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Essential dimension of algebraic groups Meyer, Aurel Nathan 2010

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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. Lötscher, 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 Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 v5 Group Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.1 p-Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2 Groups of Index Prime to p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.3 Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Forms of Algebraic Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 8 Algebraic Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.1 A Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.2 p-(FT)-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.3 The Central Split Group C(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8.4 Proof of Theorem 8.2 (a) and (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 8.5 Proof of the Additivity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 8.6 Proof of Theorem 8.2 (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 Proof of Theorem 9.1 Part (c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9.3 Proof of Theorem 9.1 Part (d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 10 Central Simple Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 10.1 Crossed Products, Splitting Groups and Fields . . . . . . . . . . . . . . . . . . . . . . 86 10.2 G-Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 10.3 An Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8.1 Indecomposable ZCp2-lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8.2 Essential dimension of tori split by Cp2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 pi0 G étale 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 étale 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 Roland Lötscher 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. 11 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 central simple algebra. In this thesis we study another such invariant and measure of complexity, called essential dimension. Let A be an algebraic object or structure. The essential dimension of A, de- noted by ed(A), is usually explained informally as ed(A) = minimal number of parameters needed to define A. The algebraic object or structure here could be almost anything in algebra or algebraic geometry. For example, A = an algebraic group, a torsor, a variety or scheme, a quadratic form, a Galois field extension, an (associative, central simple, étale) algebra, or any set which is associated in a functorial way to a field. In the next section we will make precise what we mean by “algebraic object or structure” and define essential dimension via certain functors, see Definitions 1.1 and 1.2. We will also study a variant of essential dimension which is called essential p-dimension, where p is a prime number. As essential dimension is a measure of how complex the structure A is, essential p-dimension captures this information locally or relatively to the prime p. Essential p-dimension is denoted by ed(A; p) and will be defined in Definitions 1.3 and 1.4. Essential dimension arises in many natural settings, in particular, in connec- tion with some classical open problems, such as Hilbert’s 13th problem (alge- braic version), the cyclicity problem for division algebras (Albert’s conjecture) or Serre’s Conjecture II. The notion of essential dimension was first introduced by J. Buhler and Z. Re- ichstein for finite groups [BR97] and for algebraic groups by Reichstein [Rei00]. The functorial definition of essential dimension which we will use is due to A. Merkurjev [BF03]. 1 Introduction 2 1.1 Essential Dimension - Definitions In what follows, k denotes a fixed base field and p a prime number. All fields will be assumed to be extensions of k. We denote by Ksep a separable closure and by Kalg an algebraic closure of the field K. Let F : Fields/k → Sets be a covariant functor from the category of field extensions K/k to the category of sets. The functor F formalizes what we mean by “algebraic structure” and the objects in F(K) what we mean by “algebraic object”. Before we define the essential dimension of F, let us give some natural examples of such functors (see also the list in [BF03, Example 1.1]). Examples 1.1. (a) Let X be a scheme over Speck. It is a functor Fields/k → Sets defined as X(K) = Mor(SpecK,X) = {K-rational points of X}. (b) Let G be a linear algebraic group. Define a functor TorsG : Fields/k → Sets K 7→ {G-torsors over K}/≃ See Section 3.2 for the definition of a torsor. (c) Let G be a group scheme over k. Define H1(K,G) := H1(Γ,G(Ksep)), where Γ = Gal(Ksep/K) the absolute Galois group of K and Ksep a separable closure. H1(∗,G) is called the Galois cohomology functor. If G is a smooth algebraic group, there is a well known equivalence (cf. Section 3.2) of TorsG and the Galois cohomology functor TorsG ≃ H1(∗,G) . (d) Define a functor CSAn : Fields/k → Sets K 7→ {central simple K-algebras of degree n}/≃ There are well known equivalences CSAn ≃ TorsPGLn ≃ H1(∗,PGLn), where PGLn is the projective linear group, cf. [KMRT98, VII 29] and Chapter 10. (e) The functor Br : Fields/k → Sets defined by Br(K) = Brauer group of K. 1 Introduction 3 (f) Let Curvesg : Fields/k → Sets be defined as Curvesg(K) = {isom. classes of smooth curves of genus g} . Let α ∈ F(K) be an object for some functor F : Fields/k → Sets. If K0 is a subfield of K (over k) and α is in the image of the functorial map F(K0)→ F(K), we say that α is defined over K0. Essential dimension measures the “smallest” field where α can be defined over : Definition 1.1. ed(α) = min{trdegk K0 |α ∈ Im (F(K0)→ F(K))} The essential dimension of the functor F is defined to be the maximum of essential dimensions of objects taken over all α ∈ F(K) and all fields K/k: Definition 1.2. ed(F) = maxed(α). Example 1.1. Suppose k contains an nth root of unity ξn and chark ∤ n. If a,b∈K for some K/k, let A ∈ CSAn(K) be the (isomorphism class of) the cyclic algebra A = 〈x,y|xn = a,yn = b,xy = yxξn〉 It is clear that A is defined over K0 = k(a,b) and so ed(A) ≤ 2 (in fact, equality holds if a,b are algebraically independent over k, cf. [Rei00, 9.2]). Example 1.2. Let X be the functor of rational points of an integral scheme 1.1 (a). The generic point x of X has maximal essential dimension and ed(X) = ed(x) = trdegk k(x) = dimX , [Mer09]. Remark. The definition of essential dimension depends on the base field k. Some- times we write edk instead of ed to emphasize that fact but whenever the context is clear we will omit it. One has the obvious inequality edk(∗)≥ edk′(∗) if k ⊂ k′ . (1.1) A variant of essential dimension is essential dimension relative to a prime p (also called essential dimension at p or essential p-dimension): Again, let α ∈ F(K) be an object. We allow first to pass to a finite field ex- tension L/K of degree [L : K] prime to p. Denote by αL the image of α under the natural map F(K)→ F(L). 1 Introduction 4 Definition 1.3. The essential p-dimension of α is ed(α; p) = min{ed(αL) |L/K finite of degree prime to p} In other words, we take the minimum trdegk L0 over all finite prime to p ex- tensions L/K for which there exists a diagram αL ∈ F(L) F(K) ∋ αK 9 <<yyyyyyyyy αL0 _ OO ∈ F(L0) (1.2) As before, the essential p-dimension of the functor F is the maximum of these numbers taken over all α ∈ F(K) and all fields K/k: Definition 1.4. ed(F; p) = maxed(α; p). Our main focus will be on (torsors of) algebraic groups. The essential dimen- sion of an algebraic group (or more generally group scheme) G over k is defined to be the essential dimension of the functor TorsG: Definition 1.5. ed(G) := ed(TorsG) , edp(G) := edp(TorsG) . For example, if k is an algebraically closed field of characteristic 0 then groups G of essential dimension 0 are precisely the so-called special groups, i.e., alge- braic groups G/k with the property that every G-torsor over Spec(K) is split, for every field K/k. These groups were classified by A. Grothendieck [Gro58], see also [Rei00, 5.2]. Remark. In Chapter 3.2 we will give a more geometric interpretation of essential dimension of algebraic groups and torsors. We refer to the following sources: The original papers introducing essential dimension for finite and algebraic groups are [BR97] and [Rei00]. Definitions 1.1 and 1.2 were first used in [BF03]. Essential p-dimension appeared first in [RY00]. A good introduction to essential dimension can also be found in [Rei10] or [Mer08]. 1 Introduction 5 1.2 Overview We outline the topic and results of this thesis. Essential dimension was defined in the previous section and some of the relevant notions (such as affine algebraic group, torus, lattice, etc.) will be explained in the preliminary Chapter 2. In Chapter 4 we define the p-closure of a field and discuss its properties with respect to essential dimension. The p-closure is an important technical tool which removes some of the difficulties in choosing a prime to p field extension in the definition of essential p-dimension. Ultimately we would like to be able to find the essential dimension of any given affine algebraic group. In this generality the project is far beyond our scope but as usual in mathematics, we divide it into smaller projects which are more manageable. We study groups with special properties such as finite groups, unipotent groups, algebraic tori or simple groups. From two such groups G1,G2 a new group G is obtained as an extension, i.e. a short exact sequence 1→ G1 → G→ G2 → 1 Any affine algebraic group is built up this way and one could set the following program: Determine the essential dimension of (a) finite groups, (b) unipotent groups, (c) algebraic tori, (d) simple groups. (e) Determine the behaviour of essential dimension in short exact sequences. In part (e), a simple general formula relating the essential dimensions of the groups in a short exact sequence seems to be out of reach. However, if one makes additional assumptions on the groups or the extension (such as being a direct product) some partial results can be proved, see Chapter 5. Regardless of the limitations imposed by the lack of a general formula in part (e), it is natural to try to to find the essential dimension of the four given classes of groups. Except in the “modular” case which will not be treated here, the essential dimension of unipotent groups is 0, see page 10. One of the deepest results in the theory of essential dimension is the following 1 Introduction 6 Theorem 1.1 (Karpenko-Merkurjev [KM08]). Let G be an abstract finite p-group and k be a field of characteristic 6= p containing a primitive pth root of unity. Then ed(G; p) = ed(G) = min dim(V ) , where the minimum is taken over all faithful k-representations G →֒GL(V ). Note that abstract finite groups and constant finite algebraic groups are equiv- alent. A general finite algebraic group does not have to be constant and is some- times called twisted, see Section 2.1 for some preliminaries on finite groups. First, in Chapter 6, we derive some direct consequences of Theorem 1.1. In particular we prove a new formula, Theorem 6.1, for the essential dimension of fi- nite constant p-groups in terms of subgroups of fixed index. With this tool at hand we determine the essential dimension of groups of small nilpotency class, classify finite p-groups of small essential p-dimension, Section 6.2, and we complete the computation of essential dimension of the group schemes GLn(Z) and SLn(Z) in Chapter 7. This is our first look at algebraic tori, whose automorphism group is exactly GLn(Z). Informally speaking, Theorem 1.1 means that a faithful linear representations of a p-group cannot be compressed (to an action on a lower dimensional variety), cf. Section 3.2. We will expand and generalize this result to a much larger class of groups. The context we need to work in is a class of groups which we call here (FT)-groups (for Finite by Tori -group). This class contains all algebraic tori, finite algebraic groups and extensions thereof. Natural examples of (FT)-groups are normalizers of maximal tori in reductive groups. We define (FT)-groups and describe some of their properties in Section 2.6. For many (conjecturally all) groups G of type (FT) the essential dimension is of the form ed(G) = mindimV −dimG , where V is a (generically free) linear G-representation. One reason for this is that the linear representations of (FT)-groups enjoy special properties and can be nicely interpreted in terms of certain lattices, see Sections 3.3 and 8.2. The two extreme cases of (FT)-groups, algebraic tori and finite twisted groups, are already very interesting. Algebraic tori have a very rich structure, given through their connection to integral representations of certain Galois groups and, in the words of Voskresenskii [Vos98], their study provides an amount of work sufficient for all future generations of mathematicians. Some of our main results, Theorems 8.3, 8.4 are the complete description of essential p-dimension of alge- braic tori, the absolute essential dimension of algebraic tori over a p-closed field 1 Introduction 7 and the essential p-dimension of an arbitrary finite algebraic group, all in terms of linear representations. Previously there was not much known about the essential dimension of algebraic tori and our results on tori were used by Merkurjev 11.1 to prove a strong new lower bound on the essential dimension of the projective linear group PGLn. Even in the case of (twisted) cyclic finite groups the results are new and extend and generalize (besides Theorem 1.1) work by Rost [Ros02], Bayarmagnai [Bay07] and Florence [Flo08]. As applications of Theorems 8.3, we classify tori of small essential dimen- sion and calculate the essential dimension of tori with small splitting fields, see Theorems 8.5, 8.6 and 8.8. In Chapter 9, Theorem 9.1, we completely determine the essential dimension of the normalizer of a maximal torus in the projective linear group PGLn. This normalizer is an example of an (FT)-group where again the essential dimension is determined via linear representations. Normalizers N of maximal tori in an algebraic group G allow reduction of structure from G to N, cf. [Ser97, 4.3.6], which means in many cases one can study the simpler structure of the (FT)-group N to obtain information about the group G. In particular, the essential dimension of G is always bounded from above by the essential dimension of N. Chapters 10 and 11 are devoted to the study of central simple algebras and the simple algebraic group PGLn. The essential dimension of PGLn is one of the main open problems of the theory. In Theorem 10.1 we prove a new upper bound on the essential dimension of a central simple algebra in terms of certain Galois subfields. We study several functors of central simple algebras which carry some additional structure (such as a distinguished Galois subfield or splitting group) and which allow us to interpret them in terms of (FT)-groups. Finally in Theorem 11.2 we apply these results to the so called universal division algebra UD(n) and obtain a new upper bound on the essential p-dimension of PGLn. 82 Preliminaries 2.1 Algebraic Groups By an affine (or linear) group scheme over k we mean a representable functor G : Algebras/k→ Groups. Denote the k-algebra that represents G by k[G]. It carries the structure of a Hopf-algebra and many group theoretic statements on G can be translated into algebraic statements on k[G]. G is called algebraic if k[G] is finitely generated and it is called smooth if k[G]⊗kalg is reduced. There doesn’t seem to be a consensus in the literature whether the definition of “algebraic group” includes smoothness. We will not assume smoothness in general. Good first references on affine algebraic groups are [Wat79] [Mil06]; further we refer to [DG70a], [SGA3I, SGA3II, SGA3III], [Jan03] and [KMRT98]. When we speak about the essential dimension of an affine algebraic group G, we mean the essential dimension of the functor TorsG (see Definition 1.5 and Section 3.2) and not the essential dimension of the underlying scheme (which would be just the usual dimension of the scheme, cf. [Mer08, 1.2]). In the sequel all algebraic groups will be affine and we will often omit the word affine. Examples of algebraic groups that are not affine are abelian varieties; as for their essential dimension we refer to [Bro07] and [BS08]. In the rest of this chapter we give a brief overview of some classes of groups and recall some definitions. 2.2 Finite Groups By a finite group we understand an algebraic group G over k whose algebra k[G] is finite dimensional as a k-vector space. Smooth finite groups are called étale. Equivalently, their algebra k[G] is an étale k-algebra, i.e. a product of finite separable field extensions of k. The order of a finite algebraic group G is defined as the dimension |G| := dimk k[G]. If G is smooth this number coincides with the order of the abstract 2 Preliminaries 9 group G(kalg). For every (abstract) finite group Γ there is an algebraic group G over k called constant such that G(K) = Γ for all K/k, see for example [KMRT98, VI 20]. The standard example of a non-constant group is µn, the n-th roots of unity; for example if n is odd and k = R, µn(R) consists only of the identity element {1} but µn(C) is a group of order n. A finite group which is not necessarily constant we will sometimes call twisted. This notion comes from the fact that all étale algebraic groups are obtained from abstract groups by twisting them by an action of the absolute Galois group Gal(ksep/k). Finite algebraic groups are studied in the more general context of finite flat group schemes. We refer to [Tat97] and [Sha86]. 