Cohomological invariants: exceptional groups and spin groups

102 Pages·2009·1.881 MB·English

by  Skip Garibaldi, Detlev W. Hoffmann

Cohomological Invariants:
Exceptional Groups
and Spin Groups
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M
EMOIRS
of the
American Mathematical Society
Number 937
Cohomological Invariants:
Exceptional Groups
and Spin Groups
Skip Garibaldi
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American Mathematical Society
Providence, Rhode Island
2000MathematicsSubjectClassification.
Primary11E72;Secondary12G05,20G15,17B25.
Library of Congress Cataloging-in-Publication Data
Garibaldi,Skip,1972–
Cohomological invariants : exceptional groups and spin groups / Skip Garibaldi ; with an
appendixbyDetlevW.Hoffmann.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no.937)
“Volume200,number937(secondof6numbers).”
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4404-5(alk.paper)
1.Galoiscohomology. 2.Linearalgebraicgroups. 3.Invariants. I.Title.
QA169.G367 2009
515(cid:1).23—dc22 2009008059
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Contents
List of Tables ix
Preface xi
Part I. Invariants, especially modulo an odd prime 1
1. Definitions and notations 2
2. Invariants of µ 5
n
3. Quasi-Galois extensions and invariants of Z/pZ 7
4. An example: the mod p Bockstein map 10
5. Restricting invariants 12
6. Mod p invariants of PGL 14
p
7. Extending invariants 17
8. Mod 3 invariants of Albert algebras 19
Part II. Surjectivities and invariants of E6, E7, and E8 23
9. Surjectivities: internal Chevalley modules 24
10. New invariants from homogeneous forms 29
11. Mod 3 invariants of simply connected E 31
6
12. Surjectivities: the highest root 33
13. Mod 3 invariants of E 38
7
14. Construction of groups of type E 39
8
15. Mod 5 invariants of E 44
8
Part III. Spin groups 47
16. Introduction to Part III 48
17. Surjectivities: Spin for 7≤n≤12 48
n
18. Invariants of Spin for 7≤n≤10 53
n
19. Divided squares in the Grothendieck-Witt ring 56
20. Invariants of Spin and Spin 58
11 12
21. Surjectivities: Spin 61
14
22. Invariants of Spin 65
14
23. Partial summary of results 66
Appendices 69
A. Examples of anisotropic groups of type E 70
7
B. A generalization of the Common Slot Theorem
By Detlev W. Hoffmann 73
Bibliography 77
v
vi CONTENTS
Index 81
Abstract
This volume concerns invariants of G-torsors with values in mod p Galois
cohomology—inthesenseofSerre’slecturesinthebookCohomologicalinvariants
in Galois cohomology — for various simple algebraic groups G and primes p. We
determinetheinvariantsfortheexceptionalgroupsF mod3,simplyconnectedE
4 6
mod 3, E mod 3, and E mod 5. We also determine the invariants of Spin mod
7 8 n
2 for n≤12 and construct some invariants of Spin .
14
Along the way, we prove that certain maps in nonabelian cohomology are
surjective. These surjectivities give as corollaries Pfister’s results on 10- and 12-
dimensionalquadraticformsandRost’stheoremon14-dimensionalquadraticforms.
This material on quadratic forms and invariants of Spin is based on unpublished
n
work of Markus Rost.
An appendix by Detlev Hoffmann proves a generalization of the Common Slot
Theorem for 2-Pfister quadratic forms.
ReceivedbytheeditorSeptember14,2006,andinrevisedformApril20,2007.
2000MathematicsSubjectClassification. Primary11E72;Secondary12G05,20G15,17B25.
Key words and phrases. cohomological invariant, exceptional group, spin group, Rost’s
theorem.
vii
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List of Tables
5 References for results on Invnorm(G,C) where G is exceptional and the
exponent of C is a power of a prime p 13
9 Extended Dynkin diagrams 26
12 Internal Chevalley modules corresponding to the highest root 34
16 Description of Spin for 1≤n≤6 48
n
23aExamples of inclusions for which the morphism H1 (∗,N)→H1(∗,G) is
fppf
surjective 66
23bInvariants and essential dimension of Spin for n≤14 67
n
ix