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Essential Dimension of Algebraic Groups By Essential Dimension of Algebraic Groups by Aurel Nathan Meyer B.Sc., The University of Basel, 2004 M.Sc., The University of British Columbia, 2006 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2010 c Aurel Nathan Meyer 2010 ii Abstract We study the essential dimension of linear algebraic groups. For a group G, essential dimension is a measure for the complexity of G- torsors or, more generally, the complexity of any algebraic or geometric structure with automorphism group G. This makes essential dimension a powerful invari- ant with many interesting and surprising connections to problems in algebra and geometry. We show that for various classes of groups, including finite (algebraic) groups and algebraic tori, the essential dimension is related to minimal faithful repre- sentations. In many cases this renders the exact value of the essential dimension computable and we explore several of its consequences. An important open problem is the essential dimension of the projective linear group PGLn. This topic is closely related to the structure theory of central simple algebras, which may be viewed as twisted forms of the algebra of n × n matrices. We study central simple algebras with additional structure such as a distinguished Galois subfield. We prove new bounds on the essential dimension of these alge- bras and, as a corollary, of the group PGLn. iii Preface The following results in this thesis were obtained in joint work and are published or submitted for publication. The results are reproduced in this thesis with per- mission from the coauthors and journals. All work involved in these publications was shared in equal parts between the authors. From [MR09a], coauthor: Z. Reichstein. - A variant of Theorem 5.2. - Most of Chapter 9, in particular the main Theorem 9.1. From [MR08], coauthor: Z. Reichstein. - Theorem 5.4. - Most of Chapter 6, in particular the main results Theorem 6.1 and 6.3. - Theorem 7.1. From [MR09b], coauthor: Z. Reichstein. - Theorems 10.1, 11.2. Sections 10.2, 10.3, 10.4. From [LMMR09], coauthors: R. Lotscher,¨ M. MacDonald, Z. Reichstein. - Chapters 4, Sections 5.1 and 5.3. - Most of Chapter 8, in particular Theorems 8.1, 8.5, 8.6, 8.8 and variants of Theorems 8.3, 8.4. iv Table of Contents Abstract .......................................................................... ii Preface ........................................................................... iii Table of Contents ............................................................... iv List of Tables .................................................................... vi List of Symbols ................................................................. vii Acknowledgements ............................................................. ix 1 Introduction ................................................................. 1 1.1 Essential Dimension - Definitions .................................... 2 1.2 Overview .............................................................. 5 2 Preliminaries ................................................................ 8 2.1 Algebraic Groups ...................................................... 8 2.2 FiniteGroups .......................................................... 8 2.3 Extensions ............................................................. 9 2.4 Unipotent Groups, Passage to Reductive Groups .................... 10 2.5 Algebraic Tori, Characters, Lattices .................................. 11 2.6 (FT)-Groups ........................................................... 13 2.7 (Semi-)Simple Groups ................................................ 15 3 Actions, Representations, Lattices ........................................ 16 3.1 Faithful and Generically Free Actions ............................... 16 3.2 Geometric Interpretation of Essential Dimension ................... 19 3.3 Representations of (FT)-Groups ...................................... 21 4 The p-Closure of a Field ................................................... 26 v 5 Group Extensions .......................................................... 30 5.1 p-Isogenies ............................................................ 30 5.2 Groups of Index Prime to p ........................................... 32 5.3 DirectProducts ........................................................ 35 5.4 Quotients of Large Essential Dimension ............................. 36 6 Finite Groups ............................................................... 38 6.1 Essential Dimension of Finite Constant p-Groups .................. 38 6.2 p-Groups of Essential Dimension ≤ p ............................... 44 7 GL(Z) and SL(Z) .......................................................... 