2.3 Extensions A sequence of group homomorphisms 1→ F → G φ−→ H → 1 (2.1) is short exact if φ is a quotient map with kernel F . Here we say a homomor- phism φ : G → H is a quotient map if its associated k-algebra map k[H]→ k[G] is injective, see [Wat79, 15]. Note that in general the morphism of K-points G(K) φ−→ H(K) need not be surjective for an arbitrary field K/k. If a sequence as above exists, G is called an extension of H by F . An isogeny of algebraic groups is a quotient map G→ H such that the kernel F is finite. In that situation G,H are called isogenous. If the kernel is of order prime to p we say that the isogeny is a p-isogeny. For any affine algebraic group G there is a unique connected normal (in fact characteristic) subgroup G0 with a finite étale quotient denoted by pi0 G. Thus in principal we can pass from an arbitrary group to a connected group via an extension 1→ G0 → G→ pi0 G→ 1 , (2.2) see for example [Wat79, 6.7] or [DG70a, II 5.1]. 2 Preliminaries 10 2.4 Unipotent Groups, Passage to Reductive Groups An algebraic group G is called unipotent if every linear G-representation (6= 0) has non-trivial fixed points. Alternatively, G embeds into a group of upper triangular matrices with 1 on the diagonal, cf. [DG70a, IV 2.2]. In many cases the essential dimension of a unipotent group G is zero: Proposition 2.1. Let G be a unipotent group over k. (a) Suppose chark = 0. Then ed(G; p) = ed(G) = 0 for all p. (b) Suppose chark = q > 0. Then ed(G; p) = 0 for all p 6= q. Proof. G has a central composition series with quotients isomorphic to Ga or a finite q-group (if chark = q), see [DG70a, IV 2.2.5]. Now note that H1(K,Ga) = {0}, [Ser62b, II 1.2] for any K. For part (b) we can assume K is so called p-closed (see Definition 4.1 and Lemma 4.1) and thus perfect and the Galois cohomology H1(K,F) = TorsF(K) = {0} for any finite q-group F , cf. Lemma 5.2. Part (b) now follows by induction on the length of the composition series. There are however unipotent groups in positive characteristic with non-trivial essential dimension: For example the constant group Z/p is unipotent over a field of characteristic p and its essential dimension is 1, see [BF03, Ex. 2.3]. Some new results on unipotent groups in positive characteristic can be found in a recent preprint by Tossici and Vistoli [TV10]. A smooth algebraic group G is called reductive, if the unipotent radical Ru G0kalg is trivial. If G = G0 is smooth and connected over a perfect field k, the unipotent radical Ru G is defined over k and one can pass to the reductive quotient via an extension 1→ Ru G→ G→ G/Ru G→ 1 , (2.3) Moreover, in characteristic 0, [San81, Lemma 1.13], H1(K,G) = H1(K,G/Ru G) and thus ed(G) = ed(G/Ru G) . 2 Preliminaries 11 2.5 Algebraic Tori, Characters, Lattices In this section we collect preliminary facts about groups of multiplicative type, tori, lattices and their interplay. The references are [DG70a, IV 1.], [Vos98, Sec- tion 3.4], [Wat79, 7] or [KMRT98] and [Lor05] (for lattices). An algebraic group G over a field k is said to be diagonalizable if every lin- ear G-representation can be diagonalized and G is called of multiplicative type if Gksep is diagonalizable over the separable closure ksep of k. Here, as usual, Gk′ := G×Speck Spec(k′) for any field extension k′/k. Let X(G) := Hom(Gksep ,Gm) be the character group of G and Γ = Gal(ksep/k) be the absolute Galois group of the field k. It acts continuously on X(G) which is to say the kernel is of finite index and thus Γ acts via a finite quotient Γ = Gal(K/k), for K/k some finite extension. We denote the action of Γ (or Γ) by superscripts γ χ(t) := γ ·χ(γ−1 · t) where γ ∈ Γ, χ ∈ X(T ) and t ∈ T (ksep). Conversely a finitely generated abelian group A with continuous Γ-action (= a ZΓ-module) yields a group of multiplicative type represented by the Hopf-algebra ksep[A]Γ. We denote this group by Diag(A). In fact these two operations determine an anti-equivalence of the categories of groups of multiplicative type over k and finitely generated abelian groups A with continuous Γ-action. This equivalence is exact, i.e. the sequence of ZΓ-modules 0→ A1 → A2 → A3 → 0 is exact if and only if the following sequence of algebraic groups is exact: 1→ Diag(A3)→ Diag(A2)→ Diag(A1)→ 1 . Suppose an algebraic group F over k acts on the group G of multiplicative type on the right. Then we obtain an action of the group F(ksep) on X(G) by setting f ·χ(t) := χ(t f−1) where f ∈ F(ksep), χ ∈ X(T ) and t ∈ T (ksep). This action is compatible with the Γ-action in the sense that γ f · γ χ = γ( f ·χ) . Moreover, given an algebraic group F and a Γ-compatible action of F(ksep) on X(G) one recovers the action of F on G. 2 Preliminaries 12 A smooth connected group of multiplicative type is called algebraic torus (we will usually omit the word algebraic). A diagonalizable torus is called split. A split torus is simply a product of finitely many copies of the multiplicative group Gm. The character group of a torus is a ZΓ-lattice by which we mean the following: Let R be a commutative ring and B an R-algebra (we will mostly use R =Z or the localization R =Z(p)). A B-module is called a B-lattice if it is finitely generated and projective as an R-module. For B =ZΓ this is as usual a free abelian group of finite rank with an action of Γ. Example. Let Γ1, . . . ,Γm ⊂ Γ be subgroups of finite index in the absolute Galois group of k. Let M = Z[Γ/Γ1]⊕ ·· · ⊕Z[Γ/Γm]. Γ acts on M by permuting the cosets. A ZΓ-lattice of this form is called permutation lattice (it has a basis per- muted by Γ). The torus Diag(M) is called quasi split. An alternate description is as follows: Let M be as above and E the étale algebra E = L1× ·· ·×Lm where Li = ksepΓi . Then Diag(M) is just the Weil restriction RE/k(Gm). We summarize the above in Table 2.1. Table 2.1: Dictionary of the anti-equivalence Diag.{ algebraic k-groups of multiplicative type } X(∗) ←−−−→ Diag(∗) { finitely generated abelian Γ- groups } diagonalizable ↔ trivial Γ-action connected ↔ torsion free, except allow p-torsion if chark = p smooth ↔ no p-torsion if chark = p torus (=smooth, connected) ↔ lattice (= torsion free) split torus ≃Gmn ↔ free abelian group ≃ Zn quasisplit torus ≃ RE/k(Gm) ↔ permutation lattice anisotropic torus ↔ no Γ fixed-points other than 0 F-action ↔ Γ-compatible F(ksep)-action 2 Preliminaries 13 We will sometimes pass from ZΓ-lattices to Z(p)Γ-lattices which corresponds to identifying p-isogenous tori. For a Z-module M write M(p) := Z(p)⊗M. Lemma 2.1. Let M,L be ZΓ-lattices. Then the following statements are equiva- lent: (a) L(p) ≃M(p). (b) There exists an injective map φ : L → M of ZΓ-modules with cokernel Q finite of order prime to p. (c) There exists a p-isogeny Diag(M)→ Diag(L). Proof. The equivalence (b) ⇔ (c) is clear from the anti-equivalence of Diag. The implication (b) ⇒ (a) follows from Q(p) = 0 and that tensoring with Z(p) is exact. For the implication (a) ⇒ (b) we use the fact that L and M may be considered as subsets of L(p) (resp. M(p)). The image of L under a map α : L(p) → M(p) of Z(p)Γ-modules lands in 1mM for some m ∈ N (prime to p) and the index of α(L) in 1 m M is finite and prime to p if α is surjective. Since 1 m M ≃ M as ZΓ-modules the claim follows. 2.6 (FT)-Groups We will repeatedly encounter groups for which there is a short exact sequence 1→ T → G→ F → 1 , where T is a torus and F finite. For convenience we define Definition 2.1. An extensions of a finite group by a torus is called an (FT)-group. Alternatively, (FT)-groups are characterized in the following Lemma 2.2. (a) G is an (FT)-group ⇐⇒ G is is an extension of a finite group by a multi- plicative group. (b) A smooth group G is (FT) ⇐⇒ the connected component G0 is a torus. 2 Preliminaries 14 (c) The class of (FT)-groups is closed under taking subgroups and quotients. (d) Let chark = 0. Then a group G is (FT) ⇐⇒ every subgroup of G is reduc- tive. Proof. (a) If G is an extension of a finite group by a multiplicative group M, then [DG70a, IV1.3.9] M has a maximal subtorus T = M0red which is characteristic in M and thus normal in G. Then G/T is finite and G is (FT). (b) If G is (FT), given as an extension of the finite group F by the torus T and G is smooth, then [KMRT98, 22.4] F is smooth too. It follows that G0/T = F0 is finite, smooth and connected, hence trivial. Thus T = G0. The other direction is trivial (and does not require smoothness). (c) If H ⊂ G is a subgroup where G is an extension of a finite group by the torus T , H∩T is multiplicative [SGA3II, 2.10] and H/T ∩H is finite. Then apply (a). The assertion about quotients is proved in a similar manner. (d) If G is (FT), all subgroups of G are also (FT) by (c) and thus reductive. Conversely, suppose G is not (FT). Then its connected component G0 is not a torus by (b) (since all groups in characteristic 0 are smooth). If G0 is not reductive, we are done. Otherwise we may assume G0 is semi-simple by replacing it by its derived subgroup. It then contains an almost simple group and in fact must contain a group of type A1, i.e. a form of SL2 or PGL2, cf. [Spr98, 7.2.4]. But either of these have subgroups which are not reductive. Note that part (b) is not necessarily true if G is not smooth, for example a finite connected group is (FT) but its connected component is not a torus. In general if G is (FT), G0 is an extension of a connected finite group by a torus. We will also be interested in certain subclasses of (FT)-groups. We say G is a p-(FT)-group if G is an extension of a finite p-group by a torus and we say G is a split (FT)-group if it is a semidirect product of a finite group and a torus. Note also that abelian (FT)-groups are exactly multiplicative groups, i.e. groups that are diagonalizable over kalg. In a slightly different context, supersolvable (FT)- groups were studied in [BS53]. The case where the extension is just a direct product will also be of interest. To avoid the ugly name “central split (FT) groups”, define Definition 2.2. A direct product of a finite group and a torus is called an (FxT)- group. (FT)-groups are groups for which certain representations cannot be compressed, in a sense to be made precise in Chapters 3 and 8. 2 Preliminaries 15 2.7 (Semi-)Simple Groups A non-commutative connected smooth algebraic groups is called simple if it has no non-trivial normal subgroups and it is called almost simple if it has a finite center and the quotient by the center is a simple group. A connected smooth algebraic groups G is called semi-simple if there is an isogeny 1→ F → G1×·· ·×Gr → G→ 1 with G1, ...,Gr almost simple (and F finite). If the base field is perfect, one can pass from an arbitrary smooth connected group to a semi-simple group by taking the quotient by its radical RG, the largest connected normal solvable subgroup. Almost simple groups are classified via root systems and Dynkin diagrams into four classical families An,Bn,Cn,Dn and the exceptional groups G2, F4, E6, E7, E8. We usually write in bold face the classical split almost simple groups SLn, PGLn = GLn /Gm (type An−1), SOn (type B(n−1)/2, n odd, type Dn/2, n even) and Sp2n (type Cn). “Split” here means that they contain a split maximal torus. The problem of computing the essential dimension of semi-simple groups is largely open. Known cases include the special linear group SLn, the symplectic group Sp2n which are “special” and have essential dimension 0, the special or- thogonal group SOn which has essential dimension n−1 (for n > 2), some of the spin groups, the exceptional group of type G2 and PGL4. We refer to [Rei00], [RY00], [Lem04], [Mac08], [BRV10] for these and more results (note that some restrictions on the base field k apply). The projective linear group PGLn will be discussed in detail in chapter 11. 16 3 Actions, Representations, Lattices In this chapter we collect some preliminary results on representations and essential dimension. We will see that (FT)-groups (see Section 2.6) play a special role in this context. Unless otherwise specified we will assume that the dimensions of all varieties and ranks of all modules are finite. 3.1 Faithful and Generically Free Actions Let p be a prime and k a field of characteristic different from p. Let G be an algebraic group defined over k and α : G×X → X an action of G on an algebraic variety X over k. Here, a variety over k means a (not necessarily connected) separated scheme over Speck of finite type and an action of G on X is a (regular) morphism such that α defines an action of the abstract group G(K) on the set X(K), for all K/k. Definition 3.1. (a) (Cf. [Rei00, 2.7] or [BF03, 4.10]) X (or α) is called generically free if the scheme-theoretic stabilizer StabG(x) is trivial for all points in general position. This means that there is an open dense subset U ⊂ X such that for all K/k and x ∈U(K), StabG(x) is trivial, where StabG(x) is a group scheme over K. (b) X (or α) is called p-faithful if the kernel of the action is finite of order prime to p. (c) X (or α) is called p-generically free if the kernel N of the action is finite of order prime to p and α descends to a generically free action of G/N. The following simple lemma shows that one can always pass to the algebraic closure to check generic freeness: 3 Actions, Representations, Lattices 17 Lemma 3.1. A G-variety X is generically free if and only if the Gkalg-varitey Xkalg is. Proof. Let U ⊂ Xkalg be an open dense subvariety on which Gkalg acts freely. Let Γ = Gal(ksep/k) be the absolute Galois group of k. Its action on X(ksep) extends to an action on X(kalg). If U(kalg) is not Γ-invariant, we replace it by the union of its translates by γ ∈ Γ, on which G still acts freely. Then U is defined over k, see [Spr98, 11.2.8]. For any K-point x ∈U(K), StabG(x) is trivial if and only if it is trivial after base extension to Kalg and the lemma follows. Remark. In characteristic 0, G acts freely on U if and only if G(kalg) acts freely on U(kalg). In positive characteristic one needs in addition certain Lie stabilizer to be trivial, see [BF03, 4.2]. Groups G for which Every p-faithful irreducible G-variety is p-generically free (3.1) will play an important role later. Note that for a group that satisfies (3.1), in particular every faithful G-action is generically free. The converse, every (p-) generically free action is (p-) faithful, is true for any group. The following lemma is known to the experts, cf. [MR09a, 2.1]; [LMMR09, 7.1]. For completeness we include a proof which does not depend on the base field. Lemma 3.2. Let k be of characteristic different from p. (a) If G is finite it satisfies (3.1). (b) If G is a torus it satisfies (3.1). (c) If G satisfies (3.1) and H is a finite p-group then G×H satisfies (3.1). Proof. (a) If the kernel of the action is finite of order prime to p, we can factor out the kernel and the group is still finite, so it suffices to prove that every faithful action of a finite group is generically free. Recall [SGA3II, VIII 6.5(e)] that for an algebraic group H over k which acts on a k-variety X , the fixed points XH , defined as XH(K) = {x ∈ X(K) |hx = x} is a closed subscheme of X . Let Y = ⋃ H XH where H ranges over all nontrivial subgroups of G. If the G-action on X is faithful then Y 6= X and G acts freely on the dense open X \Y . (b) As in (a) we can factor out the finite kernel of the action first and the quotient will still be a torus. Thus we can assume the action is faithful and by the 3 Actions, Representations, Lattices 18 lemma above, G is diagonalizable. If the action is given by a linear representation V then G acts on V via characters χ . Take a basis of V such that each basis element v is in a character space i.e. t · v = χ(t)v for some character. Then we can remove the elements in V which have some 0 coefficient with respect to this basis. Since the G-action is faithful, the remaining elements form a non-empty open G-invariant subspace of V and G acts freely on it. For the general case, after applying an equivariant normalization we may as- sume first that X is normal. Reducing it further, by [Sum74, Theorem 1, Corollary 2] we may assume that X is affine the action is linear. Then proceed as before. (c) Assume again that k is algebraically closed and so H is constant and smooth (since chark 6= p). Since the kernel of the action has order prime to p it is a subgroup of G and the quotient of G by the kernel acts generically freely on X . Let U be a closed subvariety of X such that G acts freely on X \U . Replacing it by the translates of h ∈ H(k), we can assume U is H-invariant. The closed sub- varieties U 〈h〉 for h ∈ H(k) are G-invariant. After removing them, the remaining dense open subvariety is generically free for the G×H-action. Remark. As a variant of part (c) one can prove that every faithful action of a smooth group of multiplicative type is generically free. In general, algebraic groups do not satisfy (3.1), even if they are of type (FT), as the following simple example of a split (FT)-group shows. Example 3.1. Let G = Gm⋊Z/2 (≃ O2) where σ ∈ Z/2 acts by σ t = t−1. Let ρ : G→GL2 be given by t 7→ ( t 0 0 t−1 ) , σ 7→ ( 0 1 1 0 ) . Clearly ρ is faithful but for any v = (x,y) in general position, g = (x/y,σ) ∈ G is in the stabilizer (here g ∈ G(K), t ∈ T (K) and v ∈ K2 for some field K/k). We recall the following criterion from [LR00]: Lemma 3.3. Let N be a normal subgroup of G and X a G-variety. Then X is generically free as a G-variety if an only if it is generically free as an N-variety and X/N is generically free as a G/N-variety. Here X/N denotes the rational quotient. If N acts generically freely on X one can replace X by some open U ⊂ X on which the action is free and the quotient U/N in the category of schemes exists, [BF03, 4.7]. For future reference we record the following easy corollary. 3 Actions, Representations, Lattices 19 Corollary 3.1. Let G be an (FT)-group given as an extension of a finite group F by a torus T . Let W be a faithful representation of F and V be a representation of G whose restriction to T is faithful. Then V ×W is a generically free representation of G. Proof. We view W as a representation of G via the natural projection G→G/T = F . The corollary follows from the lemma in combination with Proposition 3.2. 