CONTRIBUTIONS TO THE ESSENTIAL DIMENSION OF FINITE AND ALGEBRAIC GROUPS Inauguraldissertation zur Erlangung der Wurde¨ eines Doktors der Philosophie vorgelegt der Philosophisch-Naturwissenschaftlichen Fakult¨at der Universit¨at Basel von Roland L¨otscher aus Ruswil LU Basel, 2010 Originaldokument gespeichert auf dem Dokumentenserver der Universit¨at Basel edoc.unibas.ch Dieses Werk ist unter dem Vertrag Creative Commons Namensnennung-Keine kommerzielle Nutzung- ” Keine Bearbeitung 2.5 Schweiz“ lizenziert. Die vollst¨andige Lizenz kann unter creativecommons.org/licences/by-nc-nd/2.5/ch eingesehen werden. genehmigt von der Philosophisch-Naturwissenschaftlichen Fakult¨at auf Antrag von Prof. Dr. H. Kraft Prof. Dr. Z. Reichstein Basel, den 30. M¨arz 2010 Prof. Dr. E. Parlow, Dekan ii Namensnennung-Keine kommerzielle Nutzung-Keine Bearbeitung 2.5 Schweiz Sie dürfen: das Werk vervielfältigen, verbreiten und öffentlich zugänglich machen Zu den folgenden Bedingungen: Namensnennung. 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Die Commons Deed ist eine Zusammenfassung des Lizenzvertrags in allgemeinverständlicher Sprache: http://creativecommons.org/licenses/by-nc-nd/2.5/ch/legalcode.de Haftungsausschluss: Die Commons Deed ist kein Lizenzvertrag. Sie ist lediglich ein Referenztext, der den zugrundeliegenden Lizenzvertrag übersichtlich und in allgemeinverständlicher Sprache wiedergibt. Die Deed selbst entfaltet keine juristische Wirkung und erscheint im eigentlichen Lizenzvertrag nicht. Creative Commons ist keine Rechtsanwaltsgesellschaft und leistet keine Rechtsberatung. Die Weitergabe und Verlinkung des Commons Deeds führt zu keinem Mandatsverhältnis. Quelle: http://creativecommons.org/licenses/by-nc-nd/2.5/ch/ Datum: 3.4.2009 iv Contents Overview 1 Acknowledgments 2 List of publications in peer-reviewed journals and on arxiv.org including material of this thesis 3 Chapter I. Multihomogeneous covariants and the essential dimension of algebraic groups 5 1. Preliminaries 5 2. Introduction 7 3. The multihomogenization technique 9 4. Multihomogeneous invariants 15 5. Properties of multihomogeneous covariants 19 6. Covariant dimension versus essential dimension 23 7. The central extension theorem 24 8. Subgroups and direct products 26 9. A generalization of Florence’ twisting construction 27 10. Normal elementary p-subgroups 32 Bibliography 35 Chapter II. Compressions of finite group actions and covariant dimension, II 37 1. Introduction 37 2. Multihomogeneous Covariants 38 3. Covariant dimension and essential dimension 40 4. The image of a covariant 43 5. Some examples 46 6. Faithful Groups 47 7. Groups of covariant dimension 2 48 8. Errata to [KS07] 50 Bibliography 51 Chapter III. Faithful and p-faithful representations of minimal dimension 53 1. Faithful representations and the abelian socle 53 2. A generalization of Gasch¨utz’ theorem 56 3. Minimal dimension of faithful representations 58 4. Minimal p-faithful representations of extensions of p-groupsbytori 60 Bibliography 67 Chapter IV. Essential p-dimension of algebraic tori 69 v 1. Introduction 69 2. Proof of Theorem 1.2 73 3. The p-closure of a field 74 4. The group C(G) 76 5. Proof of Theorem 1.3(a) 78 6. p-isogenies 81 7. Proof of Theorem 1.3(b) 82 8. An additivity theorem 83 9. Modules and lattices 85 10. Proof of Theorem 1.3(c) 87 11. Tori of essential dimension ≤ 1 89 12. Tori split by cyclic extensions of degree dividing p2 91 Acknowledgments 94 Bibliography 95 Curriculum Vitae 98 vi Overview Essential dimension, introduced by Joe Buhler and Zinovy Reichstein and in its most general form by Alexander Merkurjev, is a measure of complexity of algebraic structures such as quadratic forms, hermitian forms, central simple algebras, ´etale algebras etc. which are defined over fields. Informally, the essential dimension of an algebraic structure is the number of parameters needed to define its objects up to isomorphism. The objects of algebraic structures are often classified by cohomology sets H1(K,G) (with respect to the fppf topology) of an algebraic group G. The set H1(K,G) can be seen as the set of isomorphy classes of G-torsors over K. The essential dimension of G-torsors (called essential dimension of G) gives an invariant of algebraic groups, which will be of primary interest in this thesis. The text is subdivided into four chapters as follows: Chapter I+II: Multihomogenization of covariants and its application to covariant and essential dimension The essential dimension of a linear algebraic group G can be expressed via G-equivariant rational maps ϕ: A(V ) 99K A(W ), so called covariants, between generically free G- modules V and W . In these two chapters we explore a new technique for dealing with covariants, called