University

LECTURE Series

Volume 2 8

Cohomological Invariant s in Galoi s Cohomolog y

Skip Garibald i Alexander Merkurje v

Jean-Pierre Serr e

American Mathematica l Societ y Providence, Rhod e Islan d

http://dx.doi.org/10.1090/ulect/028

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University

LECTURE Series

Volume 2 8

Cohomological Invariant s in Galoi s Cohomolog y

Skip Garibald i Alexander Merkurje v

Jean-Pierre Serr e

American Mathematica l Societ y Providence, Rhod e Islan d

E D I T O R I A L C O M M I T T E E

Jerry L . Bon a (Chair ) Nige l J . Hitchi n

Peter Landwebe r

2000 Mathematics Subject Classification. Primar y 12G05 , 11E72 .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/ulect-28

Library o f Congres s Cataloging-in-Publicatio n Dat a

Garibaldi, Skip , 1972 -Cohomological invariant s i n Galoi s cohomolog y / Ski p Garibaldi , Alexande r Merkurjev , Jean -

Pierre Serre . p. cm . — (Universit y lectur e serie s ; v. 28 )

Includes bibliographica l reference s an d indexes . ISBN 0-8218-3287- 5 (alk . paper ) 1. Invariants . 2 . Galoi s cohomology . 3 . Linea r algebrai c groups . 4 . Numbe r theory .

I. Merkurjev , Alexander , 1955 - II . Serre , Jean-Pierre . III . Title . IV . Universit y lectur e se -ries (Providence , R.I. ) ; 28.

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Contents

Preface vii

Cohomological invariants , Wit t invariants , an d trac e form s 1 By Jean-Pierr e Serr e Notes b y Ski p Garibald i

Rost invariant s o f simpl y connecte d algebrai c group s 10 1 By Alexande r Merkurje v With a sectio n b y Ski p Garibald i

Bibliography 15 9

Index o f notatio n 16 5

Index o f terms 16 7

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Preface

The invariant s we are going to discuss ar e the analogue s fo r Galoi s cohomolog y of th e characteristi c classe s o f topolog y (Cher n classes , Stiefel-Whitne y classes , etc.), wher e th e topologica l space s ar e replace d b y th e schem e Spec(/c ) fo r k a field. Historically , on e o f the first example s o f cohomologica l invariant s o f the typ e considered her e was the Hasse-Wit t invarian t o f quadratic forms , define d i n Witt' s seminal pape r [Wi37] . Later , mor e invariant s o f quadrati c form s wer e defined , fo r example Stiefel-Whitne y classe s an d th e Araso n invariant . Th e first wor k i n thi s volume classifie s invariant s o f quadrati c form s an d etal e algebra s wit h value s i n Galois cohomolog y modul o 2 o r i n th e Wit t ring . Th e invariant s o f som e othe r algebraic structure s ar e als o determined . A principa l too l i s th e notio n o f versa l torsor, whic h i s an analogu e o f the universa l bundl e i n topology .

For G a simple simpl y connecte d algebrai c group , Ros t prove d th e existenc e of a canonica l an d nontrivia l invarian t o f G-torsor s wit h value s i n dimension 3 Galoi s cohomology. Th e Araso n invarian t o f quadrati c form s i s a specia l cas e o f Rost' s invariant, se e p . 107 . Th e secon d wor k i n thi s volum e give s detaile d proof s o f th e existence and basi c properties o f the Ros t invariant . Thi s is the first time that mos t of thi s materia l appear s i n print .

The two parts o f this book ar e really separat e works . Th e first bega n a s lectur e notes tha t Serr e gav e a t UCL A i n Januar y 200 1 as par t o f th e Gil l Distinguishe d Lecture Series . The y hav e bee n expande d an d no w includ e man y result s tha t wer e not mentione d durin g th e lectures , especiall y o n etal e algebra s o f lo w ran k an d their application s t o Noether' s problem o n invariants . Th e second par t wa s writte n independently b y Merkurjev (wit h one section by Garibaldi). A s a consequence, th e definitions an d notation s use d i n th e tw o part s diffe r somewhat . Fo r th e reader' s convenience, w e have included commo n indexe s an d a common bibliograph y a t th e end o f the book .

