (X,Z)HffJft,aM) is strictl y effaceable , wher e a i s th e projectio n o f th e bi g etal e sit e o n th e smal l etale site . The followin g theore m i s immediat e fro m th e argument s o f Sectio n 4 . Theorem 5.1 .1 0 Let k be infinite and S^ satisfy Assumption 5.1 .6. A cohomology theory with supports h* (resp. a substratum C) satisfying etale excision COH1 (resp. etale Mayer-Viet oris SUB1 ) and the key lemma for cohomology COH2 (resp. the key lemma for substrata SUB 2) is strictly effaceable. The poin t her e i s that , i n diagram s wher e ther e i s a n excisio n ma p a s i n sub section 4.2 , on e ca n us e a homotop y invers e o f thi s ma p t o sho w tha t th e ma p on e wishes t o b e nullhomotopi c i s indee d nullhomotopic .

T h e Bloch-Ogus—Gabbe r Theore m

57

Corollary 5.1 .1 1 Let k be infinite and Sk verify Assumption 5.1 .6, and let the cohomology theory with supports h* satisfy axioms COH1 and COH2. Then, for any smooth X 6 Sk, the Cousin complexes are flasque resolutions of the Zariski sheaves H q associated to the presheaves U i— • h q(U), and the E2-terms of the spectral sequence of Remark 5.1 .3 (3) are

There i s a nee d fo r somethin g extra , lik e transfe r maps , t o dea l wit h finit e fields: se e Theore m 6.2.5 . Remarks 5.1 .1 2 (1) Th e axiom s abov e ar e muc h mor e economica l tha n thos e o f Bloch an d Ogu s in [2 , Sectio n 1 ] . Definitio n 5.1 . 1 correspond s t o axiom s (1 .1 .1 ) an d (1 .1 .2 ) of Bloch-Ogus. Axio m COH 1 correspond s t o axio m (1 .1 .3) . Axio m COH 2 has n o counterpar t i n [2] , bu t migh t b e compare d wit h [2 , (1 .5)] . O n th e other hand , w e d o no t nee d t o introduc e an y twists , no r a correspondin g homology theory . Thi s mean s tha t purity , le t alon e Poincar e duality , i s irrelevant fo r stric t effaceability . (2) However , th e "ke y lemma " w e axiomatize d i n COH 2 an d S U B 2 i s unsat isfactory, becaus e i t i s no t obviou s ho w t o chec k i t i n practice . Moreover , COH2 nee d no t impl y SUB 2 eve n fo r fibrant substrata . I n subsection s 5. 3 and 5.4 , w e introduc e stronge r axiom s tha t d o no t hav e thi s defect . 5.2 Spectra . In orde r t o includ e algebrai c X-theor y i n th e formalis m o f thi s section , i t i s necessary t o conside r substrat a wit h value s no t onl y i n complexes , bu t als o i n th e category £ o f spectra i n th e sens e o f algebrai c topology . W e refe r t o [4 , Sectio n 2] fo r th e definitio n o f suc h a suitabl e categor y £ , provide d wit h a n appropriat e closed mode l structur e (fibrations , cofibrations , wea k equivalences) . Recal l (e.g . [50, 5.32] ) tha t th e Dold-Ka n correspondenc e give s ris e t o a n embeddin g DK o f the categor y o f complexe s o f abelia n group s int o th e categor y o f spectr a suc h tha t irn(DK(C')) = H~ n(C') fo r an y comple x C o f abelia n groups . The precedin g subsectio n "extends " t o substrat a wit h value s i n spectr a b y • replacin g "complexe s o f object s o f A v b y "object s o f £ " i n Definitio n 5.1 . 1 b); • definin g h%(X) = 7r_ g (C z (X)) i n (5.1 ) ; • replacin g th e case s i n Lemm a 5.1 . 4 b ) b y "Fo r al l X G Sk, th e spectru m C(X) i s fibrant an d cofibrant. " • replacin g "Lemm a C.1 . 4 b)" i n the proof o f Lemma 5.1 . 4 b ) by "th e followin g fact: a wea k equivalenc e betwee n tw o fibrant an d cofibran t spectr a i s a homotopy equivalence" . • replacin g Remar k 5.1 . 5 (2 ) b y "Fo r an y substratu m C : Sk — * £ , ther e exis t two substrat a C\C" an d natura l transformation s C > C ^- C" suc h that fo r al l X , (a) C'{X) i s fibrant; (b) C"(X) i s fibrant an d cofibrant ; (c) tpx.tp'x a r e wea k equivalences . This follow s fro m th e folklor e result , fo r whic h w e hav e n o reference , tha t any ma p betwee n object s o f £ ca n b e factore d in a functorial way int o a

Jean-Louis Colliot-Thelene , R a y m o n d T . Hoobler , an d B r u n o K a h n

58

cofibration followe d b y a trivia l fibration , an d als o int o a trivia l cofibratio n followed b y a fibration (smal l objec t argument). " In th e nex t sections , w e shall allo w substrat a C t o tak e thei r value s i n spectra , and commen t o n thi s onl y whe n necessary .

5.3 Homotop y invariance . I n thi s subsectio n w e discus s tw o ne w axiom s C O H 3 / S U B 3 an d C O H 4 / S U B 4 fo r cohomolog y theories/substrata . A s we shall see in Sectio n 7 , the axio m C O H 3 / S U B 3 belo w i s satisfied b y many theories . Ax iom C O H 4 / S U B 4 i s auxiliar y an d merel y serve s t o giv e a smoot h proo f tha t COH3 = > COH2 (resp . S U B 3 = > SUB2 ) We assum e tha t Sk satisfie s Assumptio n 5.1 .6 . COH3 (Homotop y invarianc e fo r cohomology) . Suppos e tha t V , TT ar e a s i n axiom COH2 . The n h qF(V) -^— » hqAl (Ay ) i s a n isomorphis m fo r al l q. COH4 (Rigidit y fo r cohomology) . Le t V,7r, F b e a s i n axio m COH2 , an d le t so? soo b e th e section s a t 0 and o o o f TT. The n SQ, S^ : hq->1 ( P v )— » h qF(V) coincid e for al l q. S U B 3 (Homotop y invarianc e fo r substrata) . Le t V, TT be a s in axio m COH2 . Then C(V) -^— > C(Ay) i s a homotop y equivalence . SUB4 (Rigidit y fo r substrata) . Le t V,TT,F b e a s i n axio m COH2 , an d le t b e th e section s a t 0 an d o o o f TT. The n SQ^O O : C P i (Py )— » Cp(V) ar e homotopic. SO.SQO

Lemma 5.3. 1 Let h* be the cohomology theory with supports associated to the substratum C. a) Axiom S U B 3 implies axiom COH3 . b) If C(X) is fibrant and cofibrant for all X, then axiom COH 3 implies axiom SUB3. Proof Par t a ) : Axio m S U B 3 implie s th e sam e propert y fo r substrat a wit h support, b y th e sam e argumen t a s i n th e proo f o f Lemma 5.1 .2 . Par t b) : th e proo f is the sam e a s fo r Lemm a 5.1 .4 . • Proposition 5.3. 2 a) Axiom COH 3 implies axiom COH 2 and axiom COH 2 implies axiom COH4 . b) Axiom S U B 3 implies axiom S U B 2 and axiom S U B 2 implies axiom SUB4 . Proof Par t a) : compar e Remar k 4.1 .4 . Par t b ) i s analogou s bu t w e giv e a detailed proo f fo r th e convenienc e o f the reader . Firs t w e show tha t S U B 2 implie s SUB4. Generally , fo r x G P1(k) (resp . x G A1 (A:)), let u s denot e b y s x (resp . s fx) the sectio n o f 7 r (resp . o f TT) determine d b y x. W e ca n complet e th e diagra m o f SUB2 a s

59

The Bloch-Ogus—Gabbe r Theore m

CF(V)

CAIAK) (5-2

)

CF(V).

It i s then clea r that th e vertical compositio n i s SQ whil e the righ t compositio n i s s^. We no w sho w tha t S U B 3 implie s SUB4 . Axio m S U B 3 implie s tha t s f*x i s a homotopy invers e o f 7r * for al l x G A1(fc). Conside r no w th e inclusio n ji : A^ ^- » P Y give n b y t H- » t/(t — 1 ) . W e hav e ji(0 ) = 0 and j i ( l ) = oo , o r i n othe r words : jl 0

5Q

= S 0, j

l °S[ =

Soo .

Since S' Q an d s'l ar e homotopic , i t follow s tha t SQ and s^ ar e homotopic . Finally, w e show tha t S U B 3 implie s SUB2 . Usin g S U B 3 => SUB4, w e ma y replace, u p t o homotopy , s^ b y SQ in diagra m 5.3 , whic h the n become s obviousl y commutative. Bu t S U B 3 implie s tha t S' Q is a homotop y equivalence , henc e th e triangle o f S U B 2 i s homotop y commutativ e a s desired . • 5.4 Cohomolog y o f P 1 . I n orde r t o expres s ou r axio m o n th e cohomolog y of P , w e need t o introduc e mor e material . W e still assum e Sk t o satisf y Assumptio n 5.1.6. a) Cohomology theories. W e suppose give n a cohomolog y theor y /i* , a cohomolog y theory e * and , fo r an y (X, Z) GPfc , a ma p 1

PicX - > Rom(e

z{X)yhz{X))

which i s natura l i n {X,Z) (w e d o no t requir e thi s ma p t o b e additive) . Takin g X = P y , Z = Pjr , w e ge t a homomorphis m

hence, composin g wit h fr* , a homomorphis m eF{V) >h

vlF{Pv)

natural i n (V , F). COH5 (cohomolog y o f P 1 , cohomologica l version ) Assum e tha t V , F, TT ar e as i n axio m COH2 . The n th e natura l ma p

h%{V)®eF{V){*-^^ h^y) is a n isomorphis m fo r al l q. b) Substrata. W e suppos e give n a substratu m C , a substratu m D and , fo r an y X 6 Sk, a ma p PicX - » Rom £(D(X),C(X)) (5.3 )

60 J e a n - L o u i

s Colliot-Thelene , R a y m o n d T . Hoobler , an d Brun o K a h n

natural i n X, wher e £ i s either th e categor y o f complexe s o f object s o f ou r abelia n category A o r th e categor y o f spectr a o f subsectio n 5.2 . Takin g X = P ^ w e ge t a map (fo r spectra , i n th e stabl e homotop y category ) D{K)

i w

] c{ply)

hence, composin g wit h if*, a ma p (fo r spectra , i n th e stabl e homotop y category ) D(V) ^ CiP'y) natural i n V. SUB5 (cohomolog y o f P 1 , substratu m version ) Assum e tha t V , fr are a s i n axiom COH2. The n the natural map (fo r spectra, i n the stable homotopy category )

c(v)eD(v) ( **,av)> C(P^) is a homotop y equivalence . (To b e correct , w e shoul d us e wedg e V rathe r tha n direc t su m 0 i n S U B 5 when C an d D ar e give n b y spectra. ) Remarks 5.4. 1 (1) Th e ma p (5.4 ) induces , b y functoriality , a ma p o n cone s PicX - > Rom

£{Dz{X),Cz{X))

for an y (X , Z) £ T V Henc e w e ge t a ma p (fo r spectra , i n th e stabl e homo topy category )

cF(v) e DF(V) ^ ^ c

l n{v v)

generalizing tha t o f axio m SUB 5, an d th e latte r implie s b y th e usua l argu ment (cf . proo f o f Lemm a 5.1 .2 ) tha t thi s generalize d ma p i s a homotop y equivalence a s well . (2) Axio m COH5 implie s that th e cohomology theory e * is uniquely determine d by ft* u p t o isomorphism . Fo r example , e * verifie s Zarisk i (resp . etale ) excision i f h* does . Similarly , axio m SUB 5 implie s tha t D i s uniquel y determined b y C u p t o homotopy . Bu t th e actio n o f Pi c i s no t determine d by thes e axiom s i n a n obviou s way . Lemma 5.4. 2 Let h* be the cohomology theory with supports associated to the substratum C and e * the cohomology theory with supports associated to the substratum D. a) Axiom SUB 5 implies axiom COH5 . b) If C(X) is fibrant and cofibrant for all X, then axiom COH 5 implies axiom SUB5. Proof a

) follow s fro m Remar k 5.4. 1 (1 ) ; b ) i s prove n a s i n Lemm a 5.1 . 4

b). • Proposition 5.4. 3 a) Axiom COH 5 implies axiom COH2 . b) Axiom SUB 5 implies axiom SUB2 . Proof Fo r a) , compar e proo f o f Lemm a 4.1 .3 . Par t b ) i s simila r bu t w e giv e a detaile d proof , a s i n subsection 5.3 . B y Remar k 5.4.1 , w e are reduce d t o checkin g that i n th e diagra m

T h e Bloch-Ogus—Gabbe r T h e o r e m

61

C A i(AiO

CF(V)

CF{V)®DF{V)

(

^ ' a v ) . Cpi(Pl>

)

the tw o path s fro m CF{V) 0 Dp{V) t o C A i (Ay ) ar e homotopic . I t i s enoug h to chec k thi s o n bot h component s CF(V) an d DF(V). O n CF(V) thi s i s trivia l (the tw o path s ar e actuall y equal) . O n DF(V), th e tw o path s ar e nullhomotopic , because th e pull-back s o f (9(1 ) b y So o and j ar e bot h trivial . • 5.5 Generatin g ne w theorie s ou t o f old . Th e following remark s sho w how to construc t som e strictl y effaceabl e cohomolog y theorie s an d substrata . Her e sub strata tak e thei r value s eithe r i n C(A), wher e A i s a suitabl e abelia n category , o r in th e categor y o f spectr a £ o f subsectio n 5.2 . (1) Le t Sk = Var/k, le t h* (resp . C) b e a cohomolog y theor y wit h support s (resp. a substratum) ove r Sfc , and le t T G Var/k. Defin e a new cohomolog y theory wit h support s h T (resp . substratu m C T) b y

(hTyz(x) = h*

ZXkT(xxkT)

(resp. CT(X) =

C(Xx

kT)).