3.2 Geometric Interpretation of Essential Dimension With the notion of a generically free action we are able to give a more geometric definition of essential dimension of an algebraic group, cf. [Rei00]. Histori- cally this approach preceded the functorial definition via the functors H1(∗,G) or TorsG. Let us recall first the definition of a G-torsor which can be thought of as a principal homogeneous space in the category of schemes over some base. Let X be a scheme over Y such that the morphism X → Y is f pp f (faithfully flat of finite presentation ). If Y = SpecK the spectrum of a field, faithfully flat is automatic and of finite presentation means that X is of finite type i.e. algebraic [Wat79, 13], [DG70a, I 2.2, I 3]. X is called a pseudo-G-torsor over Y if it is endowed with a G-action such that X → Y is G-invariant and X ×Y G → X ×Y X (x,g) 7→ (x,x ·g) is an isomorphism (here we assume G is a scheme over Y , otherwise if Y is defined over k replace G by GY ). A pseudo-G-torsor is a torsor if it is locally trivial in the f pp f -topology, i.e. there is a f pp f -covering (Yi →Y ) such that XYi ≃GYi for each i. For details we refer to [SGA3I, Exp. IV 5], [DG70a, III 4] or [BF03, 4.5]. In most of the cases here the base Y = SpecK will be the spectrum of a field K/k. Then locally trivial in the f pp f -topology simply means that there is a field K′ ⊂ Kalg such that XK′ ≃ GK′ [Wat79, 18.4] or what amounts to the same, X(K′) 6= /0. Moreover, if G is smooth, K′ can be chosen to be separable over K and thus X is a Ksep/K-form of GK . These Ksep/K-forms are classified by the Galois cohomology H1(K,G) which explains the equivalence of functors TorsG = H1(∗,G). 3 Actions, Representations, Lattices 20 Let X be an algebraic variety over k on which G acts generically freely. Then on an open U ⊂ X the categorical quotient U/G exists and gives rise to a G-torsor U →U/G, cf. [BF03, 4.7]. Alternatively we can work with a rational quotient X/G, i.e. a model for k(X)G and a rational map X //___ X/G. Rosenlicht’s theorem, cf. [Ros56] , [Ros63], links these two approaches; see also [Rei00, 2.3]. Without loss of generality assume now that X/G is a categorical quotient and X → X/G a G-torsor, otherwise replace X by U . A compression of a G-torsor X → X/G is a diagram X //____  X ′  X/G //___ X ′/G (3.2) where X ′→ X ′/G is G-torsor, the top horizontal map is G-equivariant rational and the bottom horizontal map is rational. Definition 3.2 ([Rei00, 3.1], [BF03, 6.8]). Let X be as above. The essential di- mension of X (or X → X/G) is defined as the minimal dimension of X ′/G such that there exists a compression (3.2). For any K-point SpecK → X/G we can take the fiber X ′ //  X  SpecK // X/G to obtain an element α = [X ′ → SpecK] in TorsG(K). In particular if K = k(X/G) = k(X)G is a field, we can take the generic fiber over the generic point and we have trdegk K = dimX/G = dimX −dimG. Recall that X is called primi- tive if K = k(X/G) = k(X)G is a field or equivalently G transitively permutes the components of X , see [Rei00, 2.1]. The following Lemma from [BF03, 6.11] asserts that the two definitions 1.1 and 3.2 of essential dimension of G-torsors agree: Lemma 3.4. Let X be a primitive generically free G variety, K = k(X)G and α = [X ′→ SpecK] the generic fiber. Then ed(X) = ed(α). 3 Actions, Representations, Lattices 21 Moreover, we have the obvious upper bound ed(X) = ed(α)≤ trdegk K = dimX −dimG. The essential dimension of G, ed(G) = ed(TorsG), is the maximal essential dimension of a G-torsor α or equivalently the maximal essential dimension of a primitive generically free G-variety. This maximum is attained in the case where α is a so called versal G-torsor in the sense of [GMS03, Section I.5]; see also [Dun09, 2.1]. If X is a primitive generically free G variety such that X → X/G is versal we obtain ed(G) = ed(X)≤ dimX −dimG. This is in particular the case when X =V is a generically free linear representation ρ : G → GL(V ), by [GMS03, I.5.4]. The following inequality will be used often in the sequel (for part (a) see [Rei00, 3.4] or [BF03, 4.11]): Lemma 3.5. (a) Let ρ : G→GL(V ) be a generically free G-representation. Then ed(G; p)≤ ed(G)≤ dim(V )−dim(G). (b) Let ρ : G→GL(V ) be a p-generically free G-representation. Then ed(G; p)≤ dim(V )−dim(G). Proof. For part (b), let N be the kernel of ρ . It is a finite group of order prime to p. In Chapter 5, Theorem 5.1 we will show that the essential p-dimension does not change by factoring out a finite prime to p normal subgroup. Then, together with part (a) we have ed(G; p) = ed(G/N; p)≤ dim(V )−dim(G) . 3.3 Representations of (FT)-Groups In this section we will collect some results about representations of (FT)-groups G. As we saw in Example 3.1 there are faithful representations of (FT)-groups that are not generically free. 3 Actions, Representations, Lattices 22 Proposition 3.1. Let Γ = Gal(ksep/k) and G a smooth group of multiplicative type over k with chark 6= p. There is a 1:1 correspondence between linear representa- tions ρ : G→GL(V ) and maps of ZΓ-lattices φ : L→ X(G) with L permutation. Write L = Z[Λ] where Λ is a basis permuted by Γ. Then in the above correspon- dence, (a) dimV = rk L = |Λ|. (b) ρ is generically free ⇐⇒ ρ is faithful ⇐⇒ φ is surjective. (c) Assume G is a torus. Then ρ is p-generically free ⇐⇒ ρ is p-faithful ⇐⇒ φ has finite cokernel of order prime to p ⇐⇒ L(p) → X(G)(p) is a surjective map of Z(p)Γ-modules. Proof. Let ρ : G → GL(V ) be a representation of G. Since Gksep is diagonaliz- able, there exist characters χ1, . . . ,χn ∈ X(G) (counted with multiplicities) such that G acts on Vksep via diagonal matrices with entries χ1(g), . . . ,χn(g) (for g ∈ G(ksep)) with respect to a suitable basis of Vksep . Moreover Γ permutes the set Λ := {χ1, . . . ,χn}. Define L = Z[Λ] and a map φ : L → X(G) of ZΓ-modules by sending the basis element χi ∈Λ to itself. Clearly dimV = |Λ|= rk L. Conversely, assume we have a map L → X(G). Using the anti-equivalence Diag we obtain a homomorphism G → Diag(L). We can embed the quasi-split torus Diag(L) in GLn where n = rk L [Vos98, 6.1]. The kernel of ρ corresponds to the cokernel of φ by exactness of Diag and the first two equivalences in (b) and (c) follow from Lemma 3.2. Now consider the case where G is a torus. Assume we have a surjective map α : L → X(G)(p) of Z(p)Γ-modules where L = Z(p)[Λ] is permutation, Λ a Γ- set. Then α(Λ) ⊆ 1 m X(G) for some m ∈ N prime to p (note that 1 m X(G) can be considered as a subset of X(G)(p) since X(G) is torsion free). By construction the induced map Z[Λ]→ 1 m X(G) ≃ X(G) becomes surjective after localization at p, hence its cokernel is finite of order prime to p. Suppose now that we have an (FT)-group G which is an extension of the finite group F by T . Then, from a linear representation ρ : G→GL(V ) we can still get the map of ZΓ-lattices φ : L→ X(T ) by first restricting the representation to T . The group F(ksep) acts on X(T ) compatibly with the Γ-action, see Section 2.5. Moreover F(ksep) permutes the weight spaces Vχ of ρ restricted to Tksep , i.e. the subspaces on which Tksep acts by multiplication of χ(∗). Let Λ be the set of weights counted with multiplicities so that |Λ|= dimV . Suppose we can find bases of Vχ 3 Actions, Representations, Lattices 23 for each χ which are permuted by the F(ksep)-action up to scaling (this is trivially possible if each weight space is 1-dimensional), then we obtain a permutation action of F(ksep) on Λ and thus L = Z[Λ] is an F(ksep) permutation lattice. Fol- lowing [Sal88, sec. 2] we say ρ has a good basis if in the above correspondence L = Z[Λ] is an F(ksep) permutation lattice. The group F(ksep) acts on the kernel M = kerφ of the map φ : L→X(T ). Sim- ilarly F acts on the quasisplit torus Diag(L)⊂GL(V ) and the quotient Diag(L)/T = Diag(M). Applying Lemma 3.2 and Proposition 3.1 we obtain: Proposition 3.2. Let ρ : G → GL(V ) be a representation of the (FT)-group G which is an extension of F by T and suppose ρ has a good basis. Let φ : L→X(T ) be the associated map of F(ksep)-lattices. Then ρ is generically free if and only if φ is surjective and F(ksep) acts faithfully on kerφ . Remark. For our purposes the assumption that a representation has a good basis is mostly harmless. Indeed, one can show that a faithful or generically free repre- sentation of minimal dimension has a good basis because all the weight spaces Vχ are 1-dimensional. Corollary 3.2. Let G be an (FT)-group. Suppose every F(ksep)-invariant gener- ating set Λ of X(T ) contains ≥ d elements. If G → GL(V ) is a generically free k-representation of G then dim(V )≥ d. Of course we would like to run this argument in the other direction: For every map of ZΓ-lattices L → X(T ) with L permutation and which is also a map of F(ksep)-modules, can we construct a representation ρ : G → GL(V ) “inverting” the construction above? In Proposition 3.1 we saw that by embedding Diag(T ) in some GL(V ) we obtain a T -representation, but a priori it is not clear how to define an action of all of G on V . In the special case where G is a split (FT)-group, so that G = T ⋊F , F smooth and L is a permutation F(ksep)-lattice, F permutes the basis in V and the F- and T -actions are compatible, thus give a G-action on V . Explicitly, let L = Z[Λ] with F(ksep) permuting the finite set Λ. Let VΛ be the vectorspace (defined as a variety over k) with basis elements vλ for each λ ∈ Λ. The finite group F acts on VΛ by permuting these basis elements in the natural way, f : vλ 7→ v f ·λ . (3.3) (where f ∈ F(K) which we think of as a subgroup of F(ksep)). 3 Actions, Representations, Lattices 24 The torus T acts by the character φ(λ ) on each 1-dimensional space Span(vλ ), t : vλ 7→ φ(λ )(t)vλ (3.4) In G we have t f = f ( f−1 · t) where the action in the parenthesis defines the semi- direct product T ⋊F . One checks that ρ(t)ρ( f ) = ρ( f )ρ( f−1 · t) and so these actions extend to a linear action of G = T ⋊F . Proposition 3.3. Let Γ = Gal(ksep/k) and G a smooth split (FT)-group over k. There is a 1:1 correspondence between linear representations ρ : G → GL(V ) with good bases and maps of lattices φ : L → X(G) with compatible Γ- and F(ksep)-actions and such that L is permutation with respect to both of these ac- tions. In the above correspondence, (a) dimV = rk L. (b) ρ is generically free ⇐⇒ φ is surjective and F(ksep) acts faithfully on kerφ . (c) ρ is p-generically free ⇐⇒ φ has finite cokernel of order prime to p and F(ksep) acts faithfully on kerφ . Remark. It is possible that the proposition can be generalized to all supersolvable (FT)-groups (and thus also p-(FT)-groups) G. For such groups every representa- tion ρ : G→GL(V ) is monomial, i.e. ρ(G) is contained in the normalizer N of a maximal torus in GL(V ), and N is a split (FT)-group, cf. [BS53]. However, we haven’t checked the details of this. For later use we also record two easy corollaries. Corollary 3.3. Let Γ,G be as above and suppose there is an exact sequence of Γ and F(ksep)-lattices (the two actions being compatible) 0→M → L→ X(T )→C → 0 (a) If C = 0, L is permutation and M is F(ksep)-faithful, then ed(G)≤ rk M . 3 Actions, Representations, Lattices 25 (b) If C is finite of order prime to p, L is permutation and M is F(ksep)-faithful, then ed(G; p)≤ rk M . Corollary 3.4. Let T be a split torus, F finite and N = T ⋊F. Suppose there is a finite F-invariant set Λ⊂ X(T ). Then there is a N-representation VΛ of dimension |Λ|. VΛ is faithful restricted to T ⇐⇒ Λ generates X(T ). VΛ is generically free ⇐⇒ Λ generates X(T ) and F acts faithfully on ker[Z[Λ]→ X(T )]. Similar statements appeared in [LR00], [LRRS03] and [Lem04]. 26 4 The p-Closure of a Field Throughout this chapter we assume chark 6= p. Let K be a field extension of k and Kalg an algebraic closure. We will construct a field K(p)/K in Kalg with all finite subextensions of K(p)/K of degree prime to p and all finite subextensions of Kalg/K(p) of degree a power of p. We will call K(p) a p-closure of K (to avoid the cumbersome name prime to p closure). We will show some of the advantages in passing from K to a p-closure K(p) when working with essential p-dimension. Fix a separable closure Ksep ⊂ Kalg of K and denote Γ = Gal(Ksep/K). Recall that Γ is profinite and has Sylow-p subgroups which enjoy similar properties as in the finite case, see for example [RZ00] or [Wil98]. Let Φ be a Sylow-p subgroup of Γ and KsepΦ its fixed field. Definition 4.1. The field K(p) = {a ∈ Kalg|a is purely inseparable over KsepΦ} is called a p-closure of K. A field K is called p-closed if K=K(p). Note that K(p) is unique in Kalg only up to the choice of a Sylow-p subgroup Φ in Γ. The notion of being p-closed does not depend on this choice. Proposition 4.1. (a) K(p) is a direct limit of finite extensions Ki/K of degree prime to p. (b) Every finite extension of K(p) is separable of degree a power of p; in par- ticular, K(p) is perfect. (c) The cohomological dimension of Ψ = Gal(Kalg/K(p)) is cdq(Ψ) = 0 for any prime q 6= p. Proof. (a) First note that Ksep is the direct limit of the directed set {KsepN} over all normal subgroups N ⊂ Γ of finite index. Let L = {KsepNΦ|N normal with finite index in Γ}. 4 The p-Closure of a Field 27 This is a directed set, and since Φ is Sylow, the index of NΦ in Γ is prime to p. Therefore L consists of finite separable extensions of K of degree prime to p. Moreover, KsepΦ is the direct limit of fields L in L . If chark = 0, K(p) = KsepΦ and we are done. Otherwise suppose chark = q 6= p. Let E = {E ⊂ Kalg|E/L finite and purely inseparable for some L ∈L }. E consists of finite extensions of K of degree prime to p, because a purely insep- arable extension has degree a power of q. One can check that E forms a directed set. Finally note that if a is purely inseparable over KsepΦ with minimal polynomial xq n − l (so that l ∈KsepΦ), then l is already in some L∈L since KsepΦ is the direct limit of L . Thus a ∈ E = L(a) which is in E and we conclude that K(p) is the direct limit of E . (b) K(p) is the purely inseparable closure of KsepΦ in Kalg and Kalg/K(p) is separable, see [Win74, 2.2.20]. Moreover, Gal(Kalg/K(p)) ≃ Gal(Ksep/KsepΦ) = Φ is a pro-p group and so every finite extension of K(p) is separable of degree a power of p. (c) See [Ser97, Cor. 2, I. 3]. Remark. Given an arbitrary algebraic extension L/K there may not exist an inter- mediate subfield K ⊂ K(p) ⊂ L such that the degree of every finite subextension of K(p)/K is prime to p and the degree of every finite subextension of L/K(p) is a power of p. An intermediate subfield with these properties exists if and only if L = KsKis, where Ks (respectively, Kis) denotes the separable (purely inseparable) closure of K in L. Such fields are sometimes called balanced [Lip66]. We call a covariant functor F : Fields/k → Sets colimit-preserving if for any directed system of fields {Ki}, F(lim → Ki) = lim → F(Ki). For example if G is an algebraic group, the f pp f -cohomology functor H1f pp f (∗,G) = TorsG is colimit- preserving; see [Mar07, 2.1]. Lemma 4.1. Let F be colimit-preserving and α ∈ F(K) an object. Denote the image of α in F(K(p)) by αK(p) . (a) edk(α; p) = edk(αK(p); p) = edk(αK(p)). (b) edk(F; p) = edk(p)(F; p). 4 The p-Closure of a Field 28 Proof. (a) The inequalities ed(α; p)≥ ed(αK(p); p) = ed(αK(p)) are clear from the definition and Proposition 4.1(b) since K(p) has no finite extensions of degree prime to p. It remains to prove ed(α; p) ≤ ed(αK(p)). If L/K is finite of degree prime to p, ed(α; p) = ed(αL; p), (4.1) cf. [Mer08, Proposition 1.5] and its proof. For the p-closure K(p) this is similar and uses (4.1) repeatedly: Suppose there is a subfield K0 ⊂ K(p) and αK(p) comes from an element β ∈ F(K0), so that βK(p) = αK(p) . Write K(p) = limL , where L is a direct system of finite prime to p extensions of K. Then K0 = limL0 with L0 = {L∩K0|L ∈L } and by assumption on F, F(K0) = lim L′∈L0 F(L′). Thus there is a field L′ = L∩K0 (L ∈ L ) and γ ∈ F(L′) such that γK0 = β . Since αL and γL become equal over K(p), after possibly passing to a finite extension, we may assume they are equal over L which is finite of degree prime to p over K. Combining these constructions with (4.1) we see that ed(α; p) = ed(αL; p) = ed(γL; p)≤ ed(γL)≤ ed(αK(p)) . (b) This follows immediately from (a), taking α of maximal essential p-dimension. Remark 4.1. The Lemma shows that we can always assume that the base field is p-closed when we are computing the essential p-dimension. In particular, we may assume that k contains a primitive pth root of unity, since adjoining it results in an extension of degree prime to p (however we can’t assume that k contains primitive roots of higher powers of p in general). Proposition 4.2. Let F,G : Fields/k→ Sets be colimit-preserving functors and F→G a natural transformation. If the map F(K)→G(K) is bijective (resp. surjective) for any p-closed field extension K/k then ed(F; p) = ed(G; p) (resp. ed(F; p)≥ ed(G; p)). Proof. Assume the maps are surjective. By Proposition 4.1(a), the natural trans- formation is p-surjective, in the terminology of [Mer08], so we can apply [Mer08, Prop. 1.5] to conclude ed(F; p)≥ ed(G; p). 4 The p-Closure of a Field 29 Now assume the maps are bijective. Let α be in F(K) for some K/k and β its image in G(K). We claim that ed(α; p) = ed(β ; p). First, by Lemma 4.1 we can assume that K is p-closed and it is enough to prove that ed(α) = ed(β ). Assume that β comes from β0 ∈ G(K0) for some field K0 ⊂ K. Any finite prime to p extension of K0 is isomorphic to a subfield of K (cf. [Mer08, Lemma 6.1]) and so also any p-closure of K0 (which has the same transcendence degree over k). We may therefore assume that K0 is p-closed. By assumption F(K0)→ G(K0) and F(K) → G(K) are bijective. The unique element α0 ∈ F(K0) which maps to β0 must therefore map to α under the natural restriction map. This shows that ed(α)≤ ed(β ). The other inequality always holds and the claim follows. Taking α maximal with respect to its essential dimension, we obtain ed(F; p)= ed(α; p) = ed(β ; p)≤ ed(G; p). 