Cohomological Invariants: Exceptional Groups and Spin Groups This page intentionally left blank EMOIRS M of the American Mathematical Society Number 937 Cohomological Invariants: Exceptional Groups and Spin Groups Skip Garibaldi ­ÜÌ Ê>Ê««i`ÝÊLÞÊ iÌiÛÊ7°Êvv>® ÕÞÊÓääÊÊUÊÊ6ÕiÊÓääÊÊUÊÊ ÕLiÀÊÎÇÊ­ÃiV`ÊvÊÈÊÕLiÀîÊÊUÊÊ-- ÊääÈxÓÈÈ American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 11E72; Secondary 12G05, 20G15, 17B25. Library of Congress Cataloging-in-Publication Data Garibaldi, Skip, 1972– Cohomological invariants : exceptional groups and spin groups / Skip Garibaldi ; with an appendix by Detlev W. Hoffmann. p. cm. — (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; no. 937) “Volume 200, number 937 (second of 6 numbers).” Includes bibliographical references and index. ISBN 978-0-8218-4404-5 (alk. paper) 1. Galois cohomology. 2. Linear algebraic groups. 3. Invariants. I. Title. 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This publication is indexed in Science Citation IndexR , SciSearchR , Research AlertR , CompuMath Citation IndexR , Current ContentsR /Physical, Chemical & Earth Sciences. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 Contents List of Tables ix Preface xi Part I. Invariants, especially modulo an odd prime 1 1. Definitions and notations 2 µ 2. Invariants of n 5 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 PGLp 14 7. Extending invariants 17 8. Mod 3 invariants of Albert algebras 19 Part II. Surjectivities and invariants of E6, E7,andE8 23 9. Surjectivities: internal Chevalley modules 24 10. New invariants from homogeneous forms 29 11. Mod 3 invariants of simply connected E6 31 12. Surjectivities: the highest root 33 13. Mod 3 invariants of E7 38 14. Construction of groups of type E8 39 15. Mod 5 invariants of E8 44 Part III. Spin groups 47 16. Introduction to Part III 48 ≤ ≤ 17. Surjectivities: Spinn for 7 n 12 48 ≤ ≤ 18. Invariants of Spinn for 7 n 10 53 19. Divided squares in the Grothendieck-Witt ring 56 20. Invariants of Spin11 and Spin12 58 21. Surjectivities: Spin14 61 22. Invariants of Spin14 65 23. Partial summary of results 66 Appendices 69 A. Examples of anisotropic groups of type E7 70 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 — in the sense of Serre’s lectures in the book Cohomological invariants in Galois cohomology — for various simple algebraic groups G and primes p.We determine the invariants for the exceptional groups F4 mod 3, simply connected E6 mod 3, E7 mod 3, and E8 mod 5. We also determine the invariants of Spinn mod ≤ 2forn 12 and construct some invariants of Spin14. 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- dimensional quadratic forms and Rost’s theorem on 14-dimensional quadratic forms. This material on quadratic forms and invariants of Spinn is based on unpublished work of Markus Rost. An appendix by Detlev Hoffmann proves a generalization of the Common Slot Theorem for 2-Pfister quadratic forms. Received by the editor September 14, 2006, and in revised form April 20, 2007. 2000 Mathematics Subject Classification. Primary 11E72; Secondary 12G05, 20G15, 17B25. Key words and phrases. cohomological invariant, exceptional group, spin group, Rost’s theorem. vii This page intentionally left blank List of Tables 5 References for results on Invnorm(G, C)whereG 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 Spinn for 1 n 648 1 ∗ → 1 ∗ 23a Examples of inclusions for which the morphism Hfppf ( ,N) H ( ,G)is surjective 66 ≤ 23b Invariants and essential dimension of Spinn for n 14 67 ix This page intentionally left blank Preface These notes are divided into three parts. The first part (§§1–8) is based on material originally developed for inclusion in Serre’s lecture notes in [GMS]. (In particular, §4 on the Bockstein has hardly been changed at all.) The first part of the notes culminates with the determination of the invariants of PGLp mod p (for p prime) and the invariants of Albert algebras (equivalently, groups of type F4)mod3. The second part (§§9–15) describes a general recipe for finding a subgroup N 1 ∗ → 1 ∗ of a given semisimple group G such that the natural map Hfppf ( ,N) H ( ,G) is surjective. It is a combination of two ideas: that parabolic subgroups lead to representations with open orbits, and that such representations lead to surjective maps in cohomology. I learned the second idea from Rost [Rost 99b], but both ideas seem to have been discovered and re-discovered many times. Representation theorists will note that our computations of stabilizers N for various G and V — summarized in Table 23a—are somewhat more precise than the published tables, in that we compute full stabilizers and not just identity components. The surjectivities in cohomology are used to describe the mod 3 invariants of the simply connected split E6 and split E7’s. The last two sections of this part, 14 and 15, concern groups of type E8 relative to the prime 5. The two main results in Section 14, namely Prop. 14.7 and Th. 14.14, describe a construction of groups of type E8 and how it relates to the Rost invariant; they are contained in Chernousov’s paper [Ch 95]. The proof of Th. 14.14 given here is different and uses arguments and results from Gille’s papers [Gi 00]and [Gi 02a]. §§ The third part ( 16–23) describes the mod 2 invariants of the groups Spinn for n ≤ 12 and n = 14. It may be viewed as a fleshed-out version of Rost’s unpublished notes [Rost 99b]and[Rost 99c]. A highlight of this part is Rost’s Theorem 21.3 on 14-dimensional quadratic forms in I3. There are also two appendices. The first uses cohomological invariants to give new examples of anisotropic groups of types E7, answering a question posed by Zainoulline. The second appendix—written by Hoffmann—proves a generalization of the “common slot theorem” for 2-Pfister quadratic forms. This result is used to § construct invariants of Spin12 in 20. These are notes for a series of talks I gave in a “mini-cours” at the Universit´e d’Artois in Lens, France, in June 2006. Consequently, some material has been included in the form of exercises. Although this is a convenient device to avoid going into tangential details, no substantial difficulties are hidden in this way. The exercises are typically of the “warm up” variety. On the other end of the spectrum, I have included several open problems. “Questions” lie somewhere in between. xi xii PREFACE Acknowledgements. It is a pleasure to thank Jean-Pierre Serre and Markus Rost, both for their generosity with their results and for their comments on this note; Detlev Hoffmann for providing Appendix B; and Pasquale Mammone for his hospitality during my stay in Lens.