Assume tha t h* (resp . C) satisfie s axio m COH 1 (resp . SUB1 ) (for al l /c-schemes). The n h T (resp . C T) als o does . Thi s i s obvious . Suppose no w tha t h* (resp . C) satisfie s axio m COH i (resp . SUBi ) fo r some i G {2,3, 5}, no t onl y fo r ope n subset s o f A £ bu t fo r an y V G Var/k. Then th e sam e hold s fo r h T (resp . C T). Thi s i s equall y obvious . f

(2) Le t C — > C b e a morphis m o f substrata , an d le t C" b e th e homotop y fibre o f / . Then , fo r i = 1 ,3 , i f tw o amon g C,C',C" verif y axio m SUBi , so does th e third . Thi s i s not clea r (an d probabl y wrong ) fo r axiom s S U B 2 and SUB4 , o r fo r "strictl y effaceable" . A s fo r axio m SUB5 , th e followin g holds: Let D,D' b e th e substrat a attache d respectivel y t o C an d C i n COH5 . Assume give n a morphis m / : D — > D' suc h that , fo r an y X G S k an d a G PicX , th e diagra m D(X)

Q!*

C(X) IX

D'(X)

a*

C'(X)

62 J e a n - L o u i

s Colliot-Thelene , R a y m o n d T . Hoobler , an d Brun o K a h n f

commutes. Le t D" b e the homotopy fibre of D — > D''. Fro m the assumptio n above, w e ge t a natura l transformatio n PicX - > Hom(£> // (X),C' // (X)). Then, i f tw o amon g th e pair s (C,£>) , ( C, D')> (C",D") (togethe r wit h th e actions o f Pic ) verif y axio m COH5 , s o doe s th e third . (3) Le t (h a)aeA (resp . (C a)aeA) b e a filtered direc t syste m o f cohomolog y theories wit h support s (resp . substrata ) an d h = \imh a (resp . C = limC a ). If al l h a (resp . al l C a) verif y axio m COH i fo r som e i (resp . SUB i fo r i — 1, 3), the n s o doe s h* (resp . C). Th e sam e clai m fo r SUB 2 an d S U B 4 in th e cas e o f substrat a i s no t clear . A s fo r COH5 , w e mus t reques t tha t the D a attache d t o the C a for m a direct syste m compatibl e wit h tha t o f th e Ca vi a th e action s o f Pic . (4) Le t C b e a substratum , an d suppos e give n a direc t syste m o f substrat a • C( n ) - > C

(n+1 )

- •• •-(

n> 0 )

with a homotopy equivalenc e li m C^ -—> C. Suppos e tha t C^ and , fo r al l n, th e homotop y fibre o f C^ — » C^ n+l^ satisfie s axio m SUB i fo r i — 1 , 3 o r 5 (fo r SUB5 , w e request analogou s condition s o n th e D's , a s above) . The n so doe s C . Thi s follow s b y inductio n fro m Remark s 5. 5 an d 5.5 . 6 Universa l Exactnes s In this section , w e want t o show how strict effaceabilit y o f a substratum (rathe r than a cohomology theory ) implie s no t onl y exactness, bu t eve n universal exactness of the associate d Cousi n complexes . Recal l that a complex A i s contractible if ther e is a homotop y fro m th e identit y t o 0 on A . 6.1 Generalities . W e tak e fro m [1 9 ] th e definitio n o f universa l exactness , actually i n slightl y greate r generality : Definition 6.1 . 1 Le t A b e a n abelia n category . A comple x C o f object s o f A i s universally exact i f th e followin g conditio n i s satisfied : For any abelian category B and any additive functor T : A— > B commuting with filtering direct limits, the complex T(C') is exact. (In th e cas e A i s th e categor y o f lef t module s ove r a ring , on e shoul d compar e this notio n wit h Lazard' s pur e sequence s [32 , Chapte r I , Sectio n 2 , esp . Theore m 2.3].) Note tha t th e exactnes s o f T(C) i s automatic i f T i s exact, bu t w e only requir e it t o b e additive. Her e ar e som e examples : Examples 6.1 . 2 (1) A contractibl e comple x i s universall y exact . Indeed , an y additiv e functo r will transfor m a homotop y int o a homotopy . (2) Le t A b e a n abelia n category , an d le t B b e a n abelia n categor y satisfyin g AB5. Fo r an y filtering direc t limi t C o f universall y exac t complexes , an d for an y exac t functo r T : A— > B commutin g wit h filtering direc t limit , th e complex T(C') i s exact . (3) Le t C : 0 - > C° - > • • • -• C n~l - > C n - > 0 b e a bounde d exac t com plex, s o tha t B l(C') -^- > Z l(C') fo r al l i. Suppos e al l th e exac t sequence s 0 - > Z l{C) - • C l - > B l+l(C) - > 0 ar e filterin g direc t limit s o f spli t exac t

The Bloch-Ogus-Gabbe r Theore m 6

3

sequences. The n C i s universally exact . Thi s follow s fro m th e previou s tw o examples. Conversely: Proposition 6.1 . 3 Suppose A satisfies AB5 and any object of A is a filtering direct limit of finitely presented objects (e.g. A is the category of left modules over a ring). Then any bounded universally exact complex C of objects of A can be described as in Example 6.1 .2 (3). Recall tha t a n objec t X o f A i s o f finite presentatio n i f th e functo r Y \-> Hom(X, Y) commute s wit h direc t limits . Proof Applyin g Definitio n 6.1 . 1 wit h T — identity, w e se e tha t C i s exact . Let X b e a finitely presente d objec t o f A. Applyin g th e functo r T(M) = Hom(X , M) to C , w e se e i n particula r tha t Hom(X , C n~l) — • Hom(X, Cn) i s surjective . I t follows that , fo r an y / : X— > C n, th e pull-bac k o f the exac t sequenc e 0 -» Z n~l{C) -

> C 71 '1 -> Cn - > 0 (D

)

by / i s split . B y th e assumptio n i n Propositio n 6.1 .3 , (D ) i s a filtering direc t limi t of spli t exac t sequences , an d i n particula r i s universall y exact . Thi s no w implie s that th e sequenc e 0 -+ C ° - > •

Cn~2 -> Bn-r(C') -

»0

is universall y exact . W e ge t th e conclusio n b y inductio n o n n. • 6.2 Universa l exactnes s o f Cousi n complexes . Theorem 6.2. 1 Let Sk satisfy Assumption 5.1 .6. Let X G Sk be an affine variety, t i , . . . , t r G X a finite set of points and h* a cohomology theory with supports on Vk- Suppose that h* is given by a substratum C which is strictly effaceable at t i , . . . , t r. Then the Cousin complexes

xey(0)

xeYi

1

are universally exact, where Y = SpecOx,t

)

u...,tr-

The proo f use s th e followin g well-know n lemma : Lemma 6.2. 2 Let T be a triangulated category and A — • B— > C —> A[l] an exact triangle in T. Suppose that the map C — > A[l] is 0 . Then the map B — > C has a section. ("Every epimorphism is split".) Proof Appl y th e functor Hom(C , ?) to the triangle an d ge t a n exac t sequenc e Hom(C, B) - > Hom(C , C) -^ Hom(C , A[l]). which show s tha t Hom(C , B)— > Hom(C , C) i s surjective . Le t s : C — > B b e a n element tha t map s t o Idc- The n b y definition , s i s a section , a s wanted . • To prov e Theore m 6.2.1 , w e g o a littl e mor e carefull y tha n i n th e proo f o f Proposition 2.1 .2 . W e not e tha t th e Cousi n comple x o f Theore m 6.2. 1 i s obtaine d by pastin g togethe r complexe s

0^/£(p)(r)-> J J hl(Y)^h#l

1 (6. +1 )(Y)->0

)

64

Jean-Louis Colliot-Thelene , R a y m o n d T . Hoobler , an d Brun o K a h n

which i n tur n ar e obtaine d a s direc t limit s o f th e complexe s 0 - h" z,nw(W) -

h\ z,_z)nW(W \

Z) -* h"£ w(W) -

0 (6.2

)

coming from th e long exact cohomolog y sequence of Definition 5.1 .1 . Her e W varie s among the ope n neighbourhood s o f (£i,... , t r) an d Z C Z' C W var y amon g close d subsets o f codimension s respectivel y > p an d > p + 1 . Lemma 6.2. 3 Let £ i , . . . , £ r and Y be as in Theorem 6.2.1 . Suppose the substratum C is strictly effaceable at £ i , . . . , £ r . Then for any p > 0 and q £ Z, the complex (6.1 ) is a direct limit of split exact sequences, where /i * is the cohomology theory associated to C. In particular, it is universally exact. Proof Le t W C X b e a n ope n neighbourhoo d o f ( t i , . . . , t r ) , Z C W a closed subse t o f codimensio n > p an d tak e [/ , Z' a s give n b y Definitio n 5.1 .8 . I n the triangulate d categor y K(A) o f complexes o f objects o f A u p t o homotop y ([26 , Chapter I ] an d [20 , Sectio n 5]) , o r i n th e homotop y categor y o f £ i f C i s give n b y spectra [4] , consider th e triangl e o r fibre sequenc e (tf \ Z ) - > C ZnU(U){l] -^ f

C

z>nu(U)[l].

f

Here Cz nu{U)[l] mean s Y,Cz nu(U) i f C i s give n b y spectra . Lemm a 6.2. 2 shows tha t th e ma p C(z'-z)nu{U \ Z) ~* Cznu{U)[l] ha s a homotop y section . Correspondingly, th e sequenc e 0 - h" z,nu(U) -

h\

z,

-Z)C\U

is spli t exac t fo r al l q. An d suc h sequence s ar e cofina l i n th e direc t syste m o f complexes (6.2) . • Corollary 6.2. 4 Suppose k is infinite and Sk verifies Assumption 5.1 .6. Let h* be a cohomology theory with supports on Vk, with values in an abelian category A satisfying axiom AB5 and having a generator. If h* satisfies axioms COH 1 (etale excision) and either COH 3 (homotopy invariance) or COH 5 (cohomology of P 1 ) , and can be defined by a substratum of complexes or spectra (the latter assuming A = {abelian groups}), then the Cousin complexes of Theorem 6.2.1 are universally exact for X smooth. Proof I f h* ca n b e define d b y a substratu m o f complexes , i t ca n b e define d by a fibrant substratu m C b y Remar k 5.1 . 5 (2) . Similarly , fo r a substratu m o f spectra, i t ca n b e define d b y a fibrant an d cofibran t substratu m b y subsectio n 5.2 . By Lemma s 5.1 . 4 b) , 5.3. 1 b ) an d 5.4. 2 b) , C satisfie s axiom s SUB 1 an d eithe r S U B 3 o r SUB5 , henc e axio m SUB 2 b y Propositions 5.3. 2 an d 5.4.3 . B y Theore m 5.1.10, i t i s strictl y effacable . Th e corollar y no w follow s fro m Theore m 6.2.1 . Note tha t Theore m 6.2. 1 doe s no t cove r th e cas e o f finite fields. Fo r this , w e introduce anothe r axiom , whic h wa s alread y use d i n Sectio n 4 : COH6 Fo r an y finite field extensio n £/k an d an y (X , Z) ePfc , ther e i s given a ma p Covi/k : h*Zl(Xe) - > h* z(X) such that Cor^/f c o R e s^ = [£ : fc],where R e s^ correspond s t o extension of scalars. This ma p i s natura l i n (X, Z) EVkTheorem 6.2. 5 Let k be a finite field and h* a cohomology theory with supports on Vk, with values in an abelian category satisfying axiom AB5 and having a generator. Suppose Sk verifies Assumption 5.1 .6 and h* satisfies axioms COH1 ,

65

The Bloch-Ogus—Gabbe r Theore m

COH6 and either COH 3 or COH5, and can be defined by a substratum of complexes or spectra. Then, for any connected smooth affine X G Sk and any finite set of points t\, - • • ,tr G l , the Cousin complexes

xeYM

are universally exact, where Y = SpecOx,ti,...,t

p-

Proof Exten d C to SK for infinite algebrai c extension s K/k b y setting C(X) = limC(X 0 0/c o ^) > wher e fcois a suitabl e finit e subextensio n o f K suc h that X — XQ (g)fc0 K fo r som e Xo , an d £ runs throug h th e finit e subextension s o f K/ko. Thi s extend s h* t o a cohomology theor y wit h support s o n VK, admittin g a substratu m an d satisfyin g axiom s COH1 , COH 6 an d eithe r COH 3 o r COH5 . By Corollar y 6.2.4 , th e Cousi n complexe s o f Theore m 6.2. 1 ar e universall y exac t for K- varieties. Let T b e a n additiv e functo r wit h value s i n som e abelia n categor y satisfyin g AB5 an d whic h commute s wit h filterin g direc t limits . W e hav e t o prov e tha t th e complex 0^T(h«(Y))-^T(h$(Y))^ U

T{hl+\Y))-^...

(6.3

)

is acyclic , fo r X a smooth /c-variet y an d Y a s i n Theore m 6.2.1 . W e us e th e sam e trick a s i n Sectio n 4 . Le t pi,p 2 b e tw o differen t prime s an d K\,K AK/) i s p;-primar y torsio n an d therefor e Ker( A—> AK1 0 AK2) — 0 . Finally , sinc e K\ and K2 ar e infinite , w e have AK 1 — AK 2 — 0 as observed above . S o A — 0 , a s wa s t o b e proven . • 7 Example s 7.1 Hypercohomolog y o f sheaves . I n thi s subsectio n a s i n subsectio n 7.5 , the categor y Sk nee d no t satisf y Assumptio n 5.1 .6 . 7.1.1 Le t v be a Grothendieck topolog y o n Sk- T o a complex o f sheave s o f abelian group s C over v one ca n associat e a cohomology theor y wit h support s /i* , given b y th e v- hypercohomology o f C with supports :

/i*z(x) = e*z(x„c). 7.1.2 Le t / : C—> C be a morphism. The n / induce s a morphism / * o f associated cohomolog y theories . I f / i s a quasi-isomorphism, / * i s a n isomorphis m in th e followin g tw o cases : • C and C are bounde d below ; • fo r al l X e Sk, th e zy-cohomologica l dimensio n o f X i s finite .

66 J e a n - L o u i

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Indeed, w e hav e a morphis m o f hypercohomolog y spectra l sequences :

/* /

*

which i s a n isomorphis m o n E 2 -terms b y assumption . Here , H q{C) an d H q{C) ar e the cohomolog y sheave s o f C and C respectively . I n bot h cases , th e tw o spectra l sequences converge , henc e th e ma p o n abuttment s i s a n isomorphism . 7.1.3 A cohomolog y theor y give n b y a comple x o f z/-sheave s C can alway s b e defined b y a substratu m C o f complexe s o f abelia n group s (thi s i s i n fac t th e wa y ^-hypercohomology i s defined!) Ther e ar e severa l well-know n construction s fo r C: • Suppos e C is bounded below . Choos e a Cartan-Eilenberg injectiv e resolutio n I o f C an d defin e C(X) = Tot(I)(X), wher e Tot(I) i s th e tota l comple x associated t o th e bicomple x X. Not e tha t th e term s o f C(X) ar e injectiv e abelian group s an d C(X) i s bounde d below . Henc e C(X) i s fibrant i n th e sense o f Definitio n C.l. l (compar e Propositio n C.1 .2) . • Replac e C by a fibrant comple x o f sheaves T. B y Theore m C.3.1 , w e can d o this functoriall y i n C . Defin e no w C(X) a s .F(X) ; not e tha t C(X) i s fibrant for al l X a s a complex o f abelian groups . Thi s constructio n doe s no t requir e C to b e bounde d below . • Th e Godemen t resolution . Suppos e th e topo s associate d t o v ha s enoug h points (thi s i s the cas e fo r Zariski , Nisnevich , etale , comple x topologies) . T o C one associate s a ne w comple x o f sheave s

TCIU^YI n re feuf*(u) where n i s th e se t o f point s o f v (compar e [50 , 1 .31 ] , especiall y fo r set theoretic problems) . Th e term s o f thi s comple x ar e flabby i n th e sens e o f [36, Exampl e III . 1.9 (c)] . Iteratin g T yield s a cosimplicial comple x o f flabby sheaves X ' C, whic h i n tur n yield s a bicomple x o f flabby sheave s T*C i n th e usual way . On e define s C(X) = Tot(T*C)(X). Thi s i s essentially th e objec t denoted b y H"(X,C ) i n [50] . By the usual arguments , ther e is a commutative diagra m (fo r C bounded below) Tot{l)

Tot{T*C) in whic h th e vertica l ma p induce s a quasi-isomorphis m o n globa l sections . The last tw o constructions ar e natural in C. Al l constructions hav e the followin g virtue: i f C—> • C—> C" defines a n exact triangle in the derived categor y of ^-sheaves,