30 5 Group Extensions In this chapter we consider extensions of algebraic groups, i.e. short exact se- quences 1→ G1 → G2 → G3 → 1 , (5.1) and investigate in some special cases how the essential dimensions of two of the groups are related to the essential dimension of the third group. We assume again chark 6= p throughout this chapter. 5.1 p-Isogenies We will prove that p-isogenous groups have the same essential p-dimension. This result will play a key role in the proof of Theorem 8.2 where we need to remove finite prime to p kernels of linear representations. Theorem 5.1. Suppose G→ Q is a p-isogeny of algebraic groups over k. Then (a) For any p-closed field K containing k the natural map H1(K,G)→H1(K,Q) is bijective. (b) edk(G; p) = edk(Q; p). Remark. Since the p-closed field K is perfect (Proposition 4.1) we have H1(K,G)= H1f pp f (K,G), cf. Section 3.2, and the following does not depend on whether G is smooth or not. Example. (cf. [GR09, Remark 9.7]) Let Esc6 ,Esc7 be simply connected simple groups of type E6,E7 respectively. In [GR09, 9.4, 9.6] it is shown that if k is an algebraically closed field of characteristic 6= 2 and 3 respectively, then edk(Esc6 ;2) = 3 and edk(Esc7 ;3) = 3. These results can also be deduced from [Gar09, (11.2), Lemma 13.1]. For the adjoint groups Ead6 = Esc6 /µ3, Ead7 = Esc7 /µ2 Theorem 5.1 tells us that edk(Ead6 ;2) = 3 and edk(Ead7 ;3) = 3. 5 Group Extensions 31 We will need two lemmas. Lemma 5.1. Let N be a finite algebraic group over k. The following are equiva- lent: (a) p does not divide the order of N. (b) p does not divide the order of N(kalg). If N is also assumed to be abelian, denote by N[p] the p-torsion subgroup of N. The following are equivalent to the above conditions. (c) N[p](kalg) = {1}. (d) N[p](k(p)) = {1}. Proof. (a)⇐⇒ (b): Let N◦ be the connected component of N and Net = N/N◦ the étale quotient. Recall that the order of a finite algebraic group N over k is defined as |N|= dimk k[N] and |N|= |N◦||Net |, see for example [Tat97]. If chark = 0, N◦ is trivial, if chark = q 6= p is positive, |N◦| is a power of q. Hence N is of order prime to p if and only if the étale algebraic group Net is. Since N◦ is connected and finite, N◦(kalg) = {1} and so N(kalg) is of order prime to p if and only if the group Net(kalg) is. Then |Net | = dimk k[Net ] = |Net(kalg)|, cf. [Bou03, V.29 Corollary]. (b)⇐⇒ (c) ⇒ (d) are clear. (c)⇐ (d): Suppose N[p](kalg) is nontrivial. The Galois group Γ = Gal(kalg/k(p)) is a pro-p group and acts on the p-group N[p](kalg). The image of Γ in Aut(N[p](kalg)) is again a (finite) p-group and the size of every Γ-orbit in N[p](kalg) is a power of p. Since Γ fixes the identity in N[p](kalg), this is only possible if it also fixes at least p−1 more elements. It follows that N[p](k(p)) contains at least p elements, a contradiction. Remark. Part (d) could be replaced by the slightly stronger statement that N[p](k(p)∩ ksep) = {1}, but we won’t need this in the sequel. Lemma 5.2. Let Γ be a profinite group, G an (abstract) finite Γ-group and |Γ|, |G| coprime. Then H1(Γ,G) = {1}. The case where Γ is finite and G abelian is classical. In the generality we stated, this lemma is also known, [Ser97, I.5, ex. 2]. Since we haven’t found a proof in the literature, we outline it here. 5 Group Extensions 32 Proof. First, it is easy to reduce to the case when Γ is finite since Γ is the inverse limit of finite groups. Then consider the semidirect product G⋊Γ. Each cocycle c : γ 7→ gγ defines a section ϕc : Γ→G⋊Γ, γ 7→ (gγ ,γ). Cohomologous cocycles define G-conjugate sections and there is a bijection between H1(Γ,G) and sections {ϕ : Γ→ G⋊Γ}/G-conjugation. One of the groups has odd order and by the Feit-Thompson theorem is solv- able. Then, a theorem by Zassenhaus ([Zas58, IV. 7]) asserts that any two sections are conjugate. Proof of Theorem 5.1. (a) Let N be the kernel of G → Q and K = K(p) be a p- closed field over k. Since Ksep = Kalg, the sequence of Ksep-points 1→N(Ksep)→ G(Ksep) → Q(Ksep) → 1 is exact. By Lemma 5.1, the order of N(Ksep) is not divisible by p and therefore coprime to the order of Ψ = Gal(Ksep/K). Thus H1(K,N) = {1} by Lemma 5.2. Similarly, if cN is the group N twisted by a cocycle c : Ψ → G, cN(Ksep) = N(Ksep) is of order prime to p and H1(K, cN) = {1}. It follows that H1(K,G)→ H1(K,Q) is injective, cf. [Ser97, I.5.5]. Surjectivity is a consequence of [Ser97, I. Proposition 46] and the fact that the q-cohomological dimension of Ψ is 0 for any divisor q of |N(Ksep)|, by Proposi- tion 4.1). This concludes the proof of part (a). Part (b) immediately follows from (a) and Proposition 4.2. 5.2 Groups of Index Prime to p Let H be a closed subgroup of an algebraic group G defined over k. The homo- geneous space G/H of left cosets is an algebraic scheme (i.e. of finite type) over k, see [DG70a, III 3.5.4]). If G/H is finite over k, i.e. the algebra A of any affine open is a finite dimensional k-algebra, then this dimension does not depend on the choice of affine open, and one defines the index [G : H] = dimk A. The index can be defined in more generality for schemes over an arbitrary base scheme, see [Tat97]. Theorem 5.2. Let H be a closed subgroup of an algebraic group G defined over k. Assume that the index [G : H] is finite and prime to p. Then ed(G; p) = ed(H; p). In the case where G is finite constant, this is proved in [Mer08, Proposition 4.10]. 5 Group Extensions 33 Proof. Recall that if G is a linear algebraic group and H is a closed subgroup then ed(G; p)≥ ed(H; p)+dim(H)−dim(G) ; (5.2) for any prime p; see, [BRV10, Lemma 2.2] or [Mer08, 4.3]. Since dimH = dimG, this yields ed(G; p)≥ ed(H; p). To prove the opposite inequality, by Proposition 4.2 it suffices to show that for any p-closed field K = K(p) over k the map H1(K,H)→H1(K,G) induced by the inclusion H ⊂ G is surjective. Let X be a G-torsor over K and X/H be the natural quotient of X by the action of H. Recall that X/H is a K-form of G/H, constructed by descent or Galois twisting of G/H by X with respect to the natural G-action on G/H; see [Ser62a, 1.3.2] or [Mil80, p. 134]. For a field L/K and an L-point Spec(L)→ X/H we construct an H-torsor Y as the pullback Y - X Spec(L) ? - X/H ? Spec(K) ? In this situation the homogeneous fiber product Y ×H G is isomorphic to the re- striction XL as G-torsors. Thus we have the natural diagram H1(L,H) - H1(L,G) [Y ] - [X ]L [X ] 6 H1(K,G) 6 where [X ] and [Y ] denote the classes of X and Y in H1(K,G) and H1(L,H), re- spectively. It remains to show the existence of a K-point of X/H. Let A be the K-algebra of an affine open of X/H. By assumption, A has finite dimension. Let N be the nilradical of A. Since K is p-closed, it is perfect and thus A/N is étale, in other words a direct product of field extensions Li/K: A/N ∼= L1×·· ·×Lr; 5 Group Extensions 34 see, e.g., [Mil06, 2,19] and [Bou03, V, Theorem 4]. Note that dimK A/N divides dimK A (cf. [Tat97, Prop. 3.1]) and dimK A/N = r ∑ j=1 [L j : K]. As dimK A = [G : H] is prime to p, at least one of the [L j : K] has to be prime to p as well but that is only possible if L j = K because K is p-closed. Then the projection A→ A/N → L j = K gives the required K-point SpecK → SpecA→ X/H. Corollary 5.1. Suppose k is a field of characteristic 6= p. Let Sn be the symmetric group in n letters. Then ed(Sn; p) = [n/p]. Proof. Let m = [n/p] and let D ≃ (Z/pZ)m be the subgroup generated by the disjoint p-cycles σ1 = (1, . . . , p), . . . ,σm = ((m−1)p+1, . . . ,mp) . The inequality ed(Sn; p)≥ ed(D; p)≥ [n/p] is well known; see, [BR97, 6], [BR99, 7], or [BF03, 3.7]. The opposite inequality was first noticed by J.-P. Serre and independently R. Lötscher worked out a proof (both unpublished). The semi-direct product D⋊Sm, where Sm permutes σ1, . . . ,σm, embeds in Sn with index prime to p. By Theorem 5.2, ed(D⋊Sm; p) = ed(Sn; p) and it suffices to show that ed(D⋊Sm)≤ [n/p]. By Lemma 3.5 it is enough to construct a generically free m-dimensional representation of D⋊Sm defined over k. More- over, by Remark 4.1 we may assume that ζp ∈ k, where ζp denotes a primitive pth root of unity. Let σ∗1 , . . . ,σ∗m ⊂ X(D) be the “basis” of D dual to σ1, . . . ,σm. That is, σ∗i (σ j) = { ζp, if i = j and 1, otherwise. The Sm-invariant subset Λ = {σ∗1 , . . . ,σ∗m} of X(D) gives rise to the m-dimensional k-representation VΛ of D⋊Sm, similarly as in Corollary 3.4 and it is easily checked that this representation is generically free. Another application of Theorem 5.2 is the following: Consider the group SLnm /µm where we identify µm with < ζmI > and assume that the mth root of unity ζm is in the ground field. SLnm /µm sits between SLnm and PGLnm. 5 Group Extensions 35 Corollary 5.2. Let pr and ps be the highest power of p in n and m respectively. Then ed(SLnm /µm; p) = ed(SLpr+s /µps; p) In particular, ed(SLnm /µm; p) = 0 if p ∤ m , ed(SLnm /µm; p) = ed(PGLm; p) if p ∤ n . Proof. Apply Theorem 5.2 to 1→ µm/ps → SLnm /µps → SLnm /µm → 1 and show that H1(K,SLnm /µps) = H1(K,SLpr+s /µps) for any p-closed field K. 5.3 Direct Products Recall that for algebraic groups G,H the following inequalities hold: max{ed(G),ed(H)} ≤ ed(G×H)≤ ed(G)+ ed(H) , (5.3) see [Rei00, 3.8] and [BF03, 1.16. b)]. The same also holds for essential p- dimension. Additivity (equality on the right) was proved for constant p-groups in [KM08]. We will extend this to (FxT)-group i.e. direct products of a torus and a (twisted) finite group, see Section 2.6. We say a group is p-(FxT) if it is a direct product of a torus and a (twisted) finite p-group. Theorem 5.3. (a) Let G and H be (FxT)-groups over k. Then ed(G×H; p) = ed(G; p)+ ed(H; p). (b) Let G,H be p-(FxT)-groups over k and suppose k = k(p) is p-closed. Then ed(G×H) = ed(G)+ ed(H). The proof of this theorem will be deferred until Section 8.5 and after Theo- rem 8.2 which determines the essential dimension of (FxT)-groups in terms of representations. 5 Group Extensions 36 5.4 Quotients of Large Essential Dimension C. U. Jensen, A. Ledet and N. Yui asked if ed(G) ≥ ed(G/N) for every finite group G and normal subgroup N ⊳G; see [JLY02, p. 204]. The following theorem shows that this inequality is false in general. Theorem 5.4. Assume k contains a primitive pth root of unity. For every real number λ > 0 there exists a p-group G and a central subgroup H of G such that ed(G/H) > λ ed(G). Proof. Let Γ be the non-abelian (constant) group of order p3 given by generators x,y,z and relations xp = yp = zp = [x,z] = [y,z] = 1, [x,y] = z. Choose a mul- tiplicative character χ : H → k∗ of the subgroup A = 〈x,z〉 ≃ (Z/pZ)2 which is non-trivial on the center 〈z〉 of Γ and consider the p-dimensional induced repre- sentation IndΓA(χ). Since the center 〈z〉 of Γ does not lie in the kernel of IndΓA(χ), we conclude that IndΓA(χ) is faithful. Thus we have constructed a faithful p- dimensional representation of Γ defined over k. Consequently, by Lemma 3.5, ed(Γ)≤ p . (5.4) Taking the direct sum of n copies of this representation, we obtain a faithful representation of Γn of dimension np. Thus with (5.3) we have for any n≥ 1 edΓn ≤ np . (5.5) (We remark that both (5.4) and (5.5) are in fact equalities. Indeed, if ζp2 is a primitive root of unity of degree p2 then ed(Γ)≥ edk(ζp2)(Γ) = √ p2 +1−1 = p , where the middle equality will follow from Theorem 6.2(b). Hence, we have ed(Γ) = p. Moreover, by the additivity theorem 5.3, edΓn = n ·ed(Γ) = np. How- ever, we will only need the upper bound (5.5) in the sequel.) The center of Γ is 〈z〉; denote it by C. The center of Γn is then isomorphic to Cn. Let Hn be the subgroup of Cn consisting of n-tuples (c1, . . . ,cn) such that c1 . . .cn = 1. The center C(Γn/Hn) of Γn/Hn is clearly cyclic of order p (it is generated by the class of the element (z,1, . . . ,1) modulo Hn), and the commutator [Γn/Hn,Γn/Hn] is central. Hence, ed(Γn/Hn)≥ edk(ζ 2)(Γn/Hn) = √ p2n +1−1 = pn , (5.6) 5 Group Extensions 37 where the middle equality follows from a theorem by Brosnan, Reichstein and Vistoli, cf. Theorem 6.2(b). Setting G = Γn and H = Hn, and comparing (5.5) with (5.6), we see that the desired inequality ed(G/H) > λ ed(G) holds for suit- ably large n. 38 6 Finite Groups 6.1 Essential Dimension of Finite Constant p-Groups An important result in the theory of essential dimension was Karpenko and Merkur- jev’s Theorem 1.1 on constant finite p-groups. In this chapter we will explore some of the consequences of this theorem. First we will give a new explicit formula for ed(G; p) in terms of certain sub- groups. For the rest of this chapter we assume that the base field k satisfies char(k) 6= p and k contains ζ , (6.1) where ζ is a primitive pth root of unity if p ≥ 3 and a primitive 4th root of unity if p = 2. Note that for p = 2, Theorem 1.1 is true over fields containing only a primitive second root of unity. However, the properties of p-groups we need in the sequel (all representations are monomial, cf. Lemma 6.1) require the stronger assumption. We don’t know if Theorem 6.1 below is true if k contains no primitive 4th root of unity. For a finite group H, we will denote the intersection of the kernels of all mul- tiplicative characters χ : H → k∗ by H ′. In particular, if k contains an eth root of unity, where e is the exponent of H, then H ′ = (H,H), the commutator subgroup. Given a p-group G, we set Z(G) to be the center of G and C(G) := {g ∈ Z(G) |gp = 1} (6.2) to be the p-torsion subgroup of Z(G). We will view C(G) and its subgroups as Fp-vector spaces, and write “dimFp” for their dimensions. We further set Ki := ⋂ [G:H]=pi H ′ and Ci := Ki∩C(G) . (6.3) for every i≥ 0, K−1 := G and C−1 := K−1∩C(G) = C(G). 6 Finite Groups 39 Theorem 6.1. Let G be a p-group, k be a base field satisfying (6.1) and ρ : G →֒ GL(V ) be a faithful linear k-representation of G. Then (a) ρ has minimal dimension among the faithful linear representations of G defined over k if and only if for every i≥ 0 the irreducible decomposition of ρ has exactly dimFp Ci−1−dimFp Ci irreducible components of dimension pi, each with multiplicity 1. (b) ed(G; p) = ed(G) = ∑∞i=0(dimFp Ci−1−dimFp Ci)pi. Note that Ki = Ci = {1} for large i (say, if pi ≥ |G|), so only finitely many terms in the above infinite sum are non-zero. Remark. In [Mer09, 4.3] Merkurjev proves a similar formula for tori derived from our result Theorem 8.3. Theorem 8.3 in turn is based on Karpenko and Merkur- jev’s Theorem 1.1. Theorem 6.1 will be needed in the classification of p-groups of essential di- mension ≤ p (Theorem 6.3), Theorem 5.4, as well as the calculation of essential dimension of SL(Z) (Theorem 7.1). First we write down an easy but useful corol- lary. Corollary 6.1. Let G be a p-group, and k as in (6.1). (a) If C(G)⊂ Ki then ed(G) is divisible by pi+1. (b) If C(G)⊂ G′ then ed(G) is divisible by p. (c) If C(G)⊂G(i), where G(i) denotes the ith derived subgroup of G, then ed(G) is divisible by pi. Proof. (a) C(G) ⊂ Ki implies C−1 = C0 = · · · = Ci. Hence, in the formula of Theorem 6.1(b) the p0, p1, . . . , pi terms appear with coefficient 0. All other terms are divisible by pi+1, and part (a) follows. (b) is an immediate consequence of (a), since K0 = G′. (c) By [Hup67, Theorem V.18.6] G(i) is contained in the kernel of every pi−1- dimensional representation of G. Lemma 6.1 below now tells us that G(i) ⊂ Ki−1 and part (c) follows from part (a). 6 Finite Groups 40 To make notation easier we write C = C(G) for the p-torsion subgroup C(G) of the center of G. An important role in the proof Theorem 6.1 will be played by C and by the descending sequences K−1 = G⊃ K0 ⊃ K1 ⊃ K2 ⊃ . . . and C−1 = C ⊃C0 ⊃C1 ⊃C2 ⊃ . . . of characteristic subgroups of G defined in (6.3). We will repeatedly use the well-known fact that A normal subgroup N of G is trivial if and only if N∩C is trivial. (6.4) We begin with three elementary lemmas. Lemma 6.1. Ki = ⋂ dim(ρ)≤pi ker(ρ), where the intersection is taken over all irre- ducible representations ρ of G of dimension ≤ pi. Proof. Let j≤ i. Recall that every irreducible representation ρ of G of dimension p j is induced from a 1-dimensional representation χ of a subgroup H ⊂ G of index p j; see [LGP86, (II.4)] for p≥ 3 (cf. also [Vol63]) and [LGP86, (IV.2)] for p = 2. (Note that our assumption (6.1) on the base field k is crucial here. In the case where k = C a more direct proof can be found in [Ser77, Section 8.5]). Thus ker(ρ) = ker(IndGH χ) = ⋂ g∈G gker(χ)g−1, and since each gker(χ)g−1 contains (gHg−1)′, we see that ker(ρ)⊃K j ⊃Ki. The opposite inclusion is proved in a similar manner. Lemma 6.2. Let G be a finite group, C be a central subgroup of exponent p and ρ : G→GL(V ) an irreducible representation of G. Then (a) ρ(C) consists of scalar matrices. In other words, the restriction of ρ to C decomposes as χ⊕ . . .⊕χ (dim(V ) times), for some multiplicative charac- ter χ : C →Gm. We will refer to χ as the character associated to ρ . (b) Ci =⋂dim(ρ)≤pi ker(χρ), where the intersection is taken over all irreducible G-representations ρ of dimension ≤ pi and χρ : C →Gm denotes the char- acter associated to ρ . In particular, if dim(ρ) ≤ pi then χρ vanishes on Ci. 6 Finite Groups 41 Proof. (a) follows from Schur’s lemma. (b) By Lemma 6.1 Ci = C∩ ⋂ dim(ρ)≤pi ker(ρ) = ⋂ dim(ρ)≤pi (C∩ker(ρ)) = ⋂ dim(ρ)≤pi ker(χρ) . Lemma 6.3. Let G be a p-group and ρ = ρ1 ⊕ . . .⊕ ρm be the direct sum of the irreducible representations ρi : G → GL(Vi). Let χi := χρi : C → Gm be the character associated to ρi. (a) ρ is faithful if and only if χ1, . . . ,χm span C∗ as an Fp-vector space. (b) Moreover, if ρ is of minimal dimension among the faithful representations of G then χ1, . . . ,χm form an Fp-basis of C∗. Proof. (a) By (6.4), ker(ρ) is trivial if and only if ker(ρ)∩C = ∩mi=1 ker(χi) is trivial. On the other hand, ∩mi=1 ker(χi) is trivial if and only if χ1, . . . ,χm span C∗. (b) Assume the contrary, say χm is a linear combination of χ1, . . . ,χm−1. Then part (a) tells us that ρ1⊕ . . .