multihomogenization. This technique was jointly introduced with Hans- peter Kraft and Gerald Schwarz in an already published paper [KLS], which forms the second chapter. Applications of the multihomogenization technique to the essential dimension of al- gebraic groups are given by results on the essential dimension of central extensions, di- rect products, subgroups and the precise relation of essential dimension and covariant dimension (which is a variant of the former with polynomial covariants). Moreover the multihomogenization technique allows one to extend a twisting construction introduced by Matthieu Florence from the case of irreducible representations to completely re- ducible representations. This relates Florence’s work on the essential dimension of cyclic p-groups to recent stack theoretic approaches by Patrick Brosnan, Angelo Vistoli and Zinovy Reichstein and Nikita Karpenko and Alexander Mekurjev. A portion of the first chapter is accepted for publication in Transformation groups [L2]. An earlier and larger version with the same title is published on Arxiv [L1]. Chapter III: Faithful and p-faithful representations of minimal dimension The study of essential dimension of finite and algebraic groups is closely related to the study of its faithful resp. generically free representations. In general the essential dimen- sion of an algebraic group is bounded above by the least dimension of a generically free representation minus the dimension of the algebraic group. In some prominent cases this upper bound or a variant of it is strict. In this chapter we are guided by the following general questions: What do faithful representations of the least possible dimension look like? How can they be constructed? How are they related to faithful representations of minimal dimension of subgroups? In particular we investigate representations of an extension G of a p-group by a torus (over an algebraically closed field). We will show that such an algebraic group G has a finite p-subgroup F such that the least dimension of a representation of G with kernel finite and of order prime to p is equal to the least dimension of a faithful representation of 1 F . This result has been established in a combined effort with Zinovy Reichstein and Mark MacDonald. Along the way we also compute the minimal number of irreducible representations needed to construct a faithful representation and therewith generalize a classical theorem of Wolfgang Gaschutz¨ (Endliche Gruppen mit treuen absolut-irreduziblen Darstel- lungen, Math. Nachr. 12 (1954), 253-255). Chapter IV: Essential p-dimension of algebraic tori This chapter is joint work with Mark MacDonald, Aurel Meyer and Zinovy Reichstein and is submitted for publication [LMMR]. We study a variant of essential dimension which is relative to a prime number p. This variant, called essential p-dimension, disregards effects resulting from other primes than p. In a recent paper Nikita Karpenko and Alexander Merkurjev have computed the essential dimension of p-groups. We extend their result and find the essential p-dimension for a class of algebraic groups, which includes all algebraic tori and twisted finite p-groups. Acknowledgments I would like to thank my adviser Hanspeter Kraft for introducing me to the subject of covariant and essential dimension and for guiding me during my thesis. He carefully read this manuscript and several preprints for publication. I wish to thank him for his tremendous support and for sharing his insights both into the mathematical and non-mathematical world. While I was away in Vancouver for one month he took over my duties as a teaching assistant. Thanks a lot! Many thanks go to Zinovy Reichstein from whom I learnt a lot about essential dimension and who agreed to referee this thesis. He greatly helped me to organize my stay in Vancouver and to reach the frontiers of the research in essential dimension. I would also like to thank the hosting math department of the University of British Columbia and to Kay Behrend, who allowed me to use his office during my stay in Vancouver. During my thesis I received financial support from a Schweizer Nationalfonds grant. My stay in Vancouver was funded by the Freie Akademische Gesellschaft (FAG) of Basel. It was a great pleasure to collaborate with Gerald Schwarz and Hanspeter Kraft and with Aurel Meyer, Mark MacDonald and Zinovy Reichstein. I found it very motivating to do mathematics together, be it at the blackboard or via email. Special thanks go to Aurel, with whom I had lots of useful discussions ever since. I enjoyed to take part in two conferences related to essential dimension, one in Lens in 2008, the other in Banff in 2009. I want to thank the organisers for inviting me and taking care of everything.