67

The Bloch-Ogus—Gabbe r Theore m

then s o does C'{X) — * C(X) — » C"{X), fo r al l X, i n the derive d categor y o f abelia n groups, wher e C ', C , C" ar e th e associate d substrata . 7.1.4 Le t C b e associate d t o C as i n 7.1 .3 , an d suppos e w e sheafif y i t fo r th e z/-topology. I n th e first constructio n C i s already th e comple x o f sheaves Tot(X). I n the secon d one , the stal k o f T C at a point x i s homotopy equivalen t t o the constan t cosimplicial comple x o f abelia n group s define d b y C x. I n bot h cases , th e resultin g complex o f sheave s i s quasi-isomorphi c t o C . 7.1.5 Instea d o f takin g complexe s o f sheave s o f abelia n groups , on e ca n tak e complexes o f sheave s wit h value s i n a n abelia n categor y wit h enoug h injectives , o r sheaves of spectra [50 ] i n th e lin e o f 5.2 . Al l th e abov e hold s i n thes e contexts , mutatis mutandis . I n th e cas e of spectra, fo r 7.1 . 2 us e th e spectra l sequenc e o f [50 , Proposition 1 .36] . Fo r th e secon d constructio n i n 7.1 .3 , us e [50 , Definitio n 1 .33] . Nisnevich repeat s thes e construction s i n [38] , because h e use s a differen t notio n o f point o f a topo s fo r th e Nisnevic h topology . 7.1.6 Le t v' b e anothe r Grothendiec k topolog y o n Sk whic h i s finer tha n v. Then th e identit y functo r o f Sk define s a morphis m o f site s v' — > v. I f C i s a complex o f sheave s fo r th e z / topology , ther e i s a n isomorphis m

where Ra*C i s th e tota l direc t imag e o f C (i n th e derive d category) . I n th e cas e C is a shea f o f spectra , on e shoul d us e th e objec t R'aC o f [50 , def . 1 .55 ] instea d o f Ra*C, c f [50 , th. 1 .56] . S o we can view ^'-hypercohomology a s z/-hypercohomology . 7.1.7 Suppos e that v is the big Zariski site of Spec fc. Then j/-hypercohomolog y of C verifies Zarisk i excision . Similarly , suppos e tha t v i s the bi g Nisnevic h sit e Ni s on Spec/c . The n ^/-hypercohomolog y verifie s etal e excision , i.e. , axio m COH1 . This i s know n whe n C is reduce d t o a singl e shea f (fo r th e Nisnevic h case , cf . [8 , Proposition 4.4] , which applie s t o Nisnevic h cohomology ; recal l tha t th e proof s o f [36, Propositio n III . 1.27] an d [38 , Theorem 1 .27 ] hav e a gap). I n general , th e proo f follows fro m a compariso n o f convergen t hypercohomolog y spectra l sequences , a s in 7.1 .2 . Th e tw o spectra l sequence s converg e withou t boundednes s condition s o n C, because th e Zarisk i o r Nisnevic h cohomologica l dimension s o f /c-scheme s o f finite type ar e finite [38] . Se e [50 , Exampl e 1 .49 ] fo r Zarisk i excisio n i n th e cas e o f a sheaf o f spectr a (i n [38 ] Nisnevic h doe s no t giv e th e correspondin g statemen t fo r etale excisio n explicitly) . 7.1.8 B y 7.1 . 6 an d 7.1 .7 , v- hypercohomology satisfie s Zarisk i (resp . etale ) excision a s soo n a s v i s finer tha n th e Zarisk i (resp . th e Nisnevich ) topology . 7.2 Generatin g ne w theorie s ou t o f old , continued . Let C b e a s i n 7.1 .1 , le t A b e a bounde d belo w comple x o f abelia n groups , L

viewed a s a comple x o f constan t Nisnevic h sheaves , an d le t C = C (g) A (i n th e derived category) . Then , i f th e cohomolog y theor y associate d t o C verifies axio m COH3 o r COH5 , th e sam e i s tru e fo r C . I n th e cas e o f axio m COH5 , i f V i s L

a comple x o f sheave s associate d t o C , w e associat e t o C th e comple x V — V® A and tak e fo r th e actio n o f Pi c th e origina l actio n tensore d b y A. Th e clai m ca n b e justified i n a fe w steps : (a) A = Z[0] . Thi s i s trivial . (b) A = Z/n . Follow s fro m th e previou s case , ite m (2 ) o f subsectio n 5.5 , 7.1.3 an d th e exac t sequenc e 0 —» Z — > Z — • Z/ n —> 0 .

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(c) A consist s o f a single finitely generate d abelia n grou p place d i n degre e zero. Follow s fro m th e previou s cases . (d) A consist s o f a singl e abelia n grou p place d i n degre e 0 . Follow s fro m the previou s cas e an d ite m (3 ) o f subsectio n 5. 5 b y a passag e t o th e limit. (e) Th e genera l case . Follow s fro m th e previou s cas e an d ite m (4 ) o f subsection 5.5 . In cas e C i s a shea f o f spectr a a s i n 7.1 .5 , on e ha s th e sam e b y takin g fo r V th e sheaf o f spectr a C A A, wher e A i s a n arbitrar y spectrum , viewe d a s a constan t sheaf o f spectra . Th e metho d i s th e same , reducin g t o th e cas e wher e A i s a n Eilenberg-Mac Lan e spectru m b y devissag e fro m it s Postniko v tower , c f [50 , proof of Theore m 1 .46] . 7.3 Homotop y invarian t examples . In al l example s o f thi s subsection , S & = Var/k. (1) Etale cohomology with coefficients in a sheaf defined over k and torsion prime to the characteristic of k. Etal e excision follows fro m 7.1 . 8 and homotop y in variance fro m [36 , Corollary VI.4.20] . Mor e generally , b y 7.1 . 8 an d ite m (4 ) of subsectio n 5.5 , on e ma y tak e etal e hypercohomolog y o f a bounde d belo w complex o f sheave s whos e cohomolog y i s torsion prim e t o th e characteristi c of fc. (2) (ove r C: ) Classical hypercohomology with coefficients in a bounded below complex of abelian groups. Here , etal e excisio n agai n come s fro m 7.1 . 8 an d the fact that , fo r X a C-variety, th e topological space X(C) essentiall y map s to th e smal l etal e sit e o f X (cf . [SGA4-III , expos e XI , 4.0]) . Homotop y invariance i s know n fo r Z a s coefficients , an d th e genera l cas e follow s fro m Remark 7.2 . (3) (cha rA: = 0: ) De Rham cohomology. Recal l that , fo r a /c-variet y X , H2R{X/k) = M.^ &r(X, Q' X/k)-> w n e r e ^x/k ls th e ^ e R n a m complex . T o check etale excision , w e not e that , sinc e th e Q\/ k ar e coheren t sheaves , th e map s HIzar(X, tt' x,k) — • M^ is (X, fi X/fe) ar e isomorphism s [36 , Remar k III.3.8] , so we ca n appl y 7.1 .8 . Actually , sinc e char/ c = 0 , w e eve n hav e purit y [25] . Homotopy invarianc e i s prove n i n [25 , Remark, p . 54] . (4) Motivic cohomology. Le t k admi t resolutio n o f singularitie s i n th e sens e of [1 2 , Definitio n 3.4 ] (fo r example , char/ c = 0) , an d le t i > 0 . I n [49 , Section 2] , motivi c cohomolog y o f weigh t i i s define d b y H^(X 1 Z(i) ) = H^(Xccih, Z(i) cdh) for ( ^ Z) ^T^ki wher e cd h i s the Grothendiec k topolog y introduced i n [1 2 , Definitio n 3.2] , Z(i ) i s a certai n comple x o f presheave s with transfer s wit h homotop y invarian t cohomolog y presheave s i n th e sens e of [52 ] an d Z(i) cdh i s it s sheafificatio n fo r th e cd h topology . Therefor e mo tivic cohomolog y i s give n b y a substratum , i s homotop y invarian t an d sat isfies etal e excisio n sinc e th e cd h topolog y i s stronge r tha n th e Nisnevic h topology. I t als o satisfie s purit y b y [49 , Proposition 2.4] . (5) Cycle modules. Le t M * b e a cycl e modul e i n th e sens e o f [44] . Fo r Z £ Var/k an d j G Z, denot e b y C.(Z,Mj) th e (homological ) Gerste n comple x associated t o Z an d endin g with LLe x Mj (k(x)). Fo r X o f pure dimensio n d, defin e C(X) = C-(X,Mj) =

C.(X,Mj-

d)

The Bloch-Ogus—Gabbe r Theore m

69

viewed a s a cohomologica l complex . Fo r Z C X a close d subset , ther e i s a n obvious shor t exac t sequenc e o f complexe s 0 - > C.(Z,Mj+ d) - > C'(X,Mj) -

> C'(X - Z,M 3) - > 0.

So C.(Z , Mj + d) i s homotop y equivalen t t o th e homotop y fibre Cz{X). In particular , th e substratu m C verifie s etal e excisio n (axio m SU B 1 ) , an d even purity . B y a resul t o f Ros t [44 , Propositio n 8.6] , i t i s als o homotop y invariant. Therefore , th e Gerste n complexe s o n a smoot h semi-loca l schem e are universall y exac t ([44 , Theore m 6.1 ] fo r th e exactness) . Universa l exact ness actually follow s directl y fro m replacin g th e cycl e module M * by T o M*, where T i s a give n additiv e functo r whic h commute s wit h filtering direc t limits. This exampl e include s a s a specia l cas e Milnor' s if-groups , Milnor' s if-groups mo d m , m-torsio n i n Milnor' s i^-group s . . . (6) Algebraic G- (— K1-Jtheory. Thi s i s th e onl y cas e i n thi s lis t o f example s where th e substratu m i s give n b y spectra , no t complexes . Etal e excisio n COH1 i s implied b y the muc h stronger localization theorem o f Quillen, aki n to purit y [43 , Proposition 7.3.2] . Homotop y invarianc e COH 3 als o follow s from Quille n [43 , Propositio n 7.4.1 ] . B y smashin g th e algebrai c G-theor y spectrum b y th e Moor e spectru m M ( Z / n ) , w e ge t th e case o f algebrai c G-theory wit h coefficient s Z/ n (compar e subsectio n 7.2) . 7.4 N o n homotop y invarian t examples . I n all examples of this subsection , Sk — Var/k, excep t i n Example s (4 ) an d (5 ) wher e Sk = Sm/k. (1) Etale (hyper-) cohomology with bounded below coefficients coming from k. As above , etal e excisio n follow s fro m 7.1 .8 . Axio m COH 5 i s prove n i n Appendix A . Mor e precisely , i n subsectio n A.2 , w e defin e a n etal e shea f Q/Z(—1) (ove r th e bi g etal e sit e o f SpecZ ) an d a ma p PicX - > T(X , R H o m „ ( Q / Z ( - 1 l ) [ - 3 ] , Z) ) (7.

)

for an y schem e X. Le t no w Co be a bounde d belo w comple x o f sheave s ove r the small etal e sit e o f Spec/c , an d C its invers e imag e t o th e big etale site . Let /i * b e th e cohomolog y theor y wit h support s define d b y C an d e * th e L

cohomology theor y wit h support s define d b y C 0 Q/Z ( —1)[—3]. Th e ma p (7.1) induce s a map Pic X— » Hom(e , h), an d w e show in subsection A. 3 tha t this ma p satisfie s axio m COH5 . (2) Hodge and de Rham cohomology in any characteristic. Etal e excisio n i s seen a s above . Axio m COH 5 i s du e t o Illusie : i f h* i s th e cohomolog y theory associate d t o th e Zarisk i shea f Q lx,k, the n axio m COH 5 hold s fo r h* wit h e * associate d t o ft^J-1 ] (i.e. , e 3z(X) = H Jfx(XZar,fi^)). Her e the ma p Pi c X— > Hom(e , h) i s give n b y cup-produc t wit h th e first Cher n class, define d throug h th e ma p H^ ar(X, (D* x) —^> # z a r ( X , f}^ //c ). Usin g item (4 ) o f subsectio n 5.5 , on e ca n the n exten d axio m COH 5 t o th e d e Rham-Witt comple x itself , o r truncation s o f it . Compar e [21 , p. 22 , Proo f of (4.2.7)] . Not e tha t X nee d no t b e smoot h since , i n [SGA7 , expos e XI , Theorem 1 .1 ] , X i s arbitrary. Not e als o that, eve n in characteristic 0 , Hodg e cohomology i s no t homotop y invariant . I n characteristi c p on e ca n the n "escalate th e ladder " t o ge t th e sam e resul t fo r Deligne-Illusie' s Hodge-Wit t and d e Rham-Wit t cohomolog y ([28] , compar e [21 ]) .

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(3) (cha r k — p) Logarithmic Hodge- Witt and de Rham- Witt cohomology [22] . Etale excisio n i s proven a s above. Axio m COH 5 follow s fro m [21 , Theore m 1.2.1.11] and i s proven ther e i n th e spiri t o f ite m (2 ) o f subsection 5.5 , usin g the descriptio n o f th e logarithmi c d e Rham-Wit t pro-comple x a s Frobeniu s fixed point s o f the d e Rham-Wit t pro-comple x [21 , I. (1 .3.2) ] . (4) Cohomology of a torus. Le t T be a /c-torus . Conside r th e cohomolog y theor y with support s h* give n b y th e shea f associate d t o T o n th e bi g etal e sit e o f Speck. A s above , h* satisfie s etal e excision . Le t M — Hom(Gm ,T) b e th e group o f cocharacter s o f T , als o viewe d a s a bi g etal e sheaf . Le t e * b e th e cohomology theor y wit h support s give n b y e%(X) = H

1 q

f (XiuM).

Cup-product define s a ma p PicX ->Hom x (e,/i) for al l X. I t ca n b e show n tha t axio m COH 5 i s verifie d fo r /i , e an d thi s natural transformation . (5) Etale weight-two motivic cohomology. Fo r a n affin e schem e X , le t T(X , 2) b e the weight-tw o motivi c comple x introduce d b y Lichtenbau m i n [33] . Fo r X smooth, it s Zarisk i sheafificatio n r(2)z a r i s quasi-isomorphic t o r > 0 Z(2)z a n where Z(2)z ar i s the Zariski sheafification o f the complex of 7.3 (4). Th e mai n steps in the proof o f this are [1 , Theorem 7.2] , [48 ] and [53 , proof o f Proposition 4.9 and subsectio n 4.3] . (Conjecturall y T(2)z ar an d Z(2)z a r coincide) . I t is prove n i n [3 1 ] tha t th e cohomolog y theor y (X,Z) — f > H ^ ( X ^ , r ( 2 ) ^ ) satisfie s axio m COH5 , wher e T(2) ^ i s th e etal e sheafifi cation o f X i->r(X,2) . (6) Algebraic K B -theory. Her e K B denote s th e Bas s extensio n o f Quillen' s al gebraic K-theory , whic h coincide s wit h th e latte r fo r regula r Noetheria n schemes, se e [51 , Sectio n 6] . Axio m COH 1 i s on e o f th e mai n result s o f Thomason-Trobaugh: i t applie s generally to X, X' quasi-compac t an d quasi separated suc h tha t X — Z i s quasi-compac t a s wel l [51 , Theorems 7. 1 an d 7.4]. I t woul d b e wron g fo r ordinar y i^-theor y i n general . Axio m COH 5 follows fro m [51 , Theorem 7.3] . Jus t a s in Example 7. 3 (6 ) , we get algebrai c K B -theory wit h finite coefficient s b y smashin g wit h a Moor e spectrum . 7.5 Mor e o n hypercohomolog y an d excision . Thi s subsection ca n be considered a s a seque l t o subsectio n 7.1 . 7.5.1 Le t v b e a Grothendiec k topolog y o n r(x,c) -+ r(x,Tot(i)). Similarly, i f v ha s enoug h points, th e Godemen t resolutio n constructio n o f 7.1 . 3 gives a natura l transformatio n C ( X ) ^ H - p C C ) . (7.2 By analog y wit h 7.1 .4 , w e may as k th e question : Question. Whe n i s (7.2 ) a quasi-isomorphism ?