⊕ρm−1 is a faithful representation of G, contradicting the minimality of dim(ρ). We are now ready to proceed with the proof of Theorem 6.1. To simplify the notation, we will write C for C−1 = C(G) for the rest of this section. Part (b) of Theorem 6.1 is an immediate consequence of part (a) and Theorem 1.1. We will thus focus on proving part (a). In the sequel for each i≥ 0 we will set δi := dimFp Ci−1−dimFp Ci and ∆i := δ0 +δ1 + · · ·+δi = dimFp C−dimFp Ci , where the last equality follows from C−1 = C. Our proof will proceed in two steps. In Step 1 we will construct a faithful representation µ of G such that for every i≥ 0 exactly δi irreducible components of µ have dimension pi. In Step 2 we will show that dim(ρ) ≥ dim(µ) for any other faithful representation ρ of G, and moreover equality holds if and only if for every i≥ 0 ρ has exactly δi irreducible components of dimension pi. 6 Finite Groups 42 Proof of Theorem 6.1 (a). Step 1: We begin by constructing µ . By definition, C = C−1 ⊃C0 ⊃C1 ⊃ . . . , where the inclusions are not necessarily strict. Dualizing this flag of Fp-vector spaces, we obtain a flag (0) = (C∗)−1 ⊂ (C∗)0 ⊂ (C∗)1 ⊂ . . . of Fp-subspaces of C∗, where (C∗)i := {χ ∈C∗ |χ is trivial on Ci} ≃ (C/Ci)∗. Let Ass(C) ⊂ C∗ be the set of characters of C associated to irreducible repre- sentations of G, and let Assi(C) be the set of characters associated to irreducible representations of dimension pi. Lemma 6.2(b) tells us that Ass0(C)∪Ass1(C)∪·· ·∪Assi(C) spans (C∗)i for every i≥ 0. Hence, we can choose a basis χ1, . . . ,χ∆0 of (C∗)0 from Ass0(C), then complete it to a basis χ1, . . . ,χ∆1 of (C∗)1 by choosing the last ∆1−∆0 char- acters from Ass1(C), then complete this basis of (C∗)1 to a basis of (C∗)2 by choosing ∆2−∆1 additional characters from Ass2(C), etc. We stop when Ci = (0), i.e., ∆i = dimFp C. By the definition of Assi(C), each χ j is the associated character of some irre- ducible representation µ j of G. By our construction µ = µ1⊕·· ·⊕µdimFp(C) , has the desired properties. Indeed, since χ1, . . . ,χdimFp(C) form a basis of C ∗ , Lemma 6.3 tells us that µ is faithful. On the other hand, by our construction exactly δi−δi−1 = dimFp C∗i −dimFp C∗i−1 = dimFp Ci−1−dimFp Ci of the characters χ1, . . . ,χc come from Assi(C). Equivalently, exactly dimFp Ci−1−dimFp C of the irreducible representations µ1, . . . ,µc are of dimension pi. 6 Finite Groups 43 Step 2: Let ρ : G → GL(V ) be a faithful linear representation of G of the smallest possible dimension, ρ = ρ1⊕ . . .⊕ρc be its irreducible decomposition, and χi : C → Gm be the character associated to ρi. By Lemma 6.3(b), χ1, . . . ,χc form a basis of C∗. In particular, c = dimFp C and at most dimFp C−dimFp Ci of the characters χ1, . . . ,χc can vanish on Ci. On the other hand, by Lemma 6.2(b) every representation of dimension ≤ pi vanishes on Ci. Thus if exactly di of the irreducible representations ρ1, . . . ,ρc have dimension pi then d0 +d1 +d2 + . . .+di ≤ dimFp C−dimFp Ci for every i ≥ 0. For i ≥ 0, set Di := d0 + · · ·+ di = number of representations of dimension ≤ pi among ρ1, . . . ,ρc. We can now write the above inequality as Di ≤ ∆i for every i≥ 0. (6.5) Our goal is to show that dim(ρ) ≥ dim(µ) and that equality holds if and only if exactly δi of the irreducible representations ρ1, . . . ,ρdimFp(C) have dimension pi. The last condition translates into di = δi for every i ≥ 0, which is, in turn equivalent to Di = ∆i for every i≥ 0. Indeed, setting D−1 := 0 and ∆−1 := 0, we have, dim(ρ)−dim(µ) = ∞ ∑ i=0 (di−δi)pi = ∞ ∑ i=0 (Di−∆i)pi− ∞ ∑ i=0 (Di−1−∆i−1)pi = ∞ ∑ i=0 (Di−∆i)(pi− pi+1)≥ 0 , where the last inequality follows from (6.5). Moreover, equality holds if and only if Di = ∆i for every i≥ 0, as claimed. This completes the proof of Step 2 and thus of Theorem 6.1. Another application of Theorem 6.1 is the following result on p-groups of nilpotency class 2 due to Brosnan, Reichstein and Vistoli [BRV07]. For a proof via Theorem 6.1 see [MR08]. Theorem 6.2. Let G be a p-group of exponent e and k be a field of characteristic 6= p containing a primitive e-th root of unity. Suppose the commutator subgroup (G,G) is central in G. Then 6 Finite Groups 44 (a) ed(G; p) = ed(G)≤ rk Z(G)+rk )(G,G)(p⌊m/2⌋−1), where pm is the order of G/Z(G). (b) Moreover, if (G,G) is cyclic then |G/Z(G)| is a complete square and equal- ity holds in (a). That is, in this case ed(G; p) = ed(G) = √ |G/Z(G)|+ rk Z(G)−1 . Example. Recall that a p-group G is called extra-special if its center Z is cyclic of order p, and the quotient G/Z is elementary abelian. The order of an extra special p-group G is an odd power of p; the exponent of G is either p or p2; cf. [Hup67, III. 13]. Note that every non-abelian group of order p3 is extra-special. For extra- special p-groups Theorem 6.2(b) reduces to the following. Let G be an extra-special p-group of order p2m+1. Assume that the character- istic of k is different from p, that ζp ∈ k, and ζp2 ∈ k if the exponent of G is p2. Then ed(G) = pm. 6.2 p-Groups of Essential Dimension ≤ p It is a formidable task to classify finite groups of a given essential dimension. Finite groups of essential dimension 1 where described in [BR97, 6.2], [Led07] and [CHKZ08]; finite groups of essential dimension 2 over C in [Dun09]. We content ourselves here with the case where G is a p-group. Theorem 6.3. Let p be a prime, k be as in (6.1) and G be a p-group such that G′ 6= {1}. Then the following conditions are equivalent. (a) ed(G)≤ p, (b) ed(G) = p, (c) The center Z(G) is cyclic and G has a subgroup A of index p such that A′ = {1}. (d) G is isomorphic to a subgroup of Z/pα ≀Z/p = (Z/pα)p ⋊Z/p, where α ≥ 1 is an integer such that k contains a primitive pα th root of unity. 6 Finite Groups 45 Note that the assumption that G′ 6= {1} is harmless. Indeed, if G′ = {1} then by Theorem 6.1(b) ed(G) = rk (G); cf. also [BR97, Theorem 6.1] or [BF03, section 3]. Proof. Since K0 = G′ is a non-trivial normal subgroup of G, we see that K0∩Z(G) and thus C0 = K0∩C(G) is non-trivial. This means that in the summation formula of Theorem 6.1(b) at least one of the terms (dimFp Ci−1−dimFp Ci)p i with i ≥ 1 will be non-zero. Hence, ed(G) ≥ p; this shows that (a) and (b) are equivalent. Moreover, equality holds if and only if (i) dimFp C−1 = 1, (ii) dimFp C0 = 1 and (iii) C1 is trivial. We show that (i), (ii) and (iii) are equivalent to (c): Since C−1 = C(G), (i) is equivalent to Z(G) being cyclic. Now recall that we are assuming K0 = G′ 6= {1}. By (6.4) this is equivalent to C0 = K0∩C(G) 6= {1}. Since C0 ⊂C−1, C0 has dimension at most 1, and (ii) follows from (i). Finally, (iii) means that K1 = ⋂ [G:H]=p H ′ (6.6) intersects C(G) trivially. Since K1 is a normal subgroup of G, (6.4) tells us that (iii) holds if and only if K1 = {1}. We claim that K1 = {1} if and only if H ′ = {1} for some subgroup H of G of index p. One direction is obvious: if H ′ = {1} for some H of index p then the intersection (6.6) is trivial. To prove the converse, assume the contrary: the intersection (6.6) is trivial but H ′ 6= {1} for every subgroup H of index p. Since every such H is normal in G (and so is H ′), (6.4) tells us that that H ′ 6= {1} if and only if H ′∩Z(G) 6= {1}. Since Z(G) is cyclic, the latter condition is equivalent to C(G)⊂ H ′. Thus C(G)⊂ K1 = ⋂ [G:H]=p H ′ , contradicting our assumption that K1 = {1}. This proves (c) is equivalent to (a) and (b). p-groups that have a faithful representation of degree p over a field k, satis- fying (6.1) are described in [LGP86, II.4, III.4, IV.2]; see also [Vol63]. They are exactly the subgroups of Z/pα ≀Z/p, where k contains a primitive root of unity of degree pα . Part (d) follows. 46 7 GL(Z) and SL(Z) 7.1 Forms of Algebraic Groups The notion of forms of an algebraic object is variation of the same principle we already encountered which relates an algebraic structure with cohomology. We use the notion of forms here since we want to work with automorphism groups of algebraic groups. The automorphism groups themselves need not be algebraic, i.e. they are not necessarily of finite type over the base field. This however is not a problem and one defines the essential dimension of an affine group scheme in the same way as we defined it for affine algebraic groups. Let G be an algebraic group over k and K/k a field. We say an algebraic K- group G′ is a L/K-form of G if G′L ≃GL, or simply a K-form of G if G′Ksep ≃GKsep . We can define a functor FormsG : Fields/k → Sets K 7→ {K-forms of G}/≃ Now let Aut(G) be the automorphism group of G. Then there is an equivalence of functors FormsG ≃ H1(∗,Aut(G)) , see [Ser62b, III 1.3]. 7.2 Forms of Algebraic Tori Define a functor Fields/k → Sets by Torin(K) = {n-dimensional K-tori}/≃ Since every n-dimensional K-torus is split over Ksep, i.e. isomorphic to Gmn and Aut(Gmn) = GLn(Z) we have equivalences Torin ≃ Forms(Gmn) ≃ H 1(∗,GLn(Z)) . 7 GL(Z) and SL(Z) 47 As a variant of this we define Tori1n(K) = {n-dimensional K-tori T | φT ⊂ SLn(Z)}/≃ where K/k is a field extension and φT : Gal(K)→GLn(Z) is the natural represen- tation of the Galois group of K on the character lattice of the torus T . A slightly different description of Tori1n is given in [FF08, 5] as the functor of pairs (T, ι) where T is an n-dimensional torus and ι : ∧n T → Gm an isomorphism. There is an equivalence Tori1n ≃ H1(∗,SLn(Z)) . Both GLn(Z) and SLn(Z) are not algebraic, but we define their essential dimension as in the algebraic case as the essential dimension of the functors H1(∗,GLn(Z)) and H1(∗,SLn(Z)) respectively. The essential dimension of GLn(Z) and SLn(Z) was first studied by G. Favi and M. Florence [FF08], who showed that ed(GLn(Z)) = n for every n ≥ 1 and ed(SLn(Z)) = n− 1 for every odd n. For even n Favi and Florence showed that ed(SLn(Z)) = n− 1 or n and left the exact value of ed(SLn(Z)) for n ≥ 4 as an open question. We answer this question and also compute the essential p- dimension of GLn(Z) and SLn(Z) for every prime p. Theorem 7.1. Suppose k is a field of characteristic 6= 2. Then (a) ed(GLn(Z);2) = ed(GLn(Z)) = n. (b) ed(SLn(Z);2) = ed(SLn(Z)) = { n−1, if n is odd, n, if n is even for any n≥ 3. Remark. For completeness we also record that ed(SL2(Z)) = 1 if k contains a 12th root of unity, see [FF08, Remark 5.5(2)]. It is interesting to note that while the value of ed(SL2(Z)) depends on the base field k, for n ≥ 3, the value of ed(SLn(Z)) does not (as long as char(k) 6= 2). Theorem 7.2. Let p be an odd prime, chark 6= p and k contains a primitive pth root of unity. ed(SLn(Z); p) = ed(GLn(Z); p) = [ n p−1 ] . G. Favi and M. Florence [FF08] showed that for Γ = GLn(Z) or SLn(Z), ed(Γ) = max{ed(F)|F finite subgroup of Γ}. (7.1) 7 GL(Z) and SL(Z) 48 A minor modification of the arguments in [FF08] shows that (7.1) holds also for essential dimension at a prime p: ed(Γ; p) = max{ed(F ; p)|F a finite subgroup of Γ}, (7.2) or, by Theorem 5.2 equivalently ed(Γ; p) = max{ed(F ; p)|F a finite p-subgroup of Γ}, (7.3) The finite groups F that Florence and Favi used to find the essential dimen- sion of GLn(Z) and SLn(Z) (n odd) are (Z/2Z)n and (Z/2Z)n−1 respectively. Thus ed(GLn(Z);2) = ed(GLn(Z)) = n for every n ≥ 1 and ed(SLn(Z);2) = ed(SLn(Z)) = n−1 if n is odd. Our proof of Theorem 7.1 will rely on Corollary 6.1. Proof of Theorem 7.1. We assume that n = 2d ≥ 4 is even. To prove Theorem 7.1 it suffices to find a 2-subgroup F of SLn(Z) of essential dimension n. Diagonal matrices and permutation matrices generate a subgroup of GLn(Z) isomorphic to µn2 ⋊Sn. The determinant function restricts to a homomorphism det : µn2 ⋊Sn → µ2 sending ((ε1, . . . ,εn),τ)) ∈ µn2 ⋊Sn to the product ε1ε2 · . . . · εn · sign(τ). Let Pn be a Sylow 2-subgroup of Sn and Fn be the kernel of det : µn2 ⋊Pn → µ2 . By construction Fn is a finite 2-group contained in SLn(Z). Theorem 7.1 is now a consequence of the following proposition. Proposition 7.1. If char(k) 6= 2 then ed(F2d) = 2d for any d ≥ 2. To prove the proposition, let D2d = {diag(ε1, . . . ,ε2d) |each εi =±1 and ε1ε2 · · ·ε2d = 1} be the subgroup of “diagonal” matrices contained in F2d . Since D2d ≃ µ2d−12 has essential dimension ≥ 2d− 1, we see that ed(F2d) ≥ ed(D2d) = 2d−1. On the other hand the inclusion F2d ⊂ SL2d(Z) gives rise to a 2d-dimensional representation of F2d , which remains faithful over any field k of characteristic 6= 2. Hence, ed(F2d)≤ 2d. We thus conclude that ed(F2d) = 2d−1 or 2d. (7.4) 7 GL(Z) and SL(Z) 49 Using elementary group theory, one easily checks that C(F2d)⊂ [F2d,F2d]⊂ F ′2d . (7.5) Thus, if k′ ⊃ k is a field as in (6.1), edk′(F2d) is even by Corollary 6.1; since ed(F2d) ≥ edk′(F2d), (7.4) now tells us that ed(F2d) = 2d. This completes the proof of Proposition 7.1 and thus of Theorem 7.1. Remark. (a) The assumption that d ≥ 2 is essential in the proof of the inclu- sion (7.5). In fact, F2 ≃ Z/4Z, so (7.5) fails for d = 1. (b) Note that for any integers m,n≥ 2, Fm+n contains the direct product Fm× Fn. Thus ed(Fm+n)≥ ed(Fm×Fn) = ed(Fm)+ ed(Fn) , where the last equality follows from Theorem 1.1. Thus Proposition 7.1 only needs to be proved for d = 2 and 3 (or equivalently, n = 4 and 6); all other cases are easily deduced from these by applying the above inequality recursively, with m = 4. In particular, the group-theoretic inclusion (7.5) only needs to be checked for d = 2 and 3. Somewhat to our surprise, this reduction does not appear to simplify the proof of Proposition 7.1 presented above to any significant degree. Proof of Theorem 7.2. First we let Γ = GLn(Z). Maximal p-subgroups of GLn(Z) were described by Abold and Plesken [AP78]. Let m = [n/(p−1)] and denote by Cp the cyclic group of order p and Pm a Sylow-p subgroup of the symmetric group Sm. By [AP78, Satz], every p-subgroup H of Γ is Q-conjugate to a sub- group of the wreath product G = Cp ≀Pm = C mp ⋊Pm. Since ed(H; p) ≤ ed(G; p) = ed(G), with (7.3) we have ed(Γ; p) = ed(G). It re- mains to show that ed(G) = m = [n/(p−1)]. Write m in the p-adic expansion (omitting 0 terms): m = ar pr +ar−1 pr−1 + ...+a1 p+a0 with 0 < ai < p for all i. Then Pm splits up into a direct product of Ppi and using that Cp ≀Ppi ≃Ppi+1 , G≃ (Ppr+1)ar × (Ppr)ar−1 × ...× (Pp2)a1 × (Pp)a0 7 GL(Z) and SL(Z) 50 The essential dimension of Ppi+1 is ed(Ppi+1; p) = ed(Spi+1; p) = pi, see Corol- lary 5.1 . Since G is a p-group, by additivity Theorem 5.3, the essential dimension of G is the sum edp(G) = ar edp(Ppr+1)+ ...+a1 edp(Pp2)+a0 edp(Pp) = m This shows that ed(GLn(Z); p)= ed(G; p)= m. To conclude the same for ed(SLn(Z); p) we need only note that G is a subgroup of SLn(Z), which is clear since every ele- ment of G has finite, odd order and therefore its determinant is equal to 1. 51 8 Algebraic Tori It was noted in Lemma 3.5 that if G is an algebraic group, every generically free linear representation ρ : G→GL(V ) gives rise to a versal G-torsor and we obtain the inequality ed(G; p)≤ ed(G)≤ dim(V )−dim(G). (8.1) We are interested in groups for which this inequality is sharp. Karpenko and Merkurjev’s Theorem 1.1 says that finite constant p-groups have this property. In this chapter we will prove similar formulas for a broader class of groups. The main result is Theorem 8.2. As consequences of this theo- rem we obtain the exact value of ed(G; p) in terms of representations or lattices, where G is an algebraic torus or a finite (twisted) group, as stated in Section 2.1. The last two sections are intended to illustrate our results by computing essential dimensions of specific algebraic tori. In Section 8.7 we classify algebraic tori T of essential p-dimension 0 and 1 and in Section 8.8 we compute the essential p- dimension of all tori T over a p-closed field k, which are split by a cyclic extension l/k of degree dividing p2. In this chapter we will often write ”edk ” instead of ”ed” since the base field will be changed sometimes and it is good to keep track of it. 8.1 A Lower Bound Let 1→C → G→ Q→ 1 (8.2) be an exact sequence of algebraic groups over k such that C is central in G and isomorphic to µrp for some r≥ 0. Given a character χ : C→ µp, we will, following [KM08], denote by Repχ the set of irreducible representations φ : G → GL(V ), defined over k, such that φ(c) = χ(c) IdV for every c ∈C. Theorem 8.1. Assume that k is a field of characteristic 6= p containing a primitive pth root of unity. Suppose a sequence of the form (8.2) satisfies the following 8 Algebraic Tori 52 condition: gcd{dim(φ) |φ ∈ Repχ}= min{dim(φ) |φ ∈ Repχ} for every character χ : C → µp. Then ed(G; p)≥mindim(ρ)−dimG , where the minimum is taken over all finite-dimensional k-representations ρ of G such that ρ|C is faithful. Proof. Denote by C∗ := Hom(C,µp) the character group of C. Let E → SpecK be a versal Q-torsor [GMS03, Example 5.4], where K/k is some field extension, and let β : C∗→ Brp(K) denote the homomorphism that sends χ ∈C∗ to the image of E ∈ H1(K,Q) in Brp(K) under the map H1(K,Q)→ H2(K,C) χ∗→ H2(K,µp) = Brp(K) given by composing the connecting map with χ∗. Then there exists a basis χ1, . . . ,χr of C∗ such that edk(G; p)≥ r ∑ i=1 Indβ (χi)−dimG, (8.3) see [Mer08, Theorem 4.8, Example 3.7]. Moreover, by [KM08, Theorem 4.4, Remark 4.5] Indβ (χi) = gcddim(ρ) , where the greatest common divisor is taken over all (finite-dimensional) represen- tations ρ of G such that ρ|C is scalar multiplication by χi. By our assumption, gcd can be replaced by min. Hence, for each i ∈ {1, . . . ,r} we can choose a represen- tation ρi of G with Indβ (χi) = dim(ρi) such that (ρi)|C is scalar multiplication by χi. Set ρ := ρ1⊕·· ·⊕ρr. The inequality (8.3) can be written as edk(G; p)≥ dim(ρ)−dimG. (8.4) Since χ1, . . . ,χr forms a basis of C∗ the restriction of ρ to C is faithful. This proves the theorem. 