46 7.1 Forms of Algebraic Groups ........................................... 46 7.2 FormsofAlgebraicTori .............................................. 46 8 Algebraic Tori .............................................................. 51 8.1 ALowerBound ....................................................... 51 8.2 p-(FT)-Groups ......................................................... 53 8.3 The Central Split Group C(G) ........................................ 56 8.4 ProofofTheorem8.2(a)and(b) ..................................... 58 8.5 Proof of the Additivity Theorem ..................................... 62 8.6 ProofofTheorem8.2(c) .............................................. 64 8.7 Tori of Essential Dimension ≤ 1 ..................................... 65 8.8 Tori Split by Cyclic Extensions of Degree Dividing p2 ............. 68 9 Normalizers of Maximal Tori ............................................. 73 9.1 First Reductions and Proof of Theorem 9.1 Parts (a) and (b) ....... 75 9.2 ProofofTheorem9.1Part(c) ........................................ 77 9.3 ProofofTheorem9.1Part(d) ........................................ 81 10 Central Simple Algebras .................................................. 84 10.1 Crossed Products, Splitting Groups and Fields ...................... 86 10.2 G-Lattices .............................................................. 89 10.3 AnUpperBound ...................................................... 91 10.4 Proofs of Theorems 10.1 and 10.2 ................................... 93 11 The Projective Linear Group PGLn ...................................... 95 Bibliography .................................................................... 99 vi List of Tables 2.1 Dictionary of the anti-equivalence Diag. ............................. 12 C 8.1 Indecomposable Z p2 -lattices......................................... 69 C 8.2 Essential dimension of tori split by p2 ............................... 71 8.3 Permutation ranks for lattices M7-M12. ............................... 72 11.1 Essential dimension of PGLn.......................................... 96 11.2 Essential p-dimension of PGLn....................................... 97 vii List of Symbols k base field K/k field extension of k Kalg algebraic closure of the field K Ksep separable closure of the field K K(p) p-closure of the field K, for a prime number p Fields/k category of field extensions of k Algebras/k category of commutative k-algebras with identity Sets category of sets Groups category of groups ed(∗) essential dimension of ∗ ed(∗; p) essential p-dimension of ∗ Diag(M) group of multiplicative type corresponding to M X(G) character module of the group G Γ = Gal(Ksep/K) absolute Galois group Groups: Ga additive group Gm multiplicative group GLn split general linear group SLn split special linear group Sp2n split symplectic linear group On split orthogonal linear group PGLn split projective linear group µn group of nth roots of unity Cn cyclic group of order n Sn symmetric group in n letters Pn Sylow-p subgroup of Sn G0 connected component of the group G π0 G etale´ quotient of the group G Ru G unipotent radical of G viii (FT)-group extension of a finite group by a torus (FxT)-group product of a torus and a finite group Algebras: UD(n) universal division algebra of degree n Mn matrix algebra of degree n Functors: H1(∗,G) Galois cohomology TorsG G-torsors FormsG forms of G Etn degree n etale´ algebras GalG G-Galois algebras Torin n-dimensional tori CSAn central simple algebras of degree n CPG/H G/H-crossed product algebras Splitn,G n-dimensional algebras split by the group G Splitn,E/k n-dimensional algebras split by the extension E/k Pairsn,G pairs of an algebra and a splitting field with group G ix Acknowledgements First and foremost I would like to thank my advisor Zinovy Reichstein for all he has done for me; for the abundance of ideas he has introduced me to and for his constant support and advice in all academic matters. I would also like to thank my teachers and committee members Kai Behrend, Patrick Brosnan, Jim Carrell and Kalle Karu, my coauthors RolandLotscher¨ and Mark MacDonald and fellow graduate student Alex Duncan for all I have learnt from them. Finally I thank the mathematics community and department at UBC for pro- viding an excellent learning and working environment and making my stay very enjoyable. 1 1 Introduction It is a fundamental problem in algebra to understand the complexity of its objects and structures and to classify them according to certain invariants. Natural ex- amples of such invariants include the dimension of a vector space, the order of a finite group, the transcendence degree of a field extension or the exponent of a
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