)

The Bloch-Ogus—Gabbe r T h e o r e m

71

This proble m ha s n o simpl e solutio n i n general ; howeve r w e shal l explai n tha t it ha s on e whe n v i s either th e Zarisk i o r th e Nisnevic h topology . In bot h cases , a n obviou s necessar y conditio n i s tha t th e cohomolog y theor y h* associate d t o C satisfie s Zarisk i (resp . etale ) excision , sinc e M'(X U)C) doe s b y 7.1.7. Th e remarkabl e fac t i s tha t thi s conditio n i s sufficient : Theorem 7.5. 1 Suppose that v — Za r (resp. v — Nisj . Then (7.2 ) is a quasi-isomorphism if and only if the cohomology theory h* associated to C satisfies Zariski (resp. etale) excision. In other terms, a cohomology theory with supports which admits a substratum made of complexes satisfies Zariski (resp. etale) excision if and only if it can be defined by Zariski (resp. Nisnevich) hypercohomology of a complex of sheaves. This theore m i s either a consequenc e o r a n eas y analogu e o f Theorem 7.5. 2 (Brown-Gersten-Thomason-Nisnevich ) Suppose that v — Za r (resp. v — Nisj and let C be a substratum of spectra over Sk- Then (7.2 ) is a quasi-isomorphism if and only if the cohomology theory h* associated to C satisfies Zariski (resp. etale) excision. In other terms, a cohomology theory with supports which admits a substratum made of spectra satisfies Zariski (resp. etale) excision if and only if it can be defined by Zariski (resp. Nisnevich) hypercohomology of a sheaf of spectra. Proof Se e [5] and [50 , 2.5] for the Zarisk i case, [38 ] for the Nisnevich case. • By 7.1 .4 , a comple x o f sheave s C defining h* can b e chose n a s th e shea f associ ated t o th e preshea f U — i > C(U), wher e C i s a substratu m definin g h*. Corollary 7.5. 3 Under the conditions of Theorem 7.5.2, there is for any X £ Sk a spectral sequence E$'q = H p(Xu,Hq) =

»h

p+q

(X),

where v — Za r or Nis and H q is the v sheaf associated to the presheaf U \-> h q(U). Proof Thi s is just th e hypercohomolog y spectra l sequenc e fo r th e cohomolog y ofETCX^C). • Example 7.5. 4 Le t / i b e a Grothendiec k topolog y o n Sk whic h i s finer tha n v, an d le t a : \x—> v b e th e correspondin g morphis m o f sites . Le t D b e a comple x of sheave s (o r shea f o f spectra ) fo r th e ^-topology , an d tak e C — Ra*V (o r R'aV). There i s a canonica l quasi-isomorphis m (o r wea k equivalence )

W{XU,C)~W(X^V) and w e recove r th e Lera y spectra l sequenc e fo r th e morphis m a. Note that , jus t a s th e spectra l sequenc e o f Exampl e 5.1 . 3 (3) , thi s spectra l sequence i s define d fo r arbitrary , no t necessaril y smooth , X G Sk- Th e tw o spec tral sequence s hav e a prior i nothin g t o d o wit h eac h other . I n othe r words , th e comment i n [50 , las t sectio n o f p . 467 ] misse s th e point . Th e Bloch-Ogus-Gabbe r theorem implie s that , whe n X i s smooth , the y hav e isomorphi c E^-terms . More over, the y actuall y coincid e i n thi s cas e fo r man y theorie s (Deligne , unpublished , cf. [2 , footnote p . 1 95] , Gillet-Soule, [1 7]) . Se e als o Paranjap e [41 ] .

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7.5.2 Le t ft* be the cohomology theor y wit h support s associate d t o some complex of Nisnevich sheave s C. A s seen above , ft* satisfies axio m COH1 . B y 7.1 .2 , ft* only depends , u p t o isomorphism , o n th e clas s o f C in th e derived category P(Nis ) of th e categor y o f Nisnevic h sheaves . Indeed , th e Nisnevic h cohomologica l dimen sion o f a schem e o f finite Krul l dimensio n i s finite [38] . Fo r th e convenienc e o f th e reader, w e reformulate axiom s COH 3 an d COH 5 purel y i n term s o f C (viewed i n D(Nis)): D 3 Le t 7 r : A\— > Spec/ c b e th e structura l map . The n C -^-> RTT*(C\AI). To formulat e axio m D5 , not e that , i f ft* satisfie s etal e excisio n an d axio m COH5, th e associate d cohomolog y theor y e * satisfie s etal e excisio n a s wel l b y Re mark 5.4. 1 (2) . I f ft* is given by a substratum C , the n e * is given by the substratu m D(X) = Ker(C(P^ ) ^ C{X)). B y Theore m 7.5.2 , bot h ft* an d e * ar e give n b y Nisnevich hypercohomolog y o f complexe s o f sheave s C and V. D 5 a ) Ther e exist s a n objec t V G P(Nis ) and , fo r al l X G S k, a ma p PicX - » H o m

p(NiS|x)(P|X,C|X)

natural i n X. b) Fo r X = Pj., the ma p o f a ) induce s a morphis m iy|pi >

[0(l)]-[0]

C|pi .

Consider th e adjoin t ma p V>

R7T*C\pi.

Then th e ma p CeV (£,Q

)

) itfr^p i

is a (quasi-)isomorphism , wher e e i s the uni t (adjunction ) map . If C is a shea f o f spectra , on e shoul d replac e RTT* and Rn* b y RT T an d RT T i n axioms D 3 an d D5 . 8 A Selectio n o f Corollarie s 8.1 Multiplyin g b y a fixed variety . Le t T b e a (no t necessaril y smooth ) kvariety. Th e followin g theore m give s concrete illustration s o f item (1 ) i n subsectio n 5.5. Theorem 8.1 . 1 Let Y be the spectrum of a semi-local ring of a smooth, connected k-variety, as in Proposition 2.1 .2. Let n be prime to char k and i G Z. Then, with notation as in Proposition 2.1 .2, there are universally exact sequences: 0 -> H«(Y x

k

T , Mf ) - H* (Y

x k T , fi*)

lrx r

(8.1)

i

II^xir( * ^n ) — reY(D

O^Gq(YxkT)^Gq(k(Y)®kT)-^ ]

JG

q^(k(x)

® k T) -±-

xeYi1)

(8.2)

The B l o c h - O g u s - G a b b e r T h e o r e m 7

3

O^K?(YxkT)-^K?'«*>T(YxkT)-^L ]

J K*f*

kT

(Y x

k

T ) -^ • • • (8.3)

where (8.1 ) i s etai e cohomology. If, moreover, T is smooth, there are universally exact sequences: 0 - H«{Y x

k

7 > f ) -U Hi(k(Y) ®

fc

T, Mf ) - ^ - > (8-4)

XGYW

O^Kq(YxkT)-^Kq(k(Y)®kT)-^U ]

JK

q^(k{x)

®

k

T) - ^ - • • • .

X&YW

(8.5) Proof Afte r ite m (1 ) o f subsectio n 5.5 , th e exactnes s o f (8.1 ) an d (8.3) , a s well a s th e sam e sequenc e a s (8.3 ) wit h G instea d o f K , follow s fro m Example s 7.3 (1 ) , 7. 3 (6 ) an d 7. 4 (6) . Universa l exactnes s follow s fro m Sectio n 6 . W e hav e (8.2) b y purit y o f G-theor y [43 , Propositio n 7.3.2] . Whe n T i s smoot h w e hav e purity fo r etal e cohomology , henc e (8.4) , an d th e if-group s wit h suppor t identif y with G-group s wit h support , henc e (8.3 ) yield s (8.5) . • R e m a r k s 8.1 . 2 (1) W e could o f course state (8.3 ) an d (8.5 ) fo r if-theor y wit h finite coefficients . (2) Th e reade r i s invite d t o appl y thi s principl e t o othe r example s (e.g . /C cohomology, compar e [7 , Theore m 5.2.5]) . (3) I n th e etal e case , th e us e o f ite m (1 ) o f subsectio n 5. 5 ca n b e replace d b y the isomorphism s

UUY x

kT,^)cM*t(Y,RM^)\T))

where f : T —> Spec/ c i s th e structura l map , notin g tha t th e comple x o f sheaves Rf^((^ l)\T) i s define d ove r k. I n Appendi x B , w e shal l prov e a n analogue o f Theore m 8.1 .1 , replacin g th e projectio n F x ^ T — > F b y a no t necessarily constan t ma p 1 ^ 7 , provide d • IT is prope r an d smooth ; • dim F = 1 . One ma y as k whethe r th e conditio n d i m F = 1 is necessary . Thi s issu e i s being investigate d b y Panin . 8.2 Galoi s action . Proposition 8.2. 1 Let Sk satisfy Assumption 5.1 .6. Let R be a ring and h* a cohomology theory with supports on Sk with values in the category of R-modules. Let X, ti,.. .,tr, Y be as in Theorem 6.2.1 . Suppose h* is given by a substratum C which is strictly effaceable at t i , . . ., t r. Finally, let M be a left R-module. Then,

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for any q,s, the complex 0^Tor?(M,h(Y))-U J}

Tor?(M,hl(Y))

- ^T

J Tor?(M,hl+ l(Y))^.- .

is exact. Proof Thi s i s a n immediat e consequenc e o f Theore m 6.2.1 . • Theorem 8.2. 2 Let £/k be a finite Galois extension and G = Gal(£/k). Assume Sk satisfies Assumption 5.1 .6. Let h* be a cohomology theory with supports satisfying axioms COH1 and either COH3 or COH5, plus COH6 if k is finite. Suppose /i * is given by a substratum C'. Let X,t\,...,t r,Ybe as in Theorem 6.2.1 , with X smooth; denote by Y^ the pull-back ofY over L Then, at least if C is given by complexes of abelian groups, the complex

0-+Hn(G,h(Ye))^ J ] H n{G,hHyt))-* I I H

1 + n(G,hl

(Ye))^---

xCY™ xeY™

is exact for all q,n > 0. Proof Conside h*e,Ce given b y

r th e ne w cohomolog y theor y wit h suppor t an d substratu m h%(X)t = h% t(Xt) C(X)e = C(X t).

Then h\ naturall y take s it s values i n th e categor y o f Z[G]-modules , an d clearl y satisfies th e sam e se t o f axiom s a s h*. I f C i s given b y complexe s o f abelia n groups , then Ct take s it s value s i n th e categor y o f complexe s o f Z[G]-modules . Th e clai m then follow s fro m Corollar y 6.2. 4 an d Theore m 6.2.5 , applying the functo r H n(G, ? ) to a universall y exac t sequenc e just a s fo r Propositio n 8.2.1 . • If C i s give n b y spectra , the n C? take s it s value s i n th e categor y £ G o f Gspectra. W e can the n ge t awa y similarl y i f functorial factorization s simila r t o thos e in £ (se e subsectio n 5.2 ) ar e availabl e i n £ G, provide d wit h a suitabl e close d mode l category structure . Thi s i s closel y relate d t o Thomason' s unfinishe d approac h t o model structure s o n functo r categories , a s outline d i n Weibe l [54] . Theorem 8.2. 2 applie s i n particula r t o etal e cohomology . I t als o applie s t o algebraic if-theor y provide d on e fixes th e remar k o f th e las t paragraph . I n th e former case , specializin g t o coefficient s twiste d root s o f unit y an d usin g purity , w e get th e following , whic h wa s neede d i n [30 ] (precisel y fo r a finite bas e field!): Corollary 8.2. 3 Let X, t\,...,t r, Y be as in Theorem 8.2.2, with X irreducible; denote by Y^ the pull-back of Y over L Then, for all q, n > 0, the complex 0 - + HniCH'iYt,^)) ^

H

n(G,H*(£&),»%))

- I I H^CH'-Heix),„%*-»))-+.•. x€YtW

is exact.



The Bloch-Ogus-Gabbe r Theore m 7

5

Remark 8.2. 4 On e migh t b e tempte d t o exten d thi s resul t t o th e cas e o f an y Galois etal e coverin g (no t onl y thos e comin g fro m th e bas e field) b y usin g Proposi tion 2.2.4 , bu t thi s fails . Th e poin t i s tha t Propositio n 2.2. 4 wil l giv e homotopies , but no t necessaril y G-equivarian t homotopies . 8.3 Zarisk i cohomolog y an d Nisnevic h cohomology . Theorem 8.3. 1 (Nisnevich [39 , Theore m 0.1 2], ) a) Let h* be a cohomology theory with supports satisfying axioms COHl, COH2 and also axiom COH6 if the base field k is finite. For i G Z, let H lZar (resp. 7Y^ isy) be the sheaf associated to the presheafU H- > hl(U) on the big Zariski (resp. Nisnevich) site of Speck. Then, for all smooth X G S& and n > 0, the natural map ^ Z a r ( ^ ^ Z a r ) ~> #Nis(^" > ^ N i s )

is bijective. This theore m applie s notabl y t o algebrai c i^-theory , etal e cohomolog y an d al l examples liste d i n subsection s 7. 3 an d 7.4 . Proof W e nee d a lemma: Lemma 8.3. 2 Let X G Sk and A4 be the category of etale morphisms U — > X. For x G X and i G Z; let h lx denote the presheaf on Ai

/ - I I h l(u)Then h x is a sheaf for the Nisnevich topology on M. Proof Fo r simplicity, le t u s write h x(U) instea d o f h x(f). I t i s enough t o show that, i f / cover s X a t x, i.e. , i f there exist s x' G f~L(x) suc h tha t k(x) -^-> /c(V) , then th e sequenc e

0->hi(X)^U ]l

hl(U)-^

[

J K(Ux

xU)

yef~Hx) ze(fxxf)-^x) is exact . Writ e f~ l(x) — the ma p

{xf} U T. Pro m etal e excisio n i t is eas y t o deduce tha t

hUx)-> hi,(u) is bijective. Thi s show s tha t y i s spli t injective . I t is now enoug h t o sho w tha t th e quotient comple x

0->0-[]^(L0^ J

] K(Ux

xU)

y^T ze(fxxf)-Hx) !

is exact, i.e. , tha t ip is injective. Bu t w e can decompos e th e se t ( / Xx f)~ 1 {x) int o {(x',x')}\J{x'} xTUTx

{x'}\JT'.