8 Algebraic Tori 53 8.2 p-(FT)-Groups Of particular interest to us will be p-(FT)-groups, i.e group G which fit into an exact sequence of the form 1→ T → G→ F → 1 , (8.5) where F is a finite p-group and T is a torus over k, see also section 2.1. For the sake of computing ed(G; p) we may assume that k is a p-closed field, see Lemma 4.1. The proofs of the following theorems will be provided in the next sections. Theorem 8.2. Suppose chark 6= p and k(p) denotes a p-closure of k. Let G be a k-group of type p-(FT). Then (a) edk(G; p) ≥ mindimρ − dimG, where the minimum is taken over all p- faithful linear representations ρ of Gk(p) over k(p). Now let G be a k-group of type p-(FxT). Then (b) equality holds in (a), and (c) over k(p) the absolute essential dimension of G and the essential p-dimension coincide: edk(p)(G) = edk(p)(G; p) = edk(G; p). If G is a p-group, a representation ρ is p-faithful if and only if it is faithful. However, for an algebraic torus, “p-faithful” cannot be replaced by “faithful”; see Remark [LMMR09, 10.3]. If G is multiplicative, the value of edk(G; p) given by Theorem 8.2 can be rewritten in terms of the character module X(T ) using Proposition 3.1. Corollary 8.1. Let G be a direct product of a torus and an abelian p-group over the p-closed field k and Γ := Gal(ksep/k) be the absolute Galois group of k = k(p). Let Γ act through a finite quotient Γ on X(G). Then edk(G) = edk(G; p) = minrk L−dimG , where the minimum is taken over all permutation ZΓ-lattices L which admit a map of ZΓ-modules to X(G) with cokernel finite of order prime to p. 8 Algebraic Tori 54 Let us rephrase the main result first for the most important case, algebraic tori. It is easy to show that if T is a torus over k such that its minimal splitting field L/k is of degree a power of p, then any T -representation over a p-closure k(p) can already be defined over k. Following [Mer09] we call a homomorphism P → X(T ) of Γ-lattices a p- presentation if it has finite cokernel of order prime to p and P is permutation. It is called minimal if the rank of P is minimal among all p-presentations. Theorem 8.3. Let T be a k-torus with minimal splitting field L/k. Assume [L : k] is a power of p and let Γ = Gal(L/k). The following numbers are equal: (a) edk(T ) (b) edk(T ; p) (c) min{dimρ|p-faithful T -representation/k}−dimT (d) min{rk kerφ |φ : P→ X(T ) a p-presentation }. (e) min{rk L}, where the minimum is taken over all exact sequences of Z(p)Γ- lattices of the form (0)→ L→ P→ X(T )(p) → (0) , with P permutation. For the equality of (d) and (e) we use Proposition 3.1, the other equalities are clear from the previous results. In many cases Theorem 8.3 renders the value of edk(T ) computable by known representation-theoretic methods, e.g., from [CR90]. We will give several examples of such computations in Sections 8.7 and 8.8. Another application was recently given by Merkurjev [Mer09], see also Theo- rem 11.1. Theorem 8.2 appears to be new even in the other extreme case where G is a twisted cyclic p-group. It extends earlier work of Rost [Ros02] (cyclic groups of order 4), Bayarmagnai [Bay07] (cyclic groups of order 8) and Florence [Flo08] (constant cyclic groups). Theorem 8.4. Let G be a finite algebraic group over a p-closed field k = k(p) of characteristic 6= p. Then G has a Sylow-p subgroup Gp defined over k and edk(G; p) = edk(Gp; p) = edk(Gp) = mindim(ρ) where the minimum is taken over all faithful representations of Gp over k. 8 Algebraic Tori 55 Proof. By assumption, Γ = Gal(ksep/k) is a pro-p group. It acts on the set of Sylow-p subgroups of G(ksep). Since the number of such subgroups is prime to p, Γ fixes at least one of them and by Galois descent one obtains a subgroup Gp of G. By Lemma 5.1, Gp is a Sylow-p subgroup of G. The first equality edk(G; p) = edk(Gp; p) is proved in Theorem 5.2. The minimal Gp-representation ρ from Theorem 8.2(b) is faithful and thus edk(Gp) ≤ dim(ρ), see for example [BF03, Prop. 4.11]. Remark. Two Sylow-p subgroups of G defined over k = k(p) do not need to be isomorphic over k. Corollary 8.2. Let A be a finite (twisted) cyclic p-group over k. Let l/k be a minimal Galois splitting field of A. Then ed(A; p) = |Gal(l/k)|p = |Gal(l(p)/k(p))|, where |Gal(l/k)|p denotes the p-primary part of |Gal(l/k)|. Note that for p = 2 we have |Gal(l/k)|p = |Gal(l/k)| above since the auto- morphism group of X(A)≃ Z/2nZ is a 2-group. Proof. The second equality follows from the properties of the p-closure. More- over l(p) is a minimal Galois splitting field of Ak(p) . Since the essential p-dimension of A does not change when passing to the p-closure, we can assume k = k(p). Set Γ = Gal(l/k) which is now automatically a p-group. By Corollary 8.1 ed(A; p) is equal to the least cardinality of a Γ-set Λ such that there exists a map φ : Z[Λ]→ X(A) of ZΓ-modules with cokernel finite of order prime to p. The group X(A) is a (cyclic) p-group, hence φ must be surjective. Moreover Γ acts faithfully on X(A). Surjectivity of φ implies that some element λ ∈ Λ maps to a generator a of X(A). Hence |Λ| ≥ |Γλ | ≥ |Γa|= |Γ|. Conversely we have a surjective homomorphism Z[Γ]→ X(A) that sends a to itself. It is natural to try to extend the formula of Theorem 8.2(b) to all groups G of type p-(FT) or all (FT) for that matter, by Theorem 5.2. Suppose that k is p-closed. Then by Theorem 8.2(a) and Lemma 3.5, mindim µ−dim(G)≤ ed(G; p)≤mindimρ−dimG , (8.6) where the two minima are taken over all p-faithful representations µ , and p- generically free representations ρ , respectively. If G is a direct product of a 8 Algebraic Tori 56 torus and a p-group, then every p-generically free representation is p-faithful, see Lemma 3.2, so in this case the lower and upper bounds coincide, yielding the exact value of edk(G; p) of Theorem 8.2(b). However, if we only assume G is a p-group extended by a torus, then faithful G-representations no longer need to be generically free, see example 3.1. We put forward the following conjecture. Conjecture. Let G be of type (FT) and k of characteristic 6= p. Then ed(G; p) = mindimρ−dimG, where the minimum is taken over all p-generically free representations ρ of Gk(p) over k(p). 8.3 The Central Split Group C(G) Let A be a group of multiplicative type and A[p] the p-torsion subgroup {a ∈ A |ap = 1} of A. Clearly A[p] is defined over k. If T is a torus it is well known how to construct a maximal split subtorus of T , see for example [Bor61, 8.15] or [Wat79, 7.4]. The following definition is a variant of this. Definition 8.1. Let A be an algebraic group of multiplicative type over k. Let ∆(A) be the Γ-invariant subgroup of X(A) generated by elements of the form x− γ(x), as x ranges over X(A) and γ ranges over Γ. Alternatively, ∆(A) = IX(A) where I is the augmentation ideal in ZΓ. Define Splitk(A) = Diag(X(A)/∆(A)) . Definition 8.2. Let G be a p-(FT)-group as in (8.5). C(G) := Splitk(Z(G)[p]) , where Z(G) denotes the centre of G. If X = X(Z(G)), we have C(G) = Diag(X/(pX + IX)) where again I is the augmentation ideal of X in ZΓ, see [Mer09, Section 4]. Lemma 8.1. Let A be an algebraic group of multiplicative type over k. (a) Splitk(A) is split over k, 8 Algebraic Tori 57 (b) Splitk(A) = A if and only if A is split over k, (c) If B is a k-subgroup of A then Splitk(B)⊂ Splitk(A). (d) For A = A1×A2, Splitk(A1×A2) = Splitk(A1)×Splitk(A2), (e) If A[p] 6= {1} and A is split over a Galois extension l/k, such that Γ = Gal(l/k) is a p-group, then Splitk(A) 6= {1}. Proof. Parts (a), (b), (c) and (d) easily follow from the definition. Proof of (e): By part (c), it suffices to show that Splitk(A[p]) 6= {1}. Hence, we may assume that A = A[p] or equivalently, that X(A) is a finite-dimensional Fp-vector space on which the p-group Γ acts. Any such action is upper-triangular, relative to some Fp-basis e1, . . . ,en of X(A); see, e.g., [Ser77, Proposition 26]. That is, γ(ei) = ei+ (Fp-linear combination of ei+1, . . . ,en) for every i = 1, . . . ,n and every γ ∈ Γ. Our goal is to show that ∆(A) 6= X(A). In- deed, every element of the form x−γ(x) is contained in the Γ-invariant submodule Span(e2, . . . ,en). Hence, these elements cannot generate all of X(A). Proposition 8.1. Suppose G is a p-(FT) group over a p-closed field k. Suppose N is a normal subgroup of G. Then the following conditions are equivalent: (a) N is finite of order prime to p, (b) N∩C(G) = {1}, (c) N∩Z(G)[p] = {1} In particular, taking N = G, we see that C(G) 6= {1} if G 6= {1}. Proof. (i) =⇒ (ii) is obvious, since C(G) is a p-group. (ii) =⇒ (iii). Assume the contrary: A := N∩Z(G)[p] 6= {1}. By Lemma 8.1 {1} 6= C(A)⊂ N∩C(Z(G)[p]) = N∩C(G) , contradicting (ii). Our proof of the implication (iii) =⇒ (i), will rely on the following Claim: Let M be a non-trivial normal finite p-subgroup of G such that the commutator (G0,M) = {1}. Then M∩Z(G)[p] 6= {1}. 8 Algebraic Tori 58 To prove the claim, note that M(ksep) is non-trivial and the conjugation action of G(ksep) on M(ksep) factors through an action of the p-group (G/G0)(ksep). Thus each orbit has pn elements for some n ≥ 0; consequently, the number of fixed points is divisible by p. The intersection (M∩Z(G))(ksep) is precisely the fixed point set for this action; hence, M∩Z(G)[p] 6= {1}. This proves the claim. We now continue with the proof of the implication (iii) =⇒ (i). For notational convenience, set T := G0. Assume that N ⊳G and N ∩Z(G)[p] = {1}. Applying the claim to the normal subgroup M := (N∩T )[p] of G, we see that (N∩T )[p] = {1}, i.e., N∩T is a finite group of order prime to p. The exact sequence 1→ N∩T → N → N → 1 , (8.7) where N is the image of N in G/T , shows that N is finite. Now observe that for every r ≥ 1, the commutator (N,T [pr]) is a p-subgroup of N ∩ T . Thus (N,T [pr]) = {1} for every r ≥ 1. We claim that this implies (N,T ) = {1} by Zariski density. If N is smooth, this is straightforward; see [Bor61, Proposition 2.4, p. 59]. If N is not smooth, note that the map c : N×T → G sending (n, t) to the commutator ntn−1t−1 descends to c : N × T → G (indeed, N ∩ T clearly commutes with T ). Since |N| is a power of p and char(k) 6= p, N is smooth over k, and we can pass to the separable closure ksep and apply the usual Zariski density argument to show that the image of c is trivial. We thus conclude that N ∩T is central in N. Since gcd(|N ∩T |,N) = 1, by [Sch80a, Corollary 5.4] the extension (8.7) splits, i.e., N ≃ (N ∩ T )×N. This turns N into a subgroup of G satisfying the conditions of the claim. Therefore N is trivial and N = N∩T is a finite group of order prime to p, as claimed. For future reference, we record the following obvious consequence of the equivalence of conditions (i) and (ii) in Proposition 8.1. Corollary 8.3. Let k = k(p) be a p-closed field and G be a p-(FT) group over k, as in (8.5). A finite-dimensional representation ρ of G defined over k is p-faithful if and only ρ|C(G) is faithful. 8.4 Proof of Theorem 8.2 (a) and (b) The key step in our proof will be the following proposition. Proposition 8.2. Let k be a p-closed field, and G be a p-(FT) group, as in (8.5). Then the dimension of every irreducible representation of G over k is a power of p. 8 Algebraic Tori 59 Assuming Proposition 8.2 we can easily complete the proof of Theorem 8.2(a) and (b). The proof of part (c) will be postponed until section 8.6. Indeed, by Proposition 4.2 we may assume that k = k(p) is p-closed. In particular, since we are assuming that char(k) 6= p, this implies that k contains a primitive pth root of unity, see Remark 4.1. Proposition 8.2 tells us that Theorem 8.1 can be applied to the exact sequence 1→C(G)→ G→ Q→ 1 . (8.8) This yields ed(G; p)≥min dim(ρ)−dim(G) , (8.9) where the minimum is taken over all representations ρ : G → GL(V ) such that ρ|C(G) is faithful. By Corollary 8.3, ρ|C(G) is faithful if and only if ρ is p-faithful, and Theorem 8.2(a) follows. Part (b) follows from our preparations in Chapter 3; Lemma 3.1 and 11.1. The rest of this section will be devoted to the proof of Proposition 8.2. We begin by settling it in the case where G is a finite p-group. Lemma 8.2. Proposition 8.2 holds if G is a finite p-group. Proof. Choose a finite Galois field extension l/k such that (i) G is constant over l and (ii) every irreducible linear representation of G over l is absolutely irreducible. Since k is assumed to be p-closed, [l : k] is a power of p. Let A := k[G]∗ be the dual Hopf algebra of the coordinate algebra of G. By [Jan03, Section 8.6] a G-module structure on a k-vector space V is equivalent to an A-module structure on V . Now assume that V is an irreducible A-module and let W ⊆ V ⊗k l be an irreducible A⊗k l-submodule. Then by [Kar89, Theorem 5.22] there exists a divisor e of [l : k] such that V ⊗ l ≃ e ( r⊕ i=1 σiW ) , where σi ∈Gal(l/k) and {σiW | 1≤ i≤ r} are the pairwise non-isomorphic Galois conjugates of W . By our assumption on k, e and r are powers of p and by our choice of l, diml W = diml(σ1W ) = . . . = diml(σrW ) is also a power of p, since it divides the order of Gl . Hence, so is dimk(V ) = diml V ⊗ l = e(diml σ1W + · · ·+ diml σrW ). Our proof of Proposition 8.2 in full generality will based on leveraging Lemma 8.2 as follows. 8 Algebraic Tori 60 Lemma 8.3. Let G be an algebraic group defined over a field k and F1 ⊆ F2 ⊆ ·· · ⊂ G be an ascending sequence of finite k-subgroups whose union ∪n≥1Fn is Zariski dense in G. If ρ : G → GL(V ) is an irreducible representation of G defined over k then ρ|Fi is irreducible for sufficiently large integers i. Proof. For each d = 1, ...,dim(V )−1 consider the G-action on the Grassmannian Gr(d,V ) of d-dimensional subspaces of V . Let X (d) = Gr(d,V )G and X (d)i = Gr(d,V )Fi be the subvariety of d-dimensional G- (resp. Fi-)invariant subspaces of V . Then X (d)1 ⊇ X (d) 2 ⊇ . . . and since the union of the groups Fi is dense in G, X (d) = ∩i≥0X (d) i . By the Noetherian property of Gr(d,V ), we have X (d) = X (d)md for some md ≥ 0. Since V does not have any G-invariant d-dimensional k-subspaces, we know that X (d)(k) = /0. Thus, X (d)md (k) = /0, i.e., V does not have any Fmd -invariant d- dimensional k-subspaces. Setting m := max{m1, . . . ,mdim(V )−1}, we see that ρ|Fm is irreducible. We now proceed with the proof of Proposition 8.2. By Lemmas 8.2 and 8.3, it suffices to construct a sequence of finite p-subgroups F1 ⊆ F2 ⊆ ·· · ⊂ G defined over k whose union ∪n≥1Fn is Zariski dense in G. In fact, it suffices to construct one p-subgroup F ′⊂G, defined over k such that F ′ surjects onto F . Indeed, once F ′ is constructed, we can define Fi ⊂ G as the subgroup generated by F ′ and T [pi], for every i≥ 0. Since ∪n≥1Fn contains both F ′ and T [pi], for every i≥ 0 it is Zariski dense in G, as desired. The following lemma, which establishes the existence of F ′, is thus the final step in our proof of Proposition 8.2 (and hence, of Theorem 8.2(a)). Lemma 8.4. Let 1→ T →G pi−→ F → 1 be an extension of a p-group F by a torus T over k. Then G has a finite p-subgroup F ′ with pi(F ′) = F. In the case where F is split and k is algebraically closed this is proved in [CGR06, p. 564]; cf. also the proof of [BS64, 5.11]. 8 Algebraic Tori 61 Proof. Denote by Ẽx1(F,T ) the group of equivalence classes of extensions of F by T . We claim that Ẽx 1 (F,T ) is torsion. Let Ex1(F,T ) ⊂ Ẽx 1 (F,T ) be the classes of extensions which have a scheme-theoretic section (i.e. G(K)→ F(K) is surjective for all K/k). There is a natural isomorphism Ex1(F,T ) ≃ H2(F,T ), where the latter one denotes Hochschild cohomology, see [DG70a, III. 6.2, Propo- sition]. By [Sch81] the usual restriction-corestriction arguments can be applied in Hochschild cohomology and in particular, m ·H2(F,T ) = 0 where m is the order of F . Now recall that M 7→ Ẽx i (F,M) and M 7→ Exi(F,M) are both de- rived functors of the crossed homomorphisms M 7→ Ex0(F,M), where in the first case M is in the category of F-module sheaves and in the second, F-module functors, cf. [DG70a, III. 6.2]. Since F is finite and T an affine scheme, by [Sch80b, Satz 1.2 & Satz 3.3] there is an exact sequence of F-module schemes 1 → T → M1 → M2 → 1 and an exact sequence Ex0(F,M1) → Ex0(F,M2) → Ẽx 1 (F,T ) → H2(F,M1) ≃ Ex1(F,M1). The F-module sequence also induces a long exact sequence on Ex(F,∗) and we have a diagram Ẽx 1 (F,T ) &&MM MMM MMM MM Ex0(F,M1) // Ex0(F,M2) 88qqqqqqqqqq &&NN NNN NNN NN Ex1(F,M1) Ex1(F,T ) 88qqqqqqqqqqq? OO An element in Ẽx 1 (F,T ) can thus be killed first in Ex1(F,M1) so it comes from Ex0(F,M2). Then kill its image in Ex1(F,T )≃H2(F,T ), so it comes from Ex0(F,M1), hence is 0 in Ẽx 1 (F,T ). In particular we see that multiplying twice by the order m of F , m2 · Ẽx 1 (F,T ) = 0. This proves the claim. Now let us consider the exact sequence 1 → N → T ×m 2 −−→ T → 1, where N is the kernel of multiplication by m2. Clearly N is finite and we have an induced exact sequence Ẽx 1 (F,N)→ Ẽx 1 (F,T ) ×m 2 −−→ Ẽx 1 (F,T ) which shows that the given extension G comes from an extension F ′ of F by N. Then G is the pushout of F ′ by N → T and we can identify F ′ with a subgroup of G. 8 Algebraic Tori 62 8.5 Proof of the Additivity Theorem Recall the earlier stated Theorem 5.3. Let k be of characteristic 6= p. (a) Let G and H be (FxT)-groups over k. Then ed(G×H; p) = ed(G; p)+ ed(H; p). (b) Let G,H be p-(FxT)-groups over k and suppose k = k(p) is p-closed. Then ed(G×H) = ed(G)+ ed(H). First we need a lemma. Let G be an algebraic group over k and C be a subgroup of G. Denote the minimal dimension of a representation ρ of G (defined over k) such that ρ|C is faithful by f (G,C). Lemma 8.5. For i = 1,2 let Gi be an algebraic group and Ci be a central subgroup of Gi. Assume that Ci is isomorphic to µrip over k for some r1,r2 ≥ 0. Then f (G1×G1;C1×C2) = f (G1;C1)+ f (G2;C2) . Our argument is a variant of the proof of [KM08, 5.1], where G is assumed to be a (constant) finite p-group and C is the socle of G. Proof. For i = 1,2 let pii : G1×G2 → Gi be the natural projection and εi : Gi → G1×G2 be the natural inclusion. If ρi is a di-dimensional k-representation of Gi whose restriction to Ci is faith- ful, then clearly ρ1◦pi1⊕ρ2◦pi2 is a d1+d2-dimensional representation of G1×G2 whose restriction to C1×C2 is faithful. This shows that f (G1×G1;C1×C2)≤ f (G1;C1)+ f (G2;C2) . To prove the opposite inequality, let ρ : G1×G2 → GL(V ) be a k-representation such that ρ|C1×C2 is faithful, and of minimal dimension d = f (G1×G1;C1×C2) 8 Algebraic Tori 63 with this property. Let ρ1,ρ2, . . . ,ρn denote the irreducible decomposition fac- tors in a decomposition series of ρ . Since C1 ×C2 is central in G1 ×G2, each ρi restricts to a multiplicative character of C1×C2 which we will denote by χi. Moreover since C1 ×C2 ≃ µr1+r2p is linearly reductive ρ|C1×C2 is a direct sum χ⊕d11 ⊕·· ·⊕χ⊕dnn where di = dimVi. It is easy to see that the following conditions are equivalent: (i) ρ|C1×C2 is faithful, (ii) χ1, . . . ,χn generate (C1×C2)∗ as an abelian group. In particular we may assume that ρ = ρ1⊕ ·· · ⊕ ρn. Since Ci is isomorphic to µrip , we will think of (C1×C2)∗ as a Fp-vector space of dimension r1 + r2. Since (i) ⇔ (ii) above, we know that χ1, . . . ,χn span (C1×C2)∗. In fact, they form a basis of (C1×C2)∗, i.e., n = r1 + r2. Indeed, if they were not linearly independent we would be able to drop some of the terms in the irreducible decomposition ρ1⊕·· ·⊕ρn, so that the restriction of the resulting representation to C1×C2 would still be faithful, contradicting the minimality of dim(ρ). We claim that it is always possible to replace each ρ j by ρ ′j, where ρ ′j is either ρ j ◦ ε1 ◦pi1 or ρ j ◦ ε2 ◦pi2 such that the restriction of the resulting representation ρ ′ = ρ ′1⊕·· ·⊕ρ ′n to C1×C2 remains faithful. Since dim(ρi) = dim(ρ ′i ), we see that dim(ρ ′) = dim(ρ). Moreover, ρ ′ will then be of the form α1 ◦pi1⊕α2 ◦pi2, where αi is a representation of Gi whose restriction to Ci is faithful. Thus, if we can prove the above claim, we will have f (G1×G1;C1×C2) = dim(ρ) = dim(ρ ′) = dim(α1)+dim(α2) ≥ f (G1,C1)+ f (G2,C2) , as desired. To prove the claim, we will define ρ ′j recursively for j = 1, . . . ,n. Suppose ρ ′1, . . . ,ρ ′j−1 have already be defined, so that the restriction of ρ ′1⊕·· ·⊕ρ ′j−1⊕ρ j · · ·⊕ρn to C1×C2 is faithful. For notational simplicity, we will assume that ρ1 = ρ ′1, . . . ,ρ j−1 = ρ ′j−1. Note that χ j = (χ j ◦ ε1 ◦pi1)+(χ j ◦ ε2 ◦pi2) . Since χ1, . . . ,χn form a basis (C1×C2)∗ as an Fp-vector space, we see that (a) χ j ◦ ε1 ◦ pi1 or (b) χ j ◦ ε2 ◦ pi2 does not lie in SpanFp(χ1, . . . ,χ j−1,χ j+1, . . . ,χn). Set ρ ′j := { ρ j ◦ ε1 ◦pi1 in case (a), and ρ j ◦ ε2 ◦pi2, otherwise. 