The secon d autho r i s gratefu l t o th e NS F fo r partia l suppor t (gran t DM S 0098111).

This boo k i s dedicated t o Marku s Rost . Th e influenc e o f hi s idea s ca n b e see n throughout.

Skip Garibald i Alexande r Merkurje v Jean-Pierr e Serr e Atlanta, Georgi a Lo s Angeles , Californi a Pari s USA US A Franc e

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Index o f notatio n

• (specia l cu p product) , 53 , 6 0 • (standard cu p product) , 1 6 U (standar d cu p product) , 15 2 (*) (propert y o f a functo r H), 10 8 (x)n ( i n t f ^ / x j ) , 1 8 • (characte r group) , 11 1 [n/2] (integra l part) , 5 8

A (functo r Fields/ ko — • Sets), 7 Affn, 2 8 Aff£, 5 6 Alb, 9 a(v), 1 9

An, 8 8

Br(F) , 15 1

C(d) (Tat e twist) , 1 8 Cent. Sirnpl n, 8 Cor (corestriction) , 20 , 3 4

Dec™, 19 , 2 4 A, 5 6 6Y, 12 6 9-K (q) (residu e o f q) , 6 4 d(q) = discriminan t o f q, 8 , 7 6

e (invarian t o f etal e algebra s o f ran k 6 an d 7), 83 , 86 , 89 , 9 0

E(2), 7 3 E6, 41, 51, 52, 150 E7, 52, 150 E8, 52,150 E(A,B), 78 e n (</>), 43, 77 Etn, 8

(F), 12 F4, 10, 41, 49-52, 67, 150 Fields / f co, 7 (Fus), 3 7

G2, 10 , 13 , 41, 51, 52, 67 , 15 0 Tk = Gal(fc s/fc), 9

# ! ( * ; , £ ) , 9 Hermn , 8 Hl{k) = H i(k,Z/2Z), 3 9 HifaC), 1 1 H(k) = H(k, Z/2Z) , 3 9 H{k,C), 1 1 hyp, 5 0

/ (augmentatio n idea l o f W(k)), 6 3 / (inerti a group) , 1 7 /«, 4 7 Inn(#), 11 9 Inv(A,C), 1 1 Inv(A,H), 7 Inv(G,C), 1 1 Inv(G, if) , 1 0 Inv n o r m , 6 8 It (tam e inerti a group) , 1 7 ^wikh 1 7

ko (groun d field), 7 fe|, 8 2 /c (algebraic closur e o f /c) , 9 ks (separabl e closur e o f k), 8 Ku 1 7 K u n r , 1 7

\P{q), 42 , 6 3 £(<?), 66

m = [n/2 ] (integra l part) , 5 8 MG, 12 1 Mm 1 7 /xy, 12 6

n Q (Ros t multiplier) , 12 2 NG, 13 3 riG (orde r o f th e Ros t invariant) , 105 , 13 0

equals Dynki n inde x gcd(n M), 13 3 Noe(G/fc0), 8 6

OQ (property) , 8 3 Oct, 9 Wn, 7 0

H (functo r Fields/ fco —> • Abelia n Groups) , 7 Pfistern, 9

166 INDEX O F NOTATIO N

VE, 10 , 8 8

Qi6 (quaternio n grou p o f orde r 16) , 89 , 9 0 9B, 5 9 Quad n , 8 Quad n 5 , 8

Rat(G/fc0), 8 6 Rep(G), 13 2 Res (restriction) , 3 3 R(G), 13 2 r(x) (residu e map) , 1 8

S L 2 ( F 7 ) , 9 0 S L 2 ( F 9 ) , 9 0 Sn, 9 0 Sn, 58 , 6 0

0G,H, 10 8 Torsorsc 9 TS (symmetri c square) , 7 4

w^l(E) (tota l Galoi s Stiefel-Whitne y class), 5 8

WGr(k) (Witt-Grothendiec k ring) , 6 3 Wi{E), 5 9 W{k) (Wit t ring) , 6 3 wn, 4 1 w{q) (tota l Stiefel-Whitne y class) , 4 1