1

Here w e not e that , fo r an y y G f~ (x)) th e scheme s x' xx y and y xx x' ar e spectra o f fields, becaus e k(x') = k(x)\ w e abbreviat e thes e scheme s b y (x',y) an d {y,x'). B y etal e excisio n again , th e map s

hi(U)^h\x,,v)(UxxU) hUU)^h\y,x>)(UxxU)

76

Jean-Louis Colliot-Thelene , R a y m o n d T . Hoobler , an d Brun o K a h n

given respectivel y b y the firs t an d the secon d projectio n ar e bijective. Th e injec tivity o f ip' follows. • Proof o f Theore m 8.3.1 . a ) Le t Xz a r b e the restrictio n o f the big Zariski sit e of X t o the category o f schemes etal e ove r X an d a : X^-ls— > Xza r b e the natura l projection. I t i s obviou s tha t OL*W Z^V = H lNis. Therefore , applyin g a * t o th e resolution (analogou s to ) (2.1 ) of HZar yield s a resolution o f W N[s. Fo r x G l, on e has clearl y a n isomorphism o f functor s

hence a * (2.1) ca n be identified t o the complex o f Nisnevich sheave s

0 - I I i™hl(X)-+ ] J M xex(°) xexw

+ ,

( * ) - - " - I I tSi 8/iS+'(X)^-...

xex(p) (8.6)

It i s clea r tha t z^ s i s a n exac t functo r fo r al l x G X. Therefore , th e n-t h Nisnevich cohomolog y o f the p-th ter m o f this comple x is

n Hz is(x,K+i(x)).

xex(p)

But th e Nisnevich cohomologica l dimensio n o f a field i s 0, hence thi s grou p i s 0 fo r n > 0 . I t follow s tha t th e term s o f (8.6 ) are acyclic . Finally , Lemm a 8.3. 2 shows tha t th e global section s o f (8.6 ) are (th e analogue of ) (2.1 ) . • Corollary 8.3. 3 Under the assumptions of Theorem 8.3.1 , the Zariski and Nisnevich Brown-Gersten spectral sequences of Corollary 7.5.3 coincide. Indeed, the y ar e compatible an d their E2-term s coincide . • 8.4 Shapiro' s lemma . Theorem 8.4. 1 Let h* be a strictly effaceable cohomology theory with supports and let i G Z. Let W denote the Zariski sheaf associated to the presheaf h 1 . Let f :Y —> X be a finite morphism, with Y smooth. Then R qf*W — 0 for q > 0. Proof W e can compute R qf*W b y using the (flasque) Cousi n resolution Cous of H l ove r Y. Bu t th e stal k o f f*Cous a t a poin t x G l i s non e othe r tha n th e "stalk" o f Cous a t / - 1 ( x ) , i.e , r(Oyj-i( x ),Coixs), whic h i s exact. • 8.5 Birationa l invariance . Theorem 8.5. 1 Let h* be a cohomology theory with supports satisfying axioms COH1, COH2 and also axiom COH6 if the base field k is finite. Let X G Sk be / l smooth and let H*(X,W) denote either of the groups H Z8iT(X,HZaiY),H^is(X, H Nis) of Theorem 8.3.1 (they coincide by this theorem). Then, for all i G Z, H°(X :Hl) is a birational invariant of smooth proper varieties X G Sk . Proof a ) By Corollary 5.1 .1 1 , the functor X H- > h l{X) satisfie s "codimensio n 1 purity" fo r regular loca l rings of a smooth variet y i n the sense of [7, Definition 2.1.4 (b)] (a cohomology class which is unramified a t points of codimension 1 is unramified everywhere locally) . Th e claim no w follows fro m [7 , Proposition 2.1 .8] . •

77

The Bloch-Ogus—Gabbe r Theore m

8.6 Rationa l invariance . Let Sk = Sm/k, an d le t h* b e a cohomolog y theor y wit h support s o n VkAssume h* satisfie s axiom s COH 1 (etal e excision ) an d COH 2 (ke y lemma) , th e latter fo r al l V £ Sm/k. I f k i s finite, assum e h* als o satisfie s axio m COH6 . W e then hav e th e followin g theorem . Theorem 8.6. 1 Let X,Y be two smooth integral k-varieties, with respective function fields fc(-X"), k(Y), and let p : X— > Y be a proper morphism. Assume that the generic fibre X^ of p is k(Y)-birational to d-dimensional projective space P/Ly) • Then, for any i G Z, the map

H°(Y,nl) ^H^X.H 1 ) is an isomorphism. Proof Fo r an y smoot h integra l k- variety Z , wit h generi c poin t 77 , we have b y definition h\[Z) = li m ti{U) ucz where U run s throug h th e nonempt y ope n subset s o f Z. W e denot e thi s grou p b y hl(k(Z)). Corollar y 5.1 .1 1 yields a n exac t sequenc e 0^#°(Z,?f)-^(/c(Z))-+ ]

JK xezw

1 +

{Z). (8.7

)

We ma y replac e Pf(y ) b y th e d-fol d self-produc t ( P L y O d i n th e assumptio n of Theore m 8.6.1 . B y hypothesis , ther e exist s a birationa l ma p ( P ^ ) d ^X over Y. Sinc e (Py) d i s regula r an d p i s proper , thi s rationa l ma p extend s t o a F-morphism

u-f->x where U is an open subset o f (Py) d containin g al l points of codimension 1 (valuative criterion o f properness , cf . [26 , Theore m II . 4.7]) . Th e exac t sequenc e (8.7 ) the n shows tha t th e restrictio n ma p #0((P^)^)^#0([/,W*) is a n isomorphism . Let C (resp. £ ) b e th e generi c poin t o f X (resp . ( P y ) commutative diagra m

d

an d U). W e hav e a

78

Jean-Louis Colliot-Thelene , R a y m o n d T . Hoobler , an d Brun o K a h n

hl(k{X)) J



l

h

(k(PY)d) =

II I

h

II

hl(X) - i l

* h\(U)

U H°{U,n

(k(P Y)d)

I

=

d Y) )

hl((P

U

H°(X,W) ^

1 i

U i

)=

H°((P

Y)

d

,Hl)

in whic h th e vertica l inclusion s follo w fro m (8.7) . Sinc e / i s birational , / * i s a n isomorphism. I t i s thus enoug h t o prov e Theore m 8.6. 1 i n th e cas e X = (P Y)d. B y induction o n d, w e ma y assum e d = 1 . We first dea l wit h th e specia l cas e Y — Spe c k. T o begin with , th e natura l ma p h\k) - ti(k(Pb) is injective . I f k i s infinite , thi s follow s fro m th e classica l sectio n argument , sinc e any ope n subse t o f P\ contain s a rationa l point . I f k i s finite, axio m COH 6 pro vides a varian t o f thi s argument , sinc e an y ope n subse t o f P\ contain s tw o close d points o f coprim e degrees . On th e othe r hand , sinc e P\ i s o f dimensio n one , w e hav e a n exac t sequenc e

h\Pl)^V(k(Pl))-> U

h^(Pl) *e(pi)(1)

hence, fro m (8.7) : H°(PlW) =

lm(V(Pl)^V(k(Pl))).

The ma p h l(P\) - » h l(k(Pl)) obviousl y factor s throug h h l(Al). B y axio m COH2, i t eve n factor s throug h h l(k), henc e Theore m 8.6. 1 i n thi s case . In th e genera l case , le t7 7 = Spe c k(Y) denot e th e generi c poin t o f Y. Not e that an y smoot h /c(Y)-variet y i s a filtering invers e limi t o f smooth fc-varieties, wit h affine transitio n morphisms : w e ma y therefor e exten d h* t o Vk(Y) (correspondin g to Sk{y) -= Sm/k(Y)) b y direc t limits . Thi s cohomolog y theor y wit h support s obviously satisfie s axio m COHl ; i t als o satisfie s COH 2 becaus e w e assume d th e original h* verified i t fo r al l smoot h varieties .

T h e Bloch-Ogus—Gabbe r T h e o r e m

79

We hav e a commutativ e diagra m wit h exac t row s H°(P\r,W)-

H°(Y,W)

H°(P'W)

II K

h\k(Y))

1+

(Y).

xZYW

To conclud e th e proof , i t i s sufficien t t o sho w tha t th e righ t vertica l ma p i n thi s diagram i s injective. Bu t thi s ma p factor s throug h

where r\ x — Spec A;(P*). Let x G Y^: w e hav e t o se e tha t th e ma p /i^ +1 (F)— » ^ ^ ( P y ) i s injective . We ma y replac e Y b y Y' = SpecCV, x . B y definition , w e hav e z where Z run s throug h prope r close d subset s o f P* . Suppose firs t tha t k(x) i s infinite. The n V\ — Z contain s a A:(a:)-rationa l point ; since P 1 (Yf) —> P 1 (k(x)) i s surjective , w e ma y lif t i t t o a sectio n 5 of P y ,— > Y'', which doe s no t mee t Z. The n th e composit e Wx+l(Y) - + ^ _ Z ( P [ , - Z ) - ^ V

+ X

\Y)

is th e identit y an d h l+l(Y)— » / i m ^ f P y / — Z ) i s spli t injective . Suppose no w tha t k(x) i s finite, henc e k i s finite. The n V\ — Z contain s i n an y case tw o close d point s x\,x V i s finite, V i s norma l an d F i s a constan t sheaf give n b y a finitely generate d Z-module . Passin g t o th e stric t henselizatio n A of V a t a geometri c poin t x , w e get a commutativ e diagra m o f cartesia n squares : Spec K'

Spec A7

V

TA

Spec ft

Spec A

V

Here K = ft(x), Spe c A' = Spe c A x v V an d Spe c ft' = Spec K Xy V , s o tha t K' = K^ AAf. Sinc e A! i s finite over A, i t i s a product o f strictly henselia n loca l ring s and K' is an Arti n loca l ring wit h th e sam e residu e fields a s A' '. Sinc e V i s normal , so ar e A' an d an y o f it s connecte d components . Le t B b e suc h a component . Sinc e 7f is proper, i t i s of finite type , henc e TTB an d ^ K{y) ar e norma l b y [EGA4 , (6.8.3)] , where fo r an y rin g R an d morphis m Spe c it! —> V, w e denot e b y TTR : XR— > Speci f the pull-bac k o f TT . I n particular , XB an d X K^ ar e normal , wher e y i s th e close d point o f Spec B. Hence , b y [1 0 , (2.3)] , w e have: (R'>(7rB)tF)ycH^XK{y),F). On th e othe r hand , lettin g K" be th e produc t o f th e residu e fields o f A f, th e mor phism X K»— > X K> is radicia l henc e induce s a n isomorphis m o f etal e cohomolog y [36, Chapte r II , Remar k 3.1 7] . I t follow s tha t tf\W{*ju)«F)~B?{%K,\F. q

Since r i s finite, R {rA)* = R q{rK)* = 0 for al l q > 0 [36 , Chapte r II , Corollar y 3.6] an d th e latte r isomorphis m implie s L*Rq(rA o itA>)*F - R q(rK o

T T ^ ^ T F.

The Bloch-Ogus—Gabbe r Theore m

81

x K,

X K.

X v>

XA> 7TA>

Spec nf

Spec n"

-+ Spec A'

+V

TA

Specie

Spec A

+

V

But usin g no w the finiteness o f Xyt— > Xy an d the induced map s an d arguing a s in [1 0 , proof o f (2.4)] , we get an isomorphis m L*R«(*A)*(T*F) ~

^(7r,)*(0*(T* n

as desired . (Se e diagram below. ) XK< • finite

XA> finite

xK

XA 7TA

Specie

Spec A

Remark A . 1.2 W e can use Artin's exampl e i n [SGA4-III , expos e XII , Section 2 ] t o sho w that , i n general , on e canno t exten d Propositio n A . 1.1 t o mor e general complexe s o f sheaves ove r X tha n thos e o f the for m D = TT*C . T o be specific, tak e V = Speci? wher e R i s a complete discret e valuatio n rin g an d X — P ^. Let Y b e the projective curv e ove r R wit h equatio n zy 2 — x(x — z)(x — irz), where 7r is a uniformizin g paramete r o f i?, that w e view a s a two-fol d coverin g Y — > X via th e functio n x/z. Le t D = r*Z[0] . The n H l(X,D') = H l(Y,Z) = 0 whil e iJH-^o, D) = H^YQ, Z) ~ Z where X 0 , Y0 ar e the special fibres o f X an d Y. A.2 A n integra l Cher n class . Definition A.2. 1 (compar e [1 3 , proof o f Lemma 2] ) Let i £ Z. a) Fo r any prime numbe r p , we denote b y Q p/Zp(i) th e extension b y 0 of the etale sheaf Q p/Zp(i) = lim/i® n fro m Spe c Z[l/p] t o Spec Z. Thi s define s a sheaf ove r the big etal e sit e o f Spec Z. b) W e define Q/Z(i ) = © Q P / Z p ( i ) . Remark. Not e tha t wit h thi s definition , Q/Z(0 ) doe s no t in general coincid e with Q/Z!

Jean-Louis Colliot-Thelene , R a y m o n d T . Hoobler , an d Brun o Kah n

82

Let Q/Z(0)[-1] - + Z (A.l

)

be th e morphism i n the derived categor y o f big etale sheave s ove r Spe c Z define d as follows : fo r a prime p, we have a n exact sequenc e o f sheaves 0 —> Z —> Z[l/p ]—• > Qp/Zp— » 0 hence a morphism Q p/Zp[—1 ]—- > Z in the derived category . Lettin g j p be th e open immersio n SpecZ[l/p ]— > SpecZ , w e get a correspondin g morphis m Q p /Z p (0)[-1] : = (j p )!Q p /Z p [-l] - (j

p ),Z

= (j P )!(j P )*Z.

Composing wit h th e adjunctio n ma p (j p)\(jp)*Z— » Z, w e ge t a morphis m Qp/Zp(0)[—1]—• > Z ove r SpecZ . Th e desired morphis m i s the sum of these morphisms fo r all primes p. Definition A.2. 2 (compar e [1 3 , Appendi x B] ) Le t A b e a schem e an d L a line bundl e o n A. T o L w e associate a morphism i n D(Xet)

Q/Z(-l)[-3] ^1Z as follows . Th e class o f L i n i7 1 (Aet,G m ) correspond s t o a morphism i n D(Xa) Z -S^ l Gm [l]. On the other hand , fo r all primes p, the sheaf G m i s p-divisible awa y fro m th e locus where p is not invertible. Thi s yield s fo r all n > 1 an isomorphism : Gm®Z/n(0)^*Z/n(l)[l]. These fi t togethe r t o give a "Kummer " isomorphis m Gm®Q/Z(0)^Q/Z(l)[l]. C\{L) i s then th e composition o f the morphisms i n the sequence Q/Z(-l)[-3] - G

m

I Q / Z ( - l ) [ - 2 ] ^ Q/Z(0)[-1

]->Z

where th e firs t morphis m i s [L] tensored b y Q/Z ( —1)[—3], th e secon d on e is the Kummer isomorphis m twiste d an d shifted an d the last morphis m i s (A.l). Fo r two line bundle s L an d V o n A, w e have C\(L®L f) — C\(L) + C\(L')\ i n particular , if L i s trivial the n C\{L) — 0. The las t clai m o f the definition i s obvious fro m th e construction o f C\ an d the additivity o f line bundl e classe s i n H 1 (Xet, G m ). A.3 Cohomolog y o f P 1 . T h e o r e m A.3. 1 (compare [SGA5 , expos e VII , Theorem 2.2.1 ] and [13], Lemma 3) Let V be as in Proposition A. 1.1 and let n : P y— > V be the natural projection. Then, for any complex of sheaves C over the small etale site ofV, there is a natural isomorphism in D(Vi t)-'

C ©C"!>Q/Z(-l)[-3 ] - ^ RTT^C. This isomorphism is the adjunction of a morphism 7r*C-0 7r*C"®7r*Q/Z(-l)[-3] - > rC in which the first component is the identity and the second one is given by tensoring (in the derived sense) the Chern class map 7r* Q / Z ( - l ) [ - 3] v definition A.2.2 by n*C.

y

'' TT* Z of

The Bloch-Ogus—Gabbe r Theore m

83

Proof B y Propositio n A . 1.1, w e hav e a n isomorphis m for an y geometri c poin t x o f V. T o prove Theore m A.3.1 , w e may therefor e assum e that V — Spe c K, wher e K, i s a separabl y close d field. W e first remark : Lemma A.3. 2 Let C be a complex of abelian groups and D be a bounded above complex of etale sheaves over P *. Then the natural morphism in the derived category of abelian groups c•®i^^(Pi,£)•)->i^^(Pi,c• !)£)• ) is an isomorphism. Proof Th e argumen t o f [36 , Chapte r VI , Lemm a 8.7] , whic h consist s o f re ducing t o th e cas e wher e C i s a singl e finitely generate d fre e Z-modul e place d i n degree 0 , applie s (compar e loc. cit., Remar k 8.1 4) . Not e tha t i t i s not necessar y t o assume tha t C i s bounde d above , sinc e th e Tor-dimensio n o f Z i s finite. Applying Lemm a A.3. 2 t o D — Z[0] ( Z place d i n degre e 0) , i t no w suffice s t o prove Theore m A.3. 1 i n th e case C — Z[0] . I n thi s case , i t follow s fro m Lemma A.3. 3 a) We have H l(V\, Z ) = 0 and H q(Pl, Q if char(n) — p > 0 . b) There is an isomorphism in D(Ab):

p /Z p )

= 0 for q > 0

K*®z[-i]-^i?r(Pi,G m ) whose first component is the adjunction of the map 7f*ft *— > G m in D((P * )^) and the second one is the adjunction ofJr*Z •

Gm (compare definition A.2.2).