8 Algebraic Tori 64 Using the equivalence of (i) and (ii) above, we see that the restriction of ρ1⊕·· ·⊕ρ j−1⊕ρ ′j⊕ρ j+1, · · ·⊕ρn to C is faithful. This completes the proof of the claim and thus of Lemma 8.5. Proof of Theorem 5.3. Part (a) follows from (b) with Lemma 4.1 and Theorem 5.2. For part (b), let C(G) be as in Definition 8.2. By Theorem 8.2(b) ed(G; p) = f (G,C(G))−dimG ; cf. Corollary 8.3. Furthermore, we have C(G1×G2)=C(G1)×C(G2); cf. Lemma 8.1(d). Applying Lemma 8.5 finishes the proof. 8.6 Proof of Theorem 8.2 (c) We will prove Theorem 8.2(c) by using the lattice point of view and the additivity theorem from Section 8.5. Let Γ be a finite group. Two ZΓ-lattices M,N are said to be in the same genus if M(p) ≃ N(p) for all primes p, cf. [CR90, 31A]. It is sufficient to check this condition for divisors p of the order of Γ. By a theorem of A.V. Roı̌ter [CR90, Theorem 31.28] M and N are in the same genus if and only if there exists a ZΓ- lattice L in the genus of the free ZΓ-lattice of rank one such that M⊕ZΓ≃ N⊕L. This has the following consequence for essential dimension: Proposition 8.3. Let T,T ′ be k-tori. If the lattices X(T ),X(T ′) belong to the same genus then H1(K,T ) = H1(K,T ′) for all field extensions of K/k. In particular edk(T ) = edk(T ′) and edk(T ;ℓ) = edk(T ′;ℓ) for all primes ℓ. Proof. Let Gal(ksep/k) act through a finite quotient Γ on X(T ) and X(T ′). By assumption there exists a ZΓ-lattice L in the genus of ZΓ such that X(T )⊕ZΓ≃ X(T ′)⊕L. The torus S = Diag(ZΓ) has H1(K,S) = {1} for all field extensions K/k. The same applies to the torus S′ := Diag(L) since L is a direct summand of ZΓ⊕ZΓ. Therefore H1(K,T ) = H1(K,T × S) = H1(K,T ′× S′) = H1(K,T ′) for all K/k. This concludes the proof. Corollary 8.4. Let k = k(p) be a p-closed field and T a k-torus. Then edk(T ) = edk(T ; p). 8 Algebraic Tori 65 Proof. The inequality edk(T ; p)≤ edk(T ) is clear. By Theorem 8.2(a) there is a p- faithful representation ρ : T →GL(V ) of minimal dimension so that edk(T ; p) = dimρ−dimT . The representation ρ can be considered as a faithful representation of the torus T ′ = T/N where N := kerρ is finite of order prime to p. By construc- tion the character lattices X(T ) and X(T ′) are isomorphic after localization at p. Since Gal(ksep/k) is a (profinite) p-group it follows that X(T ) and X(T ′) belong to the same genus. Hence by Proposition 8.3 we have edk(T ′) = edk(T ). Moreover edk(T ′)≤ dimρ−dimT ′, since ρ is a generically free representation of T ′. Proof of Theorem 8.2(c). The equality edk(p)(Gk(p) ; p) = edk(G; p) is Lemma 4.1. Now we are assuming G = T×F for a torus T and a p-group F over k, which is p- closed. Notice that a minimal p-faithful representation of F from Theorem 8.2(a) is also faithful, and therefore edk(F ; p) = edk(F). Combining this with Corol- lary 8.4 and the additivity Theorem 5.3, we see ed(T ×F)≤ ed(T )+ ed(F) = ed(T ; p)+ ed(F ; p) = ed(T ×F; p)≤ ed(T ×F). 8.7 Tori of Essential Dimension ≤ 1 Theorem 8.5. Let k = k(p) be a p-closed field, Γ = Gal(kalg/k) be the absolute Galois group of k and T be a torus over k. Then the following conditions are equivalent: (a) edk(T ) = 0. (b) edk(T ; p) = 0. (c) X(T )(p) is a Z(p)Γ-permutation module. (d) X(T ) is an invertible ZΓ-lattice (i.e. a direct summand of a permutation lattice). (e) There is a torus S over k and an isomorphism T ×S≃ RE/k(Gm), for some étale algebra E over k. (f) H1(K,T ) = {1} for any field K containing k. 8 Algebraic Tori 66 Remark. A prime p for which any of these statements fails is called a torsion prime of T . Proof. (a) ⇔ (b) by Theorem 8.2(c). (b) ⇒ (c) follows from Corollary 8.1. Indeed, edk(T ; p) = 0 implies the exis- tence of a Z(p)Γ-permutation lattice L together with a surjective homomorphism α : L → X(T )(p) such that rk L = rk X(T )(p). It follows that α is injective and X(T )(p) ≃ L. (c) ⇒ (d): Let L be a ZΓ-permutation lattice such that L(p) ≃ X(T )(p). Then by [CR90, Corollary 31.7] there is a ZΓ-lattice L′ such that L⊕L≃ X(T )⊕L′. (d) ⇔ (e): A permutation lattice P can be written as P = m⊕ i+1 Z[Γ/ΓLi], for some (separable) extensions Li/k and ΓLi = Gal(kalg/Li). Set E = L1×·· ·× Lm. The torus corresponding to P is exactly RE/k(Gm), cf. [Vos98, 3. Example 19]. (e) ⇒ (f) because H1(K,RE/k(Gm)) = {1}. (f) ⇒ (a) is obvious from the definition of edk(T ). Example. Let T be a torus over k of rank < p−1. Then edk(T ; p) = 0. This fol- lows from the fact that there is no non-trivial integral representation of dimension < p−1 of any p-group, see for example [AP78, Satz]. Thus any finite quotient of Γ = Gal(kalg/k) acts trivially on X(T ) and so does Γ. Remark. The equivalence of parts (d) and (f) can also be deduced from [CTS77, Proposition 7.4]. Theorem 8.6. Let p be an odd prime, T an algebraic torus over k, and Γ = Gal(kalg/k(p)). (a) ed(T ; p)≤ 1 iff there exists a Γ-set Λ and an m ∈ Z[Λ] fixed by Γ such that X(T )(p) ∼= Z(p)[Λ]/〈m〉 as Z(p)Γ-lattices. (b) ed(T ; p) = 1 iff m = ∑aλ λ from part (a) is not 0 and for any λ ∈Λ fixed by Γ, aλ = 0 mod p. (c) If ed(T ; p) = 1 then Tk(p) ∼= T ′×S where edk(p)(S; p) = 0 and X(T ′)(p) is an indecomposable Z(p)Γ-lattice, and edk(p)(T ′; p) = 1. 8 Algebraic Tori 67 Proof. (a) If ed(T ; p) = 1, then by Corollary 8.1 there is a map of ZΓ-lattices from Z[Λ] to X(T ) which becomes surjective after localization at p and whose kernel is generated by one element. Since the kernel is stable under Γ, any element of Γ sends a generator m to either itself or its negative. Since p is odd, m must be fixed by Γ. The ed(T ; p) = 0 case and the converse follows from Theorem 8.3 or Corol- lary 8.1. (b) Assume we are in the situation of (a), and say λ0 ∈ Λ is fixed by Γ and aλ0 is not 0 mod p. Then X(T )(p) ∼= Z(p)[Λ−{λ0}], so by Theorem 8.5 we have ed(T ; p) = 0. Conversely, assume ed(T ; p) = 0. Then by Theorem 8.5, we have an exact sequence 0 → 〈m〉 → Z(p)[Λ]→ Z(p)[Λ′]→ 0 for some Γ-set Λ′ with one fewer element than Λ. We have Ext1Γ(Z(p)[Λ′],Z(p)) = (0) by [CTS77, Key Lemma 2.1(i)] together with the Change of Rings Theorem [CR90, 8.16]; therefore this sequence splits. In other words, there exists a Z(p)Γ- module homomorphism f : Z(p)[Λ]→ Z(p)[Λ] such that the image of f is 〈m〉 and f (m) = m. Then we can define cλ ∈ Z(p) by f (λ ) = cλ m. Note that f (γ(λ )) = f (λ ) and thus cγ(λ ) = cλ (8.10) for every λ ∈ Λ and γ ∈ Γ. If m = ∑λ∈Λ aλ λ , as in the statement of the theorem, then f (m) = m translates into ∑ λ∈Λ cλ aλ = 1 . Since every Γ-orbit in Λ has a power of p elements, reducing modulo p, we obtain ∑ λ∈ΛΓ cλ aλ = 1 (mod p) . This shows that aλ 6= 0 modulo p, for some λ ∈ ΛΓ, as claimed. (c) Decompose X(T )(p) uniquely into a direct sum of indecomposable Z(p)Γ- lattices by the Krull-Schmidt theorem [CR90, Theorem 36.1]. Since ed(T ; p) = 1, and the essential p-dimension of tori is additive (Theorem 5.3), all but one of these summands are permutation Z(p)Γ-lattices. Now by [CR90, 31.12], we can lift this decomposition to X(T ) ∼= X(T ′)⊕ X(S), where ed(T ′; p) = 1 and ed(S; p) = 0. 8 Algebraic Tori 68 Example. Let E be an étale algebra over k. It can be written as E = L1×·· ·×Lm with some separable field extensions Li/k. The kernel of the norm RE/k(Gm)→ Gm is denoted by R(1)E/k(Gm). It is a torus with lattice m⊕ i=1 Z[Γ/ΓLi] / 〈1, · · · ,1〉, where Γ = Gal(ksep/k) and ΓLi = Gal(ksep/Li). Let Λ be the disjoint union of the cosets Γ/ΓLi Passing to a p-closure k(p) of k, Γk(p) fixes a λ in Λ iff [Li : k] is prime to p for some i. We thus have edk(R (1) E/k(Gm); p) = { 1, [Li : k] is divisible by p for all i = 1, ...,m 0, [Li : k] is prime to p for some i. 8.8 Tori Split by Cyclic Extensions of Degree Dividing p2 In this section we assume k = k(p) is p-closed. Over k = k(p) every torus is split by a Galois extension of p-power order. We wish to compute the essential dimension of all tori split by a Galois extension with a (small) fixed Galois group G. The following theorem tells us for which G this is feasible: Theorem 8.7 (A. Jones [Joh71]). For a p-group G there are only finitely many genera of indecomposable ZG-lattices if and only if G is cyclic of order dividing p2. Remark. For G = C2×C2 a classification of the (infinitely many) different genera of ZG-lattices has been worked out by [Naz61]. In contrast for G = Cp3 or G = Cp×Cp and p odd (in the latter case) no classification is known. We fix G = Cp2 = 〈g|gp 2 = 1〉 . We consider tori T whose character lattice X(T ) is a ZG-lattice or equivelently whose minimal splitting field is cyclic of degree dividing p2. Heller and Reiner [HR62], (see also [CR90, 34.32]) classified all indecompos- able ZG-lattices. Our goal consists in computing the essential dimension of T . By Corollary 8.4 we have edk(T ) = edk(T ; p), hence by the additivity Theorem 5.3 it will be enough to find the essential p-dimension of the tori corresponding to 8 Algebraic Tori 69 indecomposable ZG-lattices. Recall that two lattices are in the same genus if their p-localization (or equivalently p-adic completion) are isomorphic. By Proposi- tion 8.3 tori with character lattices in the same genus have the same essential p-dimension, which reduces the task to calculating the essential p-dimension of tori corresponding to the 4p+1 cases in the list [CR90, 34.32]. Denote H = Cp = 〈h|hp = 1〉 . We can consider ZH as a G-lattice with the action g ·hi = hi+1. Let δG = 1+g+ . . .+gp 2−1 δH = 1+h+ . . .+hp−1 be the “diagonals” in ZG and ZH and ε = 1+gp + . . .+gp 2−p. In table 8.1 representatives of all genera of indecomposable ZG-lattices are listed (by 〈∗〉 we mean the ZG-sublattice generated by ∗). Table 8.1: Indecomposable ZCp2-lattices. M1 = Z M2 = ZH M3 = ZH/〈δH〉 M4 = ZG M5 = ZG/〈δG〉 M6 = ZG⊕Z/〈δG− p〉 M7 = ZG/〈ε〉 M8 = ZG/〈ε−gε〉 M9,r = ZG⊕ZH/〈ε− (1−h)r〉 1≤ r ≤ p−1 M10,r = ZG⊕ZH/〈ε(1−g)− (1−h)r+1〉 1≤ r ≤ p−2 M11,r = ZG⊕ZH/〈ε− (1−h)r,δH〉 1≤ r ≤ p−2 M12,r = ZG⊕ZH/〈ε(1−g)− (1−h)r+1,δH〉 1≤ r ≤ p−2 In table 8.1 we describe ZG-lattices as quotients of permutation lattices of minimal possible rank, whereas [CR90, 34.32] describes these lattices as certain extensions 1 → L → M → N → 1 of Z[ζp2]-lattices by ZH-lattices. Therefore these two lists look differently. Nevertheless they represent the same ZG-lattices. 8 Algebraic Tori 70 We show in the example of the lattice M10,r how one can translate from one list to the other. Let Zx be a ZG-module of rank 1 with trivial G-action. We have an isomor- phism M10,r = ZG⊕ZH/〈ε(1−g)− (1−h)r+1〉 ≃ ZG⊕ZH⊕Zx/〈ε− (1−h)r− x〉 induced by the inclusion ZG⊕ZH →֒ ZG⊕ZH⊕Zx. This allows us to write M10,r as the pushout ZH h7→ε // h7→(1−h)r+x  ZG  ZH⊕Zx // M10,r Completing both lines on the right we see that M10,r is an extension 0→ ZH⊕Zx→M10,r → ZG/ZH → 0 with extension class determined by the vertical map h 7→ (1−h)r + x cf. [CR90, 8.12] and we identify (the p-adic completion of) M10,r with one of the indecom- posable lattices in the list [CR90, 34.32]. Similarly, M1, . . . ,M12,r are representatives of the genera of indecomposable ZG-lattices. Theorem 8.8. Every indecomposable torus T over k split by G has character lattice isomorphic to one of the ZG-lattices M in table 8.1 after p-localization and ed(T ) = ed(T ; p) = ed(Diag(M); p). Their essential dimensions are given in the tables ta.edtori. Proof of Proposition 8.8. We will assume p > 2 in the sequel. For p = 2 the Theorem is still true but some easy additional arguments are needed which we leave out here. The essential p-dimension of tori corresponding to M1 . . . ,M6 easily follows from the discussion in section 8.7. Let M be one of the lattices M7, . . . ,M12,r and T = DiagM the corresponding torus. We will determine the minimal rank of a permutation ZG-lattice P admitting a homomorphism P → M which becomes surjective after localization at p. Then we conclude ed(T ; p) = rk P− rk M with Corollary 8.1. 8 Algebraic Tori 71 Table 8.2: Essential dimension of tori split by Cp2 . M rk M ed(T ) M1 1 0 M2 p 0 M3 p−1 1 M4 p2 0 M5 p2−1 1 M6 p2 1 M rk M ed(T ) M7 p2− p p M8 p2− p+1 p−1 M9,r p2 p M10,r p2 +1 p−1 M11,r p2−1 p+1 M12,r p2 p We have the bounds rk M ≤ rk P≤ p2 (or p2 + p), (8.11) where the upper bound holds since every M is given as a quotient of ZG (or ZG⊕ZH). Let C = Splitk(T [p]) the finite constant group used in the proof of Theorem 8.2. The rank of C determines exactly the number of direct summands into which P decomposes. Moreover each indecomposable summand has rank a power of p. As an example, we show how to find C for M = M11,r: The relations g j · (ε− (1−h)r);δH are written out as p−1 ∑ i=0 gpi+ j− r ∑ ℓ=0 ( r ℓ ) (−1)ℓhℓ+ j, 0≤ j ≤ p−1; p−1 ∑ i=0 hi and the ksep-point of the torus are T (ksep) = { (t0, . . . , tp2−1,s0, . . . ,sp−1) | p−1 ∏ i=0 tpi+ j = r ∏ ℓ=0 s (−1)ℓ(rℓ) ℓ+ j , 0≤ j ≤ p−1; p−1 ∏ i=0 si = 1 } and C is the constant group of fixed points of the p-torsion T [p]: C(k) = {(ζ ip, . . . ,ζ ip,ζ jp, . . . ,ζ jp) | 0≤ i, j ≤ p−1}≃ µ2p. (Note that the primitive pth root of unity ζp is in k by our assumption that k is p-closed). For other lattices this is similar: C is equal to Splitk(Diag(P)[p])≃ µsp 8 Algebraic Tori 72 Table 8.3: Permutation ranks for lattices M7-M12. M rk C rk M possible rk P M7 1 p2− p p2 M8 1 p2− p+1 p2 M9,r 2 p2 p2 +1 or p2 + p M10,r 2 p2 +1 p2 +1 or p2 + p M11,r 2 p2−1 p2 +1 or p2 + p M12,r 2 p2 p2 +1 or p2 + p where M is presented as a quotient P/N of a permutation lattice P (of minimal rank) as in table 8.3 and where s denotes the number of summands in a decom- position of P. We need to exclude the possibility rk P = p2 + 1 for the lattices M = M9,r, . . . ,M12,r. We can only have the value p2 +1 if there exists a character in M which is fixed under the Galois group and nontrivial on C. The following Lemma 8.6 tells us, that such characters do not exist in either case. Hence the minimal dimension of a p-faithful representation of all these tori is p2 + p. Lemma 8.6. For i = 9, . . . ,12 and r ≥ 1 every character χ ∈ Mi,r fixed under G has trivial restriction to C. Proof. By [Hil85] the cohomology group H0(G,Mi,r) = MGi,r of G-fixed points in Mi,r is trivial for i = 11, has rank 1 for i = 9,12 and rank 2 for i = 10, respectively. They are represented by ZδH in M9,r, by Z(ε − (1− h)r) in M12,r and by Z(ε − (1−h)r)⊕ZδH in M10,r, respectively. Since all these characters are trivial on C = Splitk(Diag(ZG⊕ZH)[p]), the claim follows. 