Index o f term s

Albert algebra , 9 reduced, 4 9

Arason invarian t es, 54 , 90 , 10 8

base point , 9-1 0

classifying variety , 10 8 cohomological cycl e module , 11 0 cohomological invariant , 1 1 constant

cohomology class , 2 3 element o f th e Wit t ring , 6 5 invariant, 11 , 106

contain, 7 5 corestriction, 2 0

of a n invariant , 34 , 12 8 cubic resolvent , 7 1 cup product , 15 2

formulas, 1 6 special (•) , 53 , 6 0 standard (•) , 1 6

cycle module , 11 0 cyclotomic character , 8 4

derivation formula , 12 2 discriminant

of a polynomial , 5 6 of a quadrati c form , 8

Dynkin inde x gcdn M , 122 , 13 0 of a quasi-spli t simpl e group , 14 8 of a simpl e group , 157-15 8 of a spli t simpl e group , 13 6

Elman-Lam invariant , see en((t>) essential dimension , 13 , 32, 5 2 etale algebra , 8 exercise, 12 , 18 , 40 , 43 , 45 , 47 , 51 , 54, 56 ,

59, 61 , 64, 66 , 79 , 8 4

field o f definitio n o f a character , 13 3 first residue , 6 4 fusion, stron g contro l of , 3 7

Galois cohomology , 9 , 1 0 generic principa l homogeneou s space , 10 8

G-Galois algebra , 8 good reduction , 2 9

Hodge *-operator , 9 4 homotopy invariance , 11 2

inertia group , 1 7 invariant, 7 , 1 0

cohomological, 1 1 constant, 10 6 normalized, 10 6 of a n algebrai c group , 10 6 quadratic form , 6 8 stable, 3 6 Witt, 6 4

Killing for m (o f G 2 an d F 4 ) , 6 7

A operations, 6 3 A-ring, 6 3 liftable, 30 , 5 2 loop, 11 9

Milnor K-theory , 5 4 multiquadratic (etal e algebra) , 5 6

negligible (cohomolog y class) , 11 , 61 Noether's problem , 8 6 nondegenerate quadrati c form , 10 7 normalized

cocycle, 1 6 invariant, 11 , 57, 10 6

octonion algebra , 9 , 4 4

Pfister form , 9 , 10 7 pole (o f a cohomolog y class) , 1 9 polynome bicarre , 78 , 86 , 9 1 property OQ, 83

quaternion algebra , 3 9 quaternion grou p (Qw), 9 0

ramification locus , 5 6 rational (fiel d extension) , 8 6 reduced (Alber t algebra) , 4 9 residue

167

168 INDEX O F T E R M S

formula, 23 , 2 7 homomorphism, 110 , 15 3 of a cohomolog y class , 1 8 of a n elemen t o f th e Wit t ring , 6 4

resolvent cubic, 7 1 sextic, 7 2

restriction o f a n invariant , 3 3 Rost compatibilit y theorem , 2 9 Rost invarian t RQ, 52 , 12 9

has orde r th e Dynki n index , 13 3 Rost multiplie r n a, 12 2

second residue , 6 4 sextic resolvent , 7 2 special (ran k 3 form) , 7 7 special algebrai c group , 10 8 specialization theorem , 31 , 6 5 split (etal e algebra) , 8 stable (invariant) , 3 6 stably rationa l (fiel d extension) , 8 6 •-operator, 9 4 Stiefel-Whitney classes , 41 , 54, 5 7

total, 41 , 58 symmetric character , 13 9

tame (inertia) , 1 7 Tate twist , 1 8 Tits algebra , 13 3 torsor, 9 total Stiefel-Whitne y class , 41 , 58 trace form , 59 , 9 4 trivial (necessar y condition) , 7 6 trivial Tit s algebras , 14 8

unramified cohomological invariant , 8 7 cohomology class , 19—24 , 87 element o f a Wit t ring , 6 4 etale algebra , 5 5 r^f-module, 1 9

value of a cohomolog y class , 1 9 of a n elemen t o f th e Wit t ring , 6 5

versal (object , G-torsor) , 1 1

weakly equivalen t (r-sets) , 6 1 Weyl group , 6 0 Wit t

invariant, 6 4 ring, 6 3