1

Proof Th e vanishing o f H (P* , Z) follow s fro m th e normalit y o f P*. Suppos e char(ft) = p > 0 . W e hav e

[36, Chapte r II , Propositio n 3. 7 an d Remar k 3.8 ] an d [26 , Chapte r III , Theore m 5.1]. Usin g th e Artin-Schreie r exac t sequenc e 0 - » Z/ p - > G a —- U G a - > 1 [36 , Chapter II , Exampl e 2.1 8 (c)] , this implie s tha t Hi(PlZ/p)=0 i

f q>0

for p equa l t o th e characteristi c o f K. Usin g th e exac t sequence s 0 -> Z/p n -+ Z/p n+l ->Z/p-*0 it follow s tha t H q(V\, Z/p n) = 0 for al l n , henc e H q(Pl, Q claimed. Finally , w e hav e

p /Z p )

= 0 for q > 0, a s

( K* i f^ =0 H*(PlGm)={ PicP ^ = Z i f q=l {0 i fq >1 [36, Chapte r III , Exampl e 2.2 3 (b)] , wher e th e secon d isomorphis m i s induce d b y the degre e map . Sinc e 0(1 ) generate s PicPj. , th e ma p Z — > G m [l] define d b y it s class induce s a n isomorphis m Z = # 0 ( P ^ , Z ) — H\Pi,G

m).

84

Jean-Louis Colliot-Thelene , Raymon d T . Hoobler , an d B r u n o Kah n

inverse t o th e former . Th e las t clai m o f Lemm a A.3. 3 follows . We no w finish provin g Theore m A.3. 1 fo r C — Z[0] i n th e cas e V — Spec ft. By e.g. , [1 0 , (2.1 )] , ^ ( P * , Q ) = 0 fo r q > 0 . Fro m th e exac t sequenc e 0 -• Z - * Q - > Q/ Z - + 0 we deriv e

H^\PlQ/Z)^H^PlZ) fo

r q >1 .

In vie w o f Lemm a A.3. 3 a) , thi s implie s tha t (A.l ) induce s isomorphism s

H^iPlQ/Zm^H^PlZ) fo

r q >1.

Using th e Kumme r exac t sequenc e 1 — » ^ n— » G m -^- > G m— > 1 [36 , ch. II , example 2.1 8 (b)] , Lemm a A.3. 3 b ) implie s tha t H q(P1 K,/in) = 0 fo r q ^ 0, 2 an d n prim e t o th e characteristi c o f K; i n particular , H q(P^ Q/Z(0) ) = 0 for q ^ 0 , 2. We therefor e hav e H q(V\, Z ) = 0 for q ^ 0 , 3. Clearly, Z —> ff°(P^. , Z) is an isomorphism; i t remains to see that th e adjoin t t o C i ( 0 ( l ) ) induce s a n isomorphis m Q / Z ( - 1 ) -^- > # 3 ( P ^ , Z ) . T o d o this , w e follo w the definitio n o f C\. Accordin g t o definitio n A.2. 2 an d Lemm a A.3.2 , th e ma p Q/Z(-l)-+ff3(Pi,Z) can b e decompose d a s follows : Q/Z(-l)^i/1(^r(pi,Gm)®Q/Z(-l))^//1(P«>Gm®Q/Z(-l)) -^ H

2

(Pl,Q/Z(0)) ^

H

3

(PlZ).

By Lemma A.3.2, the second ma p is an isomorphism. Accordin g to definition A.2.2 , the third ma p is an isomorphism an d as seen above the fourth on e is an isomorphis m too. I t remain s t o se e tha t th e first ma p i s a n isomorphism . Bu t i t i s obtaine d b y tensoring th e isomorphis m o f Lemm a A.3. 3 b ) b y Q/Z ( —1). • Appendix B Th e One-Dimensiona l Cas e In thi s section , w e prov e a versio n o f Gersten' s conjectur e fo r regula r one dimensional schemes , no t necessaril y i n th e presenc e o f a bas e field. Th e proof mimics Gillet' s i n [1 6] . A s i n Sectio n 5 , w e shal l axiomatiz e th e situation . Th e axioms necessar y t o mak e th e proo f wor k tur n ou t t o b e muc h mor e costl y tha n those i n sectio n 5 . B . l Som e axioms . Le t A b e a semi-loca l principa l domain , S = Spe c A an d Ss th e categor y o f regula r scheme s separate d an d quasifinite ove r S. B y Zariski' s main Theorem , a connecte d objec t X —> • S o f Ss i s o f th e for m SpecB , wher e B i s eithe r a finite extensio n o f on e residu e field o f S o r a localizatio n o f a finite extension o f A a t som e maxima l ideals . W e give ourselve s a "cohomolog y theory " hl :S

s^A{ieZ),

a collectio n o f contravarian t functor s t o som e abelia n categor y A, satisfyin g th e following axioms : (i) Additivity . h* is additive . (ii) Transfers . Fo r a finite morphis m / : Y — • X i n Ss, ther e i s give n a ma p /* : /i*- 2 c (y) - * h*(X), wher e c = codimx Y; thi s collectio n o f maps make s h* a covarian t functor .

85

T h e Bloch-Ogus—Gabbe r T h e o r e m

(iii) Purity . Fo r X 6 Ss o f dimensio n 1 , Z —— > X a (reduced ) close d subse t o f dimension 0 and U — > X th e complementar y ope n subset , ther e is an exac t sequence • h l-2(Z) -±>

h\X) -£>

h l(U) -£+ h r~\Z) -*.-.

Moreover, i f / : X'— * X i s a finit e an d fla t map , th e squar e hl(U')

&

•* h?-\Z')

f. hl(U)

-*• W-

l

{Z)

commutes, where Z' = f~ 1 (Z)red, U' = X' — Z' an d / ' : Z' —* Z i s the ma p induced b y / (n o multiplicities! ) (iv) Actio n o f units . Fo r an y X £ Ss ther e i s a pairin g T(X,0*x) x

h*(X) - > h*

+l

(X)

which i s contravarian t i n X an d satisfie s th e projectio n formul a fo r finit e flat maps . Moreover , • fo r X, Z, U, d a s i n (iii) , we have , fo r (/ , a) € T{U, 0*x) x h*(X):

d(f-fa) = Y,v^e^ zez where i z i s th e inclusio n z •- + X (her e w e use d th e additivit y o f h*). • I n th e situatio n o f (iii) , give n / € T(X,Q X), th e followin g diagra m anticommutes: 3 hx-x{Z) hx(U) •f

ti+l{U)

•f

h\Z)

(v) Rigidity . Le t X G Ss o f dimensio n 1 , x G X a close d poin t an d X% th e henselization o f X a t x. Le t h*(X%) : = lim/i*(C7) , where U runs throug h al l etale neighbourhood s o f x. The n /i*(X£ )— > /i*(x ) i s a n isomorphism . Examples B . l . l (1) Algebrai c X-theor y verifie s al l axiom s excep t (v) ; algebrai c if-theor y wit h coefficients Z/n , wher e n i s invertible i n A , satisfie s al l axiom s includin g (v ) [47]. (2) Le t C be a comple x o f sheaves ove r th e smal l etal e sit e o f 5 ; assum e tha t it s cohomology sheave s ar e al l locall y constan t constructible , torsio n invertibl e in A. The n h*(X) = U n e Z Hl t{X,C(n)) satisfie s al l th e axioms . Axio m (i) i s a genera l propert y o f etal e cohomology . Axiom s (ii ) an d (iii ) follo w

86

Jean-Louis Colliot-Thelene , Raymon d T . Hoobler , an d B r u n o K a h n

from cohomologica l purit y i n dimensio n 1 [SGA5 , expos e I , Theore m 5.1 ] and th e existenc e o f trac e map s [SGA4-III , expos e XVIII] . Axio m (iv ) i s folklore: se e [42 , Lemma 3 ] for a detaile d proof . Axio m (v ) ca n b e deduce d from prope r bas e chang e a s i n [SGA4-III , expos e XII , cor . 5.5] . Not e tha t cohomological purit y an d prope r bas e change for complexe s o f sheaves follo w from th e sam e fo r sheave s plu s compariso n o f hypercohomolog y spectra l sequences. B.2 Th e result . Theorem B.2. 1 Let R be a ring and /i * a cohomology theory with values in R-modules, satisfying axioms (i)-(v). Let Z be the set of closed points of S and r\ its generic point. Then, for all i £ Z, the sequence 0 - > h l{S) - + ti{rj) - ^ h l~\Z) -

+0

is universally exact. Proof Fo r convenience we use ring-theoretic notation . Le t 1 Z be the radica l of A an d F it s field o f fractions , s o tha t Z — V(1 Z) an d rj = Spec F an d th e sequenc e of Theore m B.2. 1 ca n b e rewritte n 0 -• h l{A) - • h\F) -^

%

h

-\A/n) -

• 0 . (B.l

)

Write h l~1 (A/lZ) a s a direc t limi t o f finitely presente d jR-modules . Le t M be suc h a module . W e construc t a n ^-linea r ma p / M • M— > hl(F) suc h tha t d o JM — ^ ^ where i i s th e ma p M — > h l~l(A/1 Z). I n vie w o f th e lon g cohomolog y exact sequenc e o f whic h (B.l ) i s par t (axio m (hi)) , thi s wil l sho w tha t th e restric tion o f (B.l ) t o M i s a spli t exac t sequenc e o f i^-modules . Let A h b e th e henselizatio n o f A alon g 72. . The n A h split s a s a finite produc t of henselia n discret e valuatio n rings . B y axiom s (i ) an d (v) , th e natura l ma p h%-\Ah) ->

h

l l

~ {Ah/llAh)

is a n isomorphism . Consider th e commutativ e diagra m Hom H (M, h i~1 (Ah)) >

Hom#(M

ImiHomRiMTh1-1^)) •

limHom

l l

~ (Ah/lZAh))

,h

H (M,

h

l l

~ (A'/IIA'))

where A' run s throug h th e quasi-finit e yl-subalgebra s o f A h. Sinc e M i s finitely presented, th e tw o vertica l map s ar e isomorphisms , henc e s o i s als o th e botto m horizontal one . Therefor e ther e exist s a n A' an d a n iiMinea r ma p a:M -*h

l

-\A')

T h e Bloch-Ogus—Gabbe r Theore m

87

such tha t th e diagra m ti-^A'/KA')

hi-l{A/n)

M commutes.

Let A\ b e th e integra l closur e of A i n th e tota l rin g o f fraction s F' o f A'. The n Ai C A' an d A' i s a semi-localizatio n o f A\ a t som e o f its maxima l ideals . Sinc e A' is etale ove r A, F'/F i s separable an d A\ i s finite ove r A [46 , Proposition 8 , p.24] . Then th e diagra m

Cor

(B.2)

Cor hl-l{A/U)

ti{F)

commutes (axio m (iii)) . Le t H x = HA X. Writ e U x = ftW, wher e TV + K" = A K'A' = Iii A' an d 1 1 " A' = A'. The n Ai/Ki =

Ai/K' x

i4i/7e " * A'/ftA ' x Ai/K" ^

u

A/f t x 4 i / f t "

and th e composit e A/ft - • Ai/K! =

A/1 1 x Ai/n" ^

A/ 1 Z

is th e identity . Accordingly , diagra m (B.2 ) becomes : hl(F') Cor

V(F)

WJ®!) h

1 i

- {A/n)®hi-x{Alin")

Id

(B.3)

7

hl-\A/1l)

By th e Chines e remainde r theorem , choos e f £ E'* suc h that : • / = 1 (mo d ft"); • / generate s ft 7. For p e h l-l(A'), writ e / ? for it s imag e i n / ^ ( . A ' / T t A ' ) = ^^{A/Tl). Apply ing axio m (iv) , w e get : L e m m a B.2. 2 Fo r a/ / / ? e h i~1 (Af), one

-

has

df(P-(f)) = 0 ; 3"(/J • ( / ) ) = < > .

a

88 J e a n - L o u i

s Colliot-Thelene , Raymon d T . Hoobler , an d B r u n o K a h n

Corollary B.2. 3 The map f M : M - > h l(F) given by fM(a) =

Cor F / / F (cr(a) • (/))

has the required properties. Proof Thi s follow s fro m diagra m (B.2 ) an d Lemm a B.2.2 . D B.3 Corollaries . Notation B.3. 1 Fo r an y schem e X , w e denot e b y CC^(X) th e categor y o f bounded belo w complexe s o f sheave s ove r th e smal l etal e sit e o f X , whos e co homology sheave s ar e locall y constan t construc t ible, torsio n prim e t o th e residu e characteristics o f X. Corollary B.3. 2 With notation as in Theorem B.2.1 , let B be a finite, etale, Galois A-algebra, with Galois group G. Let C G CC+(B). Let E be the total ring of fractions of B. Then the complex of Z[G]-modules 0 -> ti{B) - • h\E) -?->

ti^iB/KB)

- >0

is universally exact for all i G Z. Proof Appl y Theore m B.2. 1 t o Rn*C' an d R = Z[G] , wher e n : S p e c£—> Spec A is th e natura l map . D Corollary B.3. 3 With notation as in Theorem B.2.1 , let X be a proper and smooth A-scheme, Xp its generic fibre and XQ its closed fibre. Let C G CC +{X). Then the complex 0 -+ H^XtuC) -

H\X

FitUC')

^

W-\X^enC\-l)) -

0

is universally exact for all i G Z. Proof Le t IT : X— > Spe c A b e th e structur e map . B y [36 , Corollar y VI.4.2] , Rn*C i s in CC +{A). I t i s clea r tha t H Q(F,RK*C) = H°(X F,C), sinc e Spec F - • Spec A i s a n ope n immersion . Moreover , th e prope r bas e chang e theore m [36 , Corollary VI.2.3 ] show s tha t H°(A/K,Rn*C'(-l)) ^ i f ° ( X 0 , C ( - l ) ) . Therefor e the comple x o f Corollar y B.3. 3 ca n b e rewritte n 0 —• > Hit(A,Rn+C-) -*

Hi t{F,RTr*C) - ^ H^iA/K^R^Ci-l)) -

•0

and w e ca n appl y Theore m B.2.1 . • Remark B.3. 4 W e ca n combin e corollarie s B.3. 2 an d B.3.3 .

Appendix C Unbounde d Complexe s In thi s appendix , w e exten d th e notio n o f injectiv e resolutio n fro m bounde d below t o unbounde d complexe s o f object s o f a suitabl e abelia n category . Thes e results ar e simila r t o thos e o f Spaltenstei n [45 ] (se e als o [3]) .

The Bloch-Ogus—Gabbe r T h e o r e m

89

C.l Fibran t complexes . Le t A b e a n abelia n categor y an d le t C(A) b e th e category o f complexe s o f object s o f A- W e set u p th e followin g definition : Definition C . l . l a ) A morphis m C -^D in C(A) i s a trivial cofibration if it is both a monomorphism an d a quasi-isomorphis m (i.e., i t induce s a n isomorphis m o n cohomology) . b) A n objec t F G C(A) i s fibrant i f i t ha s th e followin g propert y give n a trivia l cofibration C — > D', an y morphis m fro m C t o F extend s t o a morphis m fro m D to F\ Fibrant complexe s ar e closel y relate d wit h K-injectiv e complexe s i n th e sens e of J . Bernstein . I t ca n b e show n tha t th e latte r ar e thos e complexe s whic h ar e homotopy equivalen t t o th e forme r (compar e [45 , Proposition 1 .5]) . Proposition C.l. 2 a) If F is fibrant, then F n is injective for any n G Z. b) If F is bounded below, the converse is true. Proof a ) Le t A °- > B b e a monomorphis m i n A, an d le t / : A — > F n be a homomorphism . Le t C\D b e th e complexe s suc h tha t C l — D% — 0 fo r i ^ n , n + l , C n - C n + 1 = A, D n = D n+l = B an d th e differential s C n - + C n + 1 and D n— > D n+1 ar e give n b y th e identity . Th e monomorphis m ip induces a n ob vious monomorphis m o f acycli c complexe s C D ', an d / induce s a morphis m o f complexes / ' : C— > F ' . Applyin g th e definin g property , / ' extend s t o a morphis m f : D' —> F', whos e restrictio n t o D n = B define s a n extensio n o f / t o B. b) I t i s convenien t t o giv e a lemma : Lemma C.1 . 3 Let F G C(A) be such that F n is injective for some n G Z. Let C D be a trivial cofibration, and let f : C — • F be a homomorphism. Assume that f n~l : C 71 " 1 - • F n~l extends to f n~x : D71" 1 - > F n~l such that: (i) p- l(Bn-l(D-)) C B n-l(F-); n 1 n 1 (ii) f - (Z - (D')) c r ^ F ) . Then there exists f n : Dn— > F n extending f n such that

a) f nd = df n-1 ; b) f n(Zn(D-)) CZ

n

(F').

n

Proof W e define f first o n B n(D'), the n o n Z n(D') an d finally o n al l of D n. • O n £ n ( £ > ) , defin e f n b y f n{dy) = df n~l{y) fo r y G Dn~l. B y assumptio n (ii), thi s doe s no t depen d o n th e choic e o f y. • O n Z n(D'), defin e f n a s th e uniqu e ma p whos e restrictio n t o B n(D') i s a s above an d whos e restrictio n t o Z n(C') i s f n. Thi s i s well-define d b y th e quasi-isomorphism assumption . • O n D n, choos e fo r f n an y extensio n o f the above , applyin g th e injectivenes s of F n. One readil y check s tha t f n indee d verifie s condition s a ) an d b) . • Proof o f Propositio n C.l. 2 b) . Lemm a C.1 . 3 implie s tha t i f it s condition s a) an d b ) ar e satisfie d fo r som e n G Z the n the y ar e satisfie d fo r n + 1 . I n cas e F is bounde d below , condition s a ) an d b ) ar e triviall y satisfie d fo r n « 0 . •

90 J e a n - L o u i

s Colliot-Thelene , R a y m o n d T . Hoobler , an d Brun o K a h n

Lemma C.1 . 4 a) Let C G C(A) be acyclic and F e C(A) be fibrant. Then any morphism C — > F is homotopic to 0 . f b) Let F be fibrant and C be arbitrary. Then, any quasi-isomorphism F — » C has a homotopy left inverse. If C is itself fibrant, f is a homotopy equivalence. Proof a ) Applyin g th e definin g propert y o f "fibrant " t o th e monomorphis m F' c -> C ( / ), wher e C(f) i s the mappin g con e of / , w e get tha t th e identit y F — > F extends t o a morphis m C(f) — > F (not e that , sinc e C i s acyclic , F ^- > C(f) i s a quasi-isomorphism). Sinc e th e composit e C — > F — > C(f) i s homotopi c t o 0 , w e get tha t / itsel f i s homotopic t o 0 . b) (cf . [24 , proof o f Lemm a 4.5] ) Not e tha t th e mappin g con e C(f) i s acyclic . Applying a) , w e se e tha t th e morphis m C(f) — > F'[l] i s homotopi c t o 0 , an d th e conclusions easil y follow . • Lemma C.1 . 5 Consider a commutative square of objects ofC(A) C



4

r

^F V

D in which ip is a trivial cofibration and F' is fibrant. Let (3 : D —> F r be a morphism extending a'. Then the two morphisms a, if op : D— > F are homotopic. Proof B y assumption , a — ip o /3 factors throug h th e acycli c comple x D /C . The conclusio n no w follow s fro m Lemm a C.1 . 4 a) . • C.2 Homotop y limits . Definition C.2. 1 Le t (F n , ^n +i,n)n>o b e a projectiv e syste m o f object s o f C(A). It s homotopy limit holimF n i s th e tota l comple x associate d t o th e doubl e complex (wit h vertica l lengt h 1 )

where {Y[F n)q '•— n ^ ? ? ' ^ n e differentia l bein g define d componentwise , an d wher e Dn : UK - Tin i s define d a s ( - l ) ' ( J d - ^ + 1 , „ ) . One readil y check s that D anticommute s wit h th e differential s o f ]\ F n, s o tha t the constructio n indee d define s a complex . I t i s clea r tha t holi m i s a functor . B y definition an d th e tw o spectra l sequence s associate d t o th e doubl e complex , w e have: Proposition C.2. 2 a) There is a short exact sequence of complexes 0 -> JimF n - > holimF n - > J i m ^ f l ] - > 0 hence a long exact sequence of cohomology groups > H q(hmFn) -

> H q(ho\imFn) -

• H q~1 {mii1 Fn) -

• H q+1 {\imFn) -*

• ••

T h e Bloch-Ogus-Gabbe r Theore m 9 1

b) Assume that countable products are exact in A. Then sequences

there are short exact

0 -• J i m 1 ^ - 1 ^ ) - • H q(Yio\imFn) - > \miH q{Fn) - > 0 . D Corollary C.2. 3 Assume that countable products are exact in A. Let (C n ), (Fn) be two projective systems in C(A), and let (f n) be a morphism from (C n) to (Fn). Suppose that each f n is a quasi-isomorphism (resp. a trivial cofibration). Then holi m fn is a quasi-isomorphism (resp. a trivial cofibration). • Lemma C.2. 4 For all n > 0, let (p n : holimF n— » F n be given on (holim F n ) q — Yl^n ® n ^ ?- 1by {Pn,0), where p n is the n-th projection. Then: a) The ip n are morphisms of complexes. b) For all n > 0, ip n and tp n+i,n o F n given b y S n = (0 , ( — l )9 ^ - 1) i s a homotop y betwee n p n an d ip n+i,n ° ^n+i- D Lemma C.2. 5 Let (F n ,(/? n+1 n) n >o be as in Definition C.2.1 , and let C be in C(A). a) Let f : C— > holi m Fn be a morphism. Then ip n o f = 0 for all n if and only if f = (0 , (gn)), where, for all n, g n is a morphism from C to F n[\]. b) Let f n : C— > Fn be a family of morphisms such that, for all n > 0, f n and 9?n+i,n ° /n+ i are homotopic. Then there exists a morphism f : C' —* holi m F n such that, for any n > 0, (p n o f = f n. Proof a ) i s a simple computation, b ) Choos e for al l n a homotopy s n betwee n fn an d ipn+i,n ° /n+i- Defin e / b y

/ 9 = ((^),((-i) g 4))). One check s easil y tha t / i s a morphis m o f complexes , an d tha t th e identit y ¥n° f

= fn h o l d s . •

Proposition C.2. 6 Let (F n ,(/? n + i ; n ) be an inverse system in C(A). If Fn is fibrant, then holi m Fn is fibrant.

each

Proof Le t C -^- > D b e a trivia l cofibration , an d le t / : C— > holi m Fn b e a morphism . B y th e fibrancy o f F n, w e ma y exten d ip n o f t o a morphis m f n fo r each n. Lemm a C.1 . 5 show s tha t f n an d (/? n+i,n ° fn+i ar e homotopi c fo r an y n ; therefore, applyin g Lemm a C.2. 5 b) , w e ca n find a / : D — » holi m Fn suc h tha t ipn o / = f n fo r al l n. W e hav e ipn o (/ — / o a) — 0 fo

r al l n.

By Lemm a C.2. 5 a) , w e ca n writ e / — / o a = (0 , (#n))> wher e eac h g n i s a morphism fro m C t o F n[l\. Applyin g th e fibrancy o f F n again , w e can exten d eac h gn t o a g n : D' - > F n [l]. The n / + (0 , (^n)) : D - > holi m Fn extend s / . • Corollary C.2. 7 If f : C — > - D 2 5 a morphism between fibrant complexes, then the mapping cone of f is fibrant. •

Jean-Louis Colliot-Thelene , R a y m o n d T . Hoobler , an d B r u n o K a h n

92

C.3 Resolutions . Theorem C.3. 1 Suppose that A verifies axiom AB5 and has a generator in the sense of [23 , 1 . 5 an d 1 .6] , and that countable products are exact in A. Then there exists a functor F : C(A)— > C(A) and a natural transformation e : Id— » F such that, for any C G C(A), (i) F(C) is fibrant; (ii) ec- is a trivial cofibration. Proof B y [23 , Theore m 1 .1 0.1 ] , th e assumption s impl y th e existenc e o f a functor / : A — * A an d a natura l transformatio n7 7 : Id — > I suc h that , fo r al l AeA (i) 1 (A) i s injectiv e (ii) TJA i s a monomorphism . We firs t construc t F o n bounde d below complexes . I f C i s suc h a complex , the constructio n o f [24 , proo f o f Lemm a 4. 6 1 ) ] embed s C int o a bounde d belo w complex o f injective s b y a trivia l cofibration ; b y Propositio n C.1 . 2 b) , th e latte r complex i s fibrant. W e note tha t w e can mak e thi s constructio n functoria l b y usin g the functo r / above . Suppose no w C arbitrary . Fo r an y n G Z , le t C — • C >n b e th e canonica l truncation o f C a t leve l n. Recal l tha t C >n i s define d b y ( C q ifq>n (C->n)*=\ B*(C) if( [ O iiq

z= n - l n ) an d ec- a s th e compositio n o f thi s chain . By Propositio n C.2.6 , F(C') i s fibrant and , b y Corollar y C.2.3 , Ec- i s a trivia l cofibration, a s desired . • References [1] Bloch , S . an d Lichtenbaum , S . [1 994] , A spectral sequence for motivic cohomology, preprint . [2] Bloch , S . an d Ogus , A . [1 974] , Gersten's conjecture and the homology of schemes, Ann . Sci . Ec. Norm . Sup. , 4 . ser . 7, 1 81 -202 . [3] Bokstedt , M . an d Neeman , A . [1 993] , Homotopy limits in triangulated categories, Compositi o Math. 86 , 209-234 . [4] Bousfield , A . an d Friedlander , E . M . [1 978] , Homotopy theory of F-spaces, spectra, and bisimplicial sets, Lectur e Note s i n Mathematic s 658 , Springer-Verlag , Berlin , Heidelberg , Ne w York, 80-1 30 .

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[5] Brown , K . S . an d Gersten , S . M . [1 973] , Algebraic K-theory as generalized sheaf cohomology, Lecture Note s i n Mathematic s 341 , Springer-Verlag , Berlin , Heidelberg , Ne w York , 266 292. [6] Bourbaki , N . [1 982] , Algebr e commutative , Masson , Paris . [7] Colliot-Thelene , J.-L . [1 995] , Birational invariants, purity and the Gersten conjecture, in : Ktheory an d algebrai c geometry : connection s wit h quadrati c form s an d divisio n algebras , (W. Jaco b an d A . Rosenberg , ed.) , Proceeding s o f Symposi a i n Pur e Mathematic s 5 8 (I) , 1-64. [8] Colliot-Thelene , J.-L . an d Ojanguren , M . [1 992] , Espaces principaux homogenes localement triviaux, Publ . Math . IHE S 75 , 97-1 22 . [9] Colliot-Thelene , J.-L . an d Sansuc , J.-J . [1 987] , Principal homogeneous spaces under flasque tori: applications, J . Algebr a 1 06 , 1 48-205 . Deninger, C . [1 988] , A proper base change theorem for non-torsion sheaves in etale cohomology, J . Pur e Appl . Algebr a 50 , 231 -235 . Dutta, S . P . [1 995] , On Chow groups and intersection multiplicities of modules, II J . Alg . 171, 370-382 . Priedlander, E . an d Voevodsky , V . [1 995] , Bivariant cycle cohomology, preprint . Gabber, O . [1 993] , An injectivity property for etale cohomology, Compositi o Math . 86 , 1 -1 4 . Gabber, O . [1 994] , Gersten's conjecture for some complexes of vanishing cycles, Manuscript a Math. 85 , 323-343 . Gillet, H . [1 981 ] , Riemann-Roch theorems in higher algebraic K-theory, Adv . i n Math . 40 , 203-289. Gillet, H . [1 986] , Gersten's conjecture for the K-theory with torsion coefficients of a discrete valuation ring, J . Alg . 1 03 , 377-380 . Gillet, H . an d Soule , C . [1 981 ] , Filtrations on higher algebraic K-theory, preprint . Grayson, D . [1 978] , Projections, cycles and algebraic K-theory, Math . Ann . 234 , 69-72 . Grayson, D . [1 985] , Universal exactness in algebraic K-theory, J . Pur e Appl . Algebr a 36 , 139-141. | Grivel, P.-P . [1 987] , Categories derivees et foncteurs derives, Expos e 1 of Algebraic "D-module s (A. Borel , ed.) , Perspective s i n Math . 2 , Academi c Press , 1 -1 08 . Gros, M . [1 985] , Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique, Mem . Soc . Math . Pranc e 2 1 . Gros, M . an d Suwa , N . [1 988] , La conjecture de Gersten pour les faisceaux de Hodge-Witt logarithmiques, Duk e Math . Journal , 57 , 61 5-628 . Grothendieck, A . [1 957] , Sur quelques points d'algebre homologique, Tohok u Math . J . 9 , 1 1 9 221. Hartshorne, R . [1 966] , Residues and duality, Lec t Note s i n Math . 20 , Springer-Verlag , Berlin , Heidelberg, Ne w York . Hartshorne, R . [1 975] , On the de Rham cohomology of algebraic varieties, Publ . Math . IHE S 45, 5-99 . Hartshorne, R . [1 977] , Algebraic geometry, Springer-Verlag , Berlin , Heidelberg , Ne w York . Hu, S . T . [1 959] , Homotopy theory, Academi c Press , Ne w York . Illusie, L . [1 979] , Complexe de de Rham-Witt et cohomologie cristalline, Ann . Sci . Ec . Norm . Sup. 1 2 , 501 -661 . Iversen, B . [1 986] , Cohomology of sheaves, Springe r Verlag , Berlin . Kahn, B . [1 993] , Resultats de "purete" pour les varietes lisses sur un corps fini (appendic e a u n articl e d e J.-L . Colliot-Thelene) , Acte s d u Colloqu e d e KT-theori e algebriqu e d e Lak e Louise, decembr e 1 99 1 (P . G . Goerss , J . F . Jardine , ed.) , Algebrai c A'-theor y an d algebrai c topology, NAT O AS I Series , Ser . C . 407 , 57-62 . Kahn, B . [1 996] , Applications of weight-two motivic cohomology, Document a Math . 1 , 395 416. Lazard, D . [1 969] , Autour de la platitude, Bull . Soc . math . Fr . 97 , 81 -1 28 . Lichtenbaum, S . [1 987] , The construction of weight-two arithmetic cohomology, Invent . Math . 88, 1 83-21 5 . Matsumura, H . [1 986] , Commutative ring theory, Cambridg e Studie s i n advanced mathematic s 8, Cambridg e Universit y Press , Cambridge . McCleary, J . [1 985] , User's Guide to Spectral Sequences, Publis h o r Peris h Inc. , Wilmington , Delaware. Milne, J . [1 980] , Etale cohomology, Princeto n Universit y Press , Princeton , NJ .

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http://dx.doi.org/10.1090/fic/016/03 Fields Institut e Communication s Volume 1 6 , 1 99 7

Milnor's Conjectur e an d Galoi s Theor y I Wenfeng Ga o Department o f Mathematic s University o f Wester n Ontari o London, Ontario , Canad a N6 A 5B 7 [email protected]

Jan Mina c Department o f Mathematic s The Universit y o f Wester n Ontari o London, Ontario , Canad a N6 A 5B 7 minacOuwo.ca

Dedicated to the memory of Bob Thomason

A b s t r a c t . Merkurjev' s well-know n theore m o n th e secon d cohomolog y o f a n absolute Galoi s group G with coefficient s i n the field o f two elements, i s a stron g condition i n th e categor y o f profinit e groups . I n thi s pape r w e investigat e th e group theoreti c meanin g o f thi s conditio n i n th e cas e whe n G i s a pro-2-group . We ar e describin g G usin g generator s an d relations , a s wel l a s homomorphi c images o f G.

Introduction In Milno r [1 970] , Milnor introduce d hi s i 1 . Therefor e fro m th e extensio n o f th e absolut e Ga lois grou p o f F q usin g th e Galoi s grou p o f F q/F an d it s Hochschild-Serr e spectra l sequence we see that th e inflatio n ma p in f : H*(Gal(Fq/F), 2 )— » H*(F, 2 ) is an isomorphism. O n the other hand i f one assumes that h nq : kn(F) — • H*(Gal(F q/F), 2 ) is a n isomorphis m fo r al l n G N an d al l fields F on e ca n imitat e th e proo f o f (Ara son, Elma n an d Jaco b [1 984] , Propositio n 5. 9 an d Remar k 5.1 0 ) t o prov e tha t kn(F)— > H*(Gal(F q/F); 2) i s a n isomorphis m fo r al l fields F . I n orde r t o adap t the proo f i n Arason , Elma n an d Jaco b [1 984] , which use s th e Scharla u transfe r fo r quadratic form s on e ca n us e (Fesenk o an d Vostoko v [1 993] , Theorem 3.8) , an d th e result o f Bass, Tat e an d Kat o whic h establishe s th e nor m ma p fo r K-group s o f field extensions. Very little seems to be known about answer s to the questions above . I n the pape r Minac an d Spir a [1 996] , it wa s show n tha t ther e exist s a rather precis e relationshi p between Wit t ring s an d certai n Galoi s groups . Th e injectivit y o f h^ i n Merkurjev' s theorem play s a crucial role in Minac and Spir a [1 996] . Independently , i n Minac an d Spira [1 996] , Villegas foun d a very nic e Galois theoretic interpretatio n o f injectivit y of th e Merkurje v theorem . Villegas ' resul t i s explained i n (Mina c an d Spir a [1 996] , Corollaries 2.1 7 an d 2.1 8) . In thi s paper , buildin g o n a n observatio n i n Mina c an d Spir a [1 996] , w e offe r a ful l Galoi s theoreti c meanin g o f Merkurjev' s theore m a s wel l a s injectivit y o f h 3 under assumptio n tha t Merkurjev' s theore m i s valid. Se e Theorems 2, 3 and 4 below. We shal l prov e onl y Theorem s 2 an d 3 , i n thi s paper , an d postpon e th e proo f o f

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Theorem 4 t o a subsequen t paper . W e concentrat e o n th e Galoi s field theoreti c meaning o f Merkurjev' s theorem . Perhaps, mor e importantly , w e develo p som e ne w tool s whic h shoul d als o b e useful fo r h n,n > 3 . Let F b e a field of characteristic no t 2 and F q i s the quadrati c closur e o f F. Le t G b e the Galoi s group of F q/F. The n G is a pro-2-group. Le t {o-i} iei b e a minima l set o f generators o f G. A s i s well known w e can assum e tha t I i s well ordered . Le t S b e a fre e pro-2-grou p wit h a minima l se t o f generator s {s{}i ej. B y sendin g Si t o Gi we obtai n a continuou s projectio n TT : S—- > G. W e se t R — Ker(ir). Ou r first observation i s Theorem 1 For each nonempty set I there exists a field L such that 1. Gal(L q/L) is free pro-2-group on generators {s{\i G / } , 2. A/^- T G L, i.e., level of L denoted by s(L), is 1 . Also there exists a field L' such that 1. Gal{L' q/Lf) is free pro-2-group on generators {si\i G / } , 2. \f-\. £ V and - 1 = e\ + e\ for some ei , e2 G (Z/), z.e. , s(Z/ ) = 2 . Remarks (1) I n th e cas e # 7 = n , Theore m 1 was prove d b y Corde s [1 973 ] i n a slightl y different form . Namely , instea d o f conditio n (1 ) i n Theore m 1 , Cordes use s u{L) — 2. However , i f u(L) = 2 , the n Gal(L q/L) i s a fre e pro-2-group . Thi s i s prove d i n Lemma 2.2 . Recal l tha t u(F) = sup{dim(q): q is a n anisotropi c for m ove r F} an d s(F) = min{n\ — 1 is the su m o f n square s i n F}. (I f — 1 is not th e su m o f square s in F w e defin e s(F) = oc . Similarl y u(F) ca n b e oo. ) (Se e La m [1 973] , Chapte r

n). (2) Observ e tha t i f Gal(L q/L) i s a fre e pro-2-grou p the n necessaril y s(L) < 2 . Indeed b y Remar k (1 ) above , th e w-invarian t o f L , u(L) i s 2 . Thi s means , i n particular, tha t eac h elemen t o f L ca n b e writte n a s a su m o f tw o square s i n L. I n particular - 1 i s expressible a s a su m o f tw o squares . Thi s mean s s(L) < 2. By Theore m 1 , w e ma y assum e tha t S i s th e Galoi s grou p o f th e maxima l 2 extension o f som e bas e field L. Le t E b e th e fixed field o f R. The n E i s the Galoi s extension o f L wit h Galoi s grou p G . Le t L — l/ 1 ) C L^ C • • • C L q b e th e towe r of fields suc h tha t L^ i s th e compositu m o f al l quadrati c extension s o f L^ n~^ which i s Galoi s ove r L . W e defin e E^ l'L^ = E an d defin e ^^ n ' L ^:=the compositu m of E^ n-^L\y/a) suc h tha t a G E^'1^ an d E^-^ L\y/a)/L i s Galoi s fo r n > 2 . Since S an d G hav e th e sam e cardinalit y o f th e minima l se t o f generators , the n L ( 2 ) C E, o r L^E = E^ L\ I n thi s paper , w e sho w Theorem 2 L^E =

£;( 2'L) iff G satisfies the Merkurjev surjectivity condition.

We als o sho w a variatio n o f Villegas ' theorem . (Se e Mina c an d Spir a [1 996] , Corollary 2.1 8. ) Theorem 3 (Variation of Villegas' theorem) Assume that Gal(E/L) is isomorphic to a Galois group of the maximal 2-extension of some field F. Then G satisfies the Merkurjev injectivity condition iff Ef]L^ is the compositum of all Z/2, Z/4, D4 extensions of L sitting inside E. The followin g theore m i s proved i n Ga o and Mina c [1 997 ] (th e secon d reference) .

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Theorem 4 Assume that G satisfies both Merkurjev surjectivity and injectivity condition. Then G satisfies the Merkurjev-Suslin-Rost injectivity condition iff By Theore m 2 and Theorem 4 , it i s natural t o ask, Question 5 Let E be a Galois extension of L such that G = Gal(E/L) is isomorphic to the Galois group of the maximal 2-extension of some base field. Is Un>l? L(n+1)E = E (n,L) j Q r a We als o giv e th e group-theoretic meanin g o f Question 5 . Fo r a pro-2-grou p i7 , we se t i/( n + 1 ) = [H^) 2[H^n\H] an d H^ = H. Thi s mean s tha t H^ n+l^ i s the closed subgrou p o f H generate d b y all squares i n H^ an d all commutators wit h slots i n H^ an d H. The n H^ i s a close d norma l subgrou p o f H. W e denot e ffM th e quotient . Observ e tha t S^ = Gal(L^/L). W e se t R^^ = R an d > L i n (Mina c an d Spira [1 996] , Section 5) , it R(n+i,s) = (R(n,s)}2[ R(n,s)^ for n was prove d tha t R( 2,s^ — R D S^ if f the surjectivity par t o f Merkurjev's theore m is true. W e shall prov e tha t i?( n ' S ) c RD 5^ n+1 ^ an d Question 5 is equivalent to , Question 6 Let E be a Galois extension of L such that G = Gal(E/L) is isomorphic to the Galois group of the maximal 2-extension of some base field. Is Un>l? R(n,S) = R p 5 (n+l) J or a By Theore m 4 , we obtain Corollary 7 Assume that Gal(E/L) is realized as the Galois group of the maximal 2-extension of some field F. Then R( 3>s>> = RD 5 ( 4 ) . In Ga o and Minac [1 997 ] (th e first reference ) w e shall discus s ho w to climb fro m G^ t o G^ n+1 '. Thi s i s related t o the embedding proble m i n Galois theory . 1 Preliminarie

s

We shal l mak e th e followin g abus e o f notation : [a] mean s bot h a n elemen t o f F/F2 an d it s correspondin g elemen t (a ) i n H l{F,2) o r mor e generall y i n Hl(Gal(T/F),2) wher e T/F i s any Galois extensio n o f F whic h contain s F^ : = compositum o f all quadratic extension s o f F, and is a subfield o f Fq - the quadratic closure o f F. Thi s abus e i s not a crim e becaus e b y Kumme r theor y w e have th e canonical isomorphis m F/F 2— » H l(T, 2) . We will work in the category of pro-2-groups and make the usual conventions tha t by "subgroup " w e mean "close d subgroup" , b y "generated " w e mean "topologicall y generated" an d by "morphism " w e mean "continuou s morphism" . W e shall alway s work wit h mo d 2 cohomology excep t whe n w e explicitly mentio n othe r coefficients . For th e standar d fact s fo r Galoi s cohomolog y w e refer t o Serr e [1 995 ] an d Shat z [1972]. For a n extensio n 1 —>.A—»#—» C— * 1 of profinite group s an d a discret e Bmodule M w e have th e corresponding Lyndon-Hochschild-Serr e spectra l sequence s {E™,d™} wher e E v2q = H^{C,H q{A,M)) (Hochschil d an d Serr e [1 953] , Shat z [1972]). W e also hav e th e five-term exac t sequenc e 0 -*H\C,M A) ±^H\B,M)

^H\A,M)

C

^H

2

(C,MA) -^H

where Inf i s the inflation an d Res i s the restriction map.

2

(B,M)

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Observe tha t k nF i s isomorphi c t o th e facto r grou p o f F/F2®...®F/F2 by th e subgrou p (denote d b y M n) generate d b y [ai ] ® ... ® [an] where a\ i s a nor m in the quadrati c extensio n F( yJa~j) for som e j = 2,... , n an d [ai\ is the clas s o f ai i n F*/F* 2 for i = l,...,n . The followin g statemen t i s a n eas y bu t usefu l exercise . (Se e e.g. War e [1 978]. ) Statement Suppose that K/F is a Galois extension, G — Gal(K/F) is its Galois group and a G K. Then K(y/a)/F is Galois iff (era)/a is a square in K. In othe r word s th e latte r condition s sa y tha t [a] G (K/K2)G, th e fixed element s of K/k 2. Sinc e (K/K 2)G = H°(G,K/K 2) w e hav e a usefu l lin k betwee n Galoi s theory an d cohomology . Here is another ver y simpl e fac t whic h enable s u s to identif y [a ] and (a ) a s abov e also i n th e cas e whe n bot h H 1 (F , 2) an d F/F 2 ar e considere d wit h som e Galoi s group action . The reaso n i s nicel y explaine d i n Waterhous e [1 994] . Fo r th e reader' s conve nience, le t u s recal l thi s basi c fac t fro m Kumme r theory . W e follo w quit e closel y Waterhouse [1 994] . Consider a Galoi s extension K/F wit h Galoi s group G = Gal(K/F). The n ther e is one-to-on e correspondenc e betwee n Galoi s extension s L/K whic h Galoi s grou p Gal(L/K) i s annihilate d b y 2 an d thos e subgroup s N o f multiplicativ e subgroup s of K whic h contai n k 2. More precisel y give n a subgrou p N o f K suc h tha t K 2 C N we se t L = K[N 1 /2) and N i s recovere d fro m L a s N = {x G k \ y/x G L}. Th e grou p H = Gal (L/K) is the Pontrjagi n dua l o f Nik 2. I n fac t w e hav e th e followin g pairin g N/K2 x

Gal(L/K) - • {±1 } ,

for al l n G Af an d a G Gal(L/K) whic h i s a perfec t dualit y betwee n th e compac t group Gal(L/K) an d th e discret e grou p N/k 2. As Waterhous e pointe d out , thi s dualit y i s quit e standar d a t leas t i n th e finite case. However , relativ e version s o f Kumme r theor y ar e no t a s familia r bu t ar e equally straightforward . Suppose tha t L/K i s a s above . The n L/F i s Galoi s if f th e G-conjugate s o f elements i nT V are agai n i n N. I n tha t cas e i f i s a norma l subgrou p o f Gal(L/F), and henc e Gal(L/F) act s o n Gal(L/K) b y th e conjugatio n i n Gal(L/F). A crucia l observatio n i s that th e Kumme r pairin g i s consisten t wit h th e action s of Gal(K/F) o n N/K 2 an d Gal(L/K). Namel y le t a b e a n elemen t o f Gal(K/F) and a b e an y extensio n o f i t i n Gal(L/F). The n w e hav e fo r eac h 7 G Gal(L/K) and eac h n G iV ; (d-jd-l)(a(^)) = a-f(Vn) = d(y/n) &(y/n) \

?

h(Vn)\ = i(y/n) y/n J y/n '

In othe r word s (a[n], 7 f f _ 1 ) = ([n],7 )

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and (