73 9 Normalizers of Maximal Tori The normalizer of a torus in a reductive group is a (FT)-group, thus we can ap- ply some of the techniques from the previous chapters to obtain information on its essential dimension. Besides being of independent interest, normalizers N of maximal tori in a group G are a good source to obtain upper bounds on the essen- tial dimension of G, in fact many of the best known upper bounds on the essential dimension of semisimple groups were obtained this way; cf. [Rei00], [LR00] or [Lem04]. Indeed, if G is smooth and reductive, for any field K/k the natural map H1(K,N)→ H1(K,G) (9.1) is surjective, cf. [Ser97, III Lemma 6] or [CGR08, Cor. 5.3] for full generality. Via Propositions 4.1 and 4.2 we obtain ed(N; p)≥ ed(G; p) ed(N)≥ ed(G) , (9.2) where p is a prime different from the characteristic of k and the second inequality follows from [BF03, 1.9]. The case where G = PGLn the projective linear group, is of particular interest to us. In that case, as a maximal split torus we have the diagonal in GLn mod out by the center ∆ = Gm (of GLn). T = Gmn /∆ . (9.3) The normalizer N = NPGLn(T ) = T ⋊Sn is the semidirect product of T with the Weyl group Sn (the latter we can think of the subgroup of permutation matrices in GLn). We can interpret the functor H1(∗,N) = TorsN in a different way. Define a functor Fields/k → Sets by N(K) = { (A,E) A central simple of degree n over KE ⊂ A a maximal étale subalgebra } /≃ (9.4) 9 Normalizers of Maximal Tori 74 One checks that the automorphism group of a pair (A,E) is exactly N = T ⋊Sn and we have ed(N) = ed(N) = ed(H1(∗,N)) = ed(TorsN) . The functor N is often more accessible than H1(∗,PGLn) = CSAn because many of the standard constructions in the theory of central simple algebras depend on the choice of a maximal subfield L in a given central simple algebra A/K. Projecting a pair (A,L) to the first component, we obtain a surjective morphism of functors H1(∗,N)→ H1(∗,PGLn), which gives another interpretation of (9.1). In this section we compute the essential p-dimension of N. Theorem 9.1. Let N the normalizer of a maximal split torus in the projective linear group PGLn defined over a field k with char(k) 6= p. Then (a) ed(N; p) = [n/p], if n is not divisible by p. (b) ed(N; p) = 2, if n = p. (c) ed(N; p) = n2/p−n+1, if n = pr for some r ≥ 2. (d) ed(N; p) = pe(n− pe)−n+1, in all other cases. Here [n/p] denotes the integer part of n/p and pe denotes the highest power of p dividing n. We remark that our proof of the upper bounds on ed(N; p) in part (c) and (d) does not use the assumption that char(k) 6= p. These bounds are valid for every base field k. As explained above Theorem 9.1 yields an upper bound on the essential p- dimension of PGLn, in particular, since the computation of ed(PGLn; p) reduces to the case where n is a power of p (see Chapter 11), ed(PGLpr ; p)≤ p2r−1− pr +1 (9.5) for any field k and for any r ≥ 2. In Chapter 11 this bound will be improved. Remark. It would be of interest to prove an analogue of Theorem 9.1 in the more general setting, where N is the normalizer of a split maximal torus in an arbitrary simple (or semisimple) linear algebraic group G. The new technical difficulty one encounters in this more general setting is that the natural sequence 1→ T → N →W → 1 , 9 Normalizers of Maximal Tori 75 may not split. Here T is a split maximal torus and W = N/T is the Weyl group of G. The fact that this sequence splits for G = PGLn is an important ingredient in our proof of the upper bound on ed(N; p). 9.1 First Reductions and Proof of Theorem 9.1 Parts (a) and (b) Let Pn be a Sylow p-subgroup of Sn. Theorem 5.2 tells us that ed(N; p) = ed(T ⋊Pn; p) . Note that T ⋊Pn is a p-(FT)-group so that we have upper and lower bounds on ed(T ⋊Pn; p) given by (8.6). In all cases (a)-(d) we will construct a generically free T ⋊Pn-representation V and prove that in fact ed(T ⋊Pn) = ed(T ⋊Pn; p) = dimV −dimN. so the upper bound in (8.6) is sharp and the statement of Conjecture 8.2 is satisfied for this group. In the general cases (c) and (d) we will show that the generically free representation we construct is minimal among all faithful representations, thus our lower bound follows from Theorem 8.2. Note that dimN = dimT ⋊Pn = dimN = n−1. Also, by Remark 4.1 we may assume without loss of generality that k contains a primitive pth root of unity. We recall that the character lattice X(T ) is naturally isomorphic to {(a1, . . . ,an) ∈ Z n |a1 + · · ·+an = 0} , where we identify the character (t1, . . . , tn)→ t a1 1 . . .t an n of T = Gmn /∆ with (a1, . . . ,an) ∈ Zn. Note that (t1, . . . , tn) is viewed as an el- ement of Gmn modulo the diagonal subgroup ∆, so the above character is well defined if and only if a1 + · · ·+an = 0. An element σ of Sn (and in particular, of Pn ⊂Sn) acts on a = (a1, . . . ,an) ∈ X(T ) by naturally permuting a1, . . . ,an. For notational convenience, we will denote by ai, j the element (a1, . . . ,an) ∈ X(T ) such that ai = 1, a j =−1 and ah = 0 for every h 6= i, j. 9 Normalizers of Maximal Tori 76 We also recall that for n = pr the Sylow p-subgroup Pn of Sn can be de- scribed inductively as the wreath product Ppr ∼= Ppr−1 ≀Z/p∼= (Ppr−1) p⋊Z/p . For general n, Pn is the direct product of certain Ppr , see section 9.3. Proof of Theorem 9.1(a). Since n is not divisible by p, we may assume that Pn is contained in Sn−1, where we identify Sn−1 with the subgroup of Sn consisting of permutations σ ∈Sn such that σ(1) = 1. For the upper bound we construct a generically free linear representation V of T ⋊Pn of dimension n−1+[n/p]. Let Λ = {a1,i | i = 2, . . . ,n} and VΛ be as in Corollary 3.4. We have dimVΛ = |Λ| = n− 1 and the representation is faithful on T since Λ generates X(T ). Let W be a [n/p]-dimensional faithful linear representation of Pn constructed in the proof of Corollary 5.1 Applying Corollary 3.1, we see that V = VΛ×W is generi- cally free. This proves the upper bound. For the lower bound, note that since the natural projection p : T ⋊Pn →Pn has a section, so does the map p∗ : H1(K,T ⋊Pn)→ H1(K,Pn) of Galois coho- mology sets. Hence, p∗ is surjective for every field K/k. This implies that ed(T ⋊Pn)≥ ed(Pn; p) = ed(Sn; p) = [n/p] . by Theorem 5.2 and Corollary 5.1. Remark. We will now outline a different and perhaps more conceptual proof of the upper bound ed(N; p)≤ [n/p] of Theorem 9.1(a). As we pointed out in (9.4), ed(N; p) is the essential dimension at p of the functor N of pairs. Similarly, ed(Sn; p) is the essential dimension at p of the functor H1(∗,Sn) which can be interpreted as the functor Etn(K) = { ´Etale K-algebras of degree n } /≃ (9.6) Let α : Etn→N be the map taking an n-dimensional étale algebra L/K to (EndK(L),L). Here we embed L in EndK(L)≃ Mn(K) via the regular action of L on itself. It is easy to see that, over the p-closure K(p), α is surjective; indeed, any algebra A of degree n (n not divisible by p) is split over K(p). By Proposition 4.2, we conclude that ed(N; p) ≤ ed(Sn; p). Combining this with Corollary 5.1 yields the desired inequality ed(N; p)≤ [n/p]. 9 Normalizers of Maximal Tori 77 Proof of Theorem 9.1(b). Here n = p and Pn ≃ Z/p is generated by the p-cycle (1,2, . . . ,n). Let Λ = {a1,2, . . . ,ap−1,p,ap,1} and VΛ be as in Corollary 3.4. Let V = VΛ×L, where L is a 1-dimensional faithful representation of Pn ≃ Z/p and T ⋊Pn acts on L via the natural projection T ⋊Pn →Pn. Note that dim(V ) = |Λ|+1 = n+1. Since Λ generates X(T ), Corollary 3.1 tells us that V is a generi- cally free representation of T ⋊Pn. The lower bound follows from ed(T ⋊Pp) = ed(N; p)≥ ed(PGLp; p) = 2; see [RY00]. In order to prove Theorem 9.1(c) and (d) it suffices to establish the following proposition. Proposition 9.1. Assume that k is of characteristic 6= p and n 6= p is divisible by p. There is a generically free representation V of T ⋊Pn which is minimal among all faithful representations. Its dimension is (a) dimV = n2/p, if n = pr for some r ≥ 2. (b) dimV = pe(n− pe), if pe is the highest power of p dividing n (n 6= pe;e≥ 1). This will be proved in the next two sections. 9.2 Proof of Theorem 9.1 Part (c) Assume in this section that n = pr for some r ≥ 2. Construction of the representation For the upper bound we need to construct a generically free representation V of T ⋊Pn of dimension p2r−1. Our V will be of the form VΛ for a particular Pn- invariant Λ ⊂ X(T ), again as in Corollary 3.4. This construction (and thus the above inequality) will not require any assumption on the base field k. For notational convenience, we will subdivide the integers 1,2, . . . , pr into p “big blocks” B1, . . . ,Bp, where each Bi consists of the pr−1 consecutive integers (i−1)pr−1 +1,(i−1)pr−1 +2, . . . , ipr−1. 9 Normalizers of Maximal Tori 78 We define Λ⊂ X(T ) as the Pn-orbit of the element a1,pr−1+1 = (1,0, . . . ,0︸ ︷︷ ︸ B1 ,−1,0, . . . ,0︸ ︷︷ ︸ B2 ,0,0, . . . ,0︸ ︷︷ ︸ B3 , . . . ,0,0, . . . ,0︸ ︷︷ ︸ Bp ) in X(T ). Thus, Λ consists of elements aα ,β , subject to the condition that if α lies in the big block Bi then β has to lie in B j, where j− i ≡ 1 modulo p. There are pr choices for α . Once α is chosen, there are exactly pr−1 further choices for β . Thus |Λ|= pr · pr−1 = p2r−1 . The associated linear representation VΛ of T ⋊Pn has the desired dimension dim(VΛ) = |Λ|= p2r−1 . It remains to prove that VΛ is generically free. By Corollary 3.4 it suffices to show that (i) Λ generates X(T ) as an abelian group and (ii) the Pn action on the kernel of the natural morphism φ : Z[Λ]→ X(T ) is faithful. The elements aα ,β clearly generate X(T ) as an abelian group, as α and β range over 1,2, . . . , pr. Thus in order to prove (i) it suffices to show that SpanZ(Λ) contains every element of this form. Suppose α lies in the big block Bi and β in B j. If j− i ≡ 1 (mod p), then aα ,β lies in Λ and there is nothing to prove. If j− i ≡ 2 (mod p) then choose some γ ∈ Bi+1 (where the subscript i + 1 should be viewed modulo p) and write aα ,β = aα ,γ +aγ,β . Since both terms on the right are in Λ, we see that in this case aα ,β ∈ SpanZ(Λ). Using this argument recursively, we see that aα ,β also lies in SpanZ(Λ) if j− i≡ 3, . . . , p (mod p), i.e., for all possible i and j. This proves (i). To prove (ii), denote the kernel of φ by M. Since Pn is a finite p-group, every normal subgroup of Pn intersects the center of Pn, which we shall denote by Zn. Thus it suffices to show that Zn acts faithfully on M. Recall that Zn is the cyclic subgroup of Pn of order p generated by the product of disjoint p-cycles σ1 · . . . ·σpr−1 = (1 . . . p)(p+1 . . .2p) . . .(p r− p+1, . . . , pr) . 9 Normalizers of Maximal Tori 79 Since |Zn|= p, it either acts faithfully on M or it acts trivially, so we only need to check that the Zn-action on M is non-trivial. Indeed, Zn does not fix the non-zero element a1,pr−1+1 +apr−1+1,2pr−1+1 + · · ·+a(p−1)pr−1+1,1 ∈ Z[Λ] which lies in M. This gives the existence of the generically free representation in Proposition 9.1(a). Minimality Let q := pe, where e≥ 1 if p is odd and e≥ 2 if p = 2. (9.7) be a power of p. The specific choice of e will not be important in the sequel; in particular, the reader may assume that q = p if p is odd and q = 4, if p = 2. Whatever e we choose, q = pe will remain unchanged for the rest of this section. For the purpose of proving a lower bound we may assume that k contains a primitive qth root of unity, using the fact that if we adjoin such a root to the base field, the essential dimension can only go down, see 1.1. Let T [q] = µnq/µq be the q-torsion subgroup of T = Gmn /∆. Suppose W is a faithful G = T ⋊Pn. Restricting to the finite constant group T [q]⋊Pn we see that the dimension of a faithful G-representation is bounded from below by the minimal dimension of a faithful T [q]⋊Pn (which, by Theorem 1.1 is equal to its essential p-dimension). Thus it suffices to show that T [q]⋊Pn does not have a faithful linear repre- sentation of dimension < p2r−1. Corollary 3.2 further reduces this representation- theoretic assertion to the combinatorial statement of Proposition 9.2 below. Before stating Proposition 9.2 we recall that the character lattice of T [q] ≃ µnq/µq is Xn := {(a1, . . . ,an) ∈ (Z/qZ)n |a1 + · · ·+an = 0 in Z/qZ }, where we identify the character (t1, . . . , tn)→ t a1 1 . . .t an n of T [q] with (a1, . . . ,an) ∈ (Z/qZ)n. Here (t1, . . . , tn) stands for an element of µnq , modulo the diagonally embedded µq, so the above character is well defined if and only if a1 + · · ·+an = 0 in Z/qZ. (This is completely analogous to our description of the character lattice of T in the previous section.) Note that Xn depends on the integer q = pe, which we assume to be fixed throughout this section. 9 Normalizers of Maximal Tori 80 Proposition 9.2. Let n = pr and Pn be a Sylow p-subgroup of Sn. If Λ is a Pn-invariant generating subset of Xn then |Λ| ≥ p2r−1 for any r ≥ 1. Our proof of Proposition 9.2 will rely on the following special case of Nakayama’s Lemma [AM69, Proposition 2.8]. Lemma 9.1. Let q = pe be a prime power, M = (Z/qZ)d and Λ be a generating subset of M (as an abelian group). If we remove from Λ all elements that lie in pM, the remaining set, Λ\ pM, will still generate M. Proof of Proposition 9.2. We argue by induction on r. For the base case, set r = 1. We need to show that |Λ| ≥ p. Assume the contrary. In this case Pn is a cyclic p- group, and every non-trivial orbit of Pn has exactly p elements. Hence, |Λ| < p is only possible if every element of Λ is fixed by Pn. Since we are assuming that Λ generates Xn as an abelian group, we conclude that Pn acts trivially on Xn. This can happen only if p = q = 2. Since these values are ruled out by our definition (9.7) of q, we have proved the proposition for r = 1. In the previous section we subdivided the integers 1,2, . . . , pr into p “big blocks” B1, . . . ,Bp of length pr−1. Now we will now work with “small blocks” b1, . . . ,bpr−1 , where b j consists of the p consecutive integers ( j−1)p+1,( j−1)p+2, . . . , jp . We can identify Ppr−1 with the subgroup of Ppr that permutes the small blocks b1, . . . ,bpr−1 without changing the order of the elements in each block. For the induction step, assume r≥ 2 and consider the homomorphism Σ : Xpr → Xpr−1 given by a = (a1,a2, . . . ,apr) 7→ s = (s1, . . . ,spr−1) , (9.8) where si = a(i−1)p+1 +a(i−1)p+2 + . . .+aip is the sum of the entries of a in the ith small block bi. Thus (i) if Λ generates Xpr then Σ(Λ) generates Xpr−1 . (ii) if Λ is a Ppr-invariant subset of Xpr then Σ(Λ) is a Ppr−1-invariant subset of Xpr−1 . Let us remove from Σ(Λ) all elements which lie in pXpr−1 . The resulting set, Σ(Λ)\ pXpr−1 , is clearly Ppr−1-invariant. By Lemma 9.1 this set generates Xpr−1 . Thus by the induction assumption |Σ(Λ)\ pXpr−1| ≥ p2r−3. We claim that the fiber of each element s = (s1, . . . ,spr−1) in Σ(Λ)\ pXpr−1 has at least p2 elements in Λ. If we can show this, then we will be able to conclude that |Λ| ≥ p2 · |Σ(Λ)\ pXpr−1| ≥ p2 · p2r−3 = p2r−1 , 9 Normalizers of Maximal Tori 81 thus completing the proof of Proposition 9.2. Let σi be the single p-cycle, cyclically permuting the elements in the small block bi. To prove the claim, note that the subgroup 〈σi | i = 1, . . . , pr−1〉 ≃ (Z/pZ)p r−1 of Pn acts on each fiber of Σ. To simplify the exposition in the argument to follow, we introduce the follow- ing bit of terminology. Let us say that a ∈ (Z/qZ)n is scalar in the small block bi if all the entries of a in the block bi are the same, i.e., if a(i−1)p+1 = a(i−1)p+2 = · · ·= aip . We are now ready to prove the claim. Suppose a = (a1, . . . ,apr) ∈ Xpr lies in the preimage of s = (s1, . . . ,spr−1), as in (9.8). If a is scalar in the small block bi then clearly si = a(i−1)p+1 +a(i−1)p+2 + · · ·+aip ∈ pZ/qZ . Since we are assuming that s lies in Σ(Λ)\ pXpr−1 , s must have at least two entries that are not divisible by p, say, si and s j. (Recall that s1 + · · ·+ spr = 0 in Z/qZ, so s cannot have exactly one entry not divisible by p). Thus a is non-scalar in the small blocks bi and b j. Consequently, the elements σαi σ β j (a) are distinct, as α and β range between 0 and p− 1. All of these elements lie in the fiber of s under Σ. Therefore we conclude that this fiber contains at least p2 distinct elements. This completes the proof of the claim and thus of Proposition 9.2, Proposition 9.1(c) and Theorem 9.1(c). 9.3 Proof of Theorem 9.1 Part (d) In this section we assume that n is divisible by p but is not a power of p. We will modify the arguments of the last section. Throughout, let pe be the highest power of p dividing n. Write out the p-adic expansion n = n1 pe1 +n2 pe2 + ...+nu peu , (9.9) of n, where 1 ≤ e = e1 < e2 < ... < eu, and 1 ≤ ni < p for each i. Subdivide the integers 1, ...,n into n1 + ...+ nu blocks Bij of length pei , for j ranging over 9 Normalizers of Maximal Tori 82 1,2, ...,ni. By our assumption there are at least two such blocks. The Sylow subgroup Pn is a direct product Pn = (Ppe1 ) n1 ×·· ·× (Ppeu ) nu where each Ppei acts on one of the blocks Bij. Construction of the representation We construct a generically free representation of T ⋊Pn of dimension pe1(n− pe1). Again this construction does not require any assumption on the field k. Let Λ⊂ X(T ) be the union of the Pn-orbits of the elements a1, j+1 where j = pe1, ...,n1 pe1,n1 pe1 + pe2, .....,n− peu i.e., the union of the Pn-orbits of elements of the form (1,0 . . . ,0,−1,0, . . . ,0), where 1 appears in the first position of the first block and −1 appears in the first position of one of the other blocks. For aα ,β in Λ there are pe1 choices for α and n− pe1 choices for β . Thus dim(VΛ) = |Λ|= pe1(n− pe1) , where VΛ is the associated representation in Corollary 3.4. It is not difficult to see that Λ generates X(T ) as an abelian group. To conclude with Corollary 3.4 that VΛ is a generically free representation of T ⋊Pn, it remains to show that the Pn-action on the kernel of the natural morphism φ : Z[Λ]→ X(T ) is faithful when e1 ≥ 1. As in the previous section we only need to check that the center Zn of Pn acts faithfully on the kernel. Let σ be a non trivial element of Zn = (Zpe1 )n1 ×·· ·× (Zpeu )nu , with each Zpei cyclic of order p. Let h,h′ be in the first block B11 and l, l′ in some other block B j i (there are at least two blocks each of size at least p). The element a = ah,l −ah,l′ +ah′,l′−ah′,l lies in the kernel of φ . To fix a, σ must either (1) fix all h,h′, l, l′ or (2) σ(h) = h′,σ(h′) = h and σ(l) = l′,σ(l′) = l. Since σ is nontrivial we may choose B ji such that (1) is not possible and if p 6= 2, (2) is not possible either. If p = 2, by (14), B ji is at least of size 4 and we can choose l, l′ within B ji such that (2) does not hold. Therefore σ does not fix a nonzero element of the kernel of φ . 9 Normalizers of Maximal Tori 83 Minimality Arguing as in Section 9.2 (and using the same notation, with q = p), it suffices to show that every faithful representation of T(p)⋊Pn has dimension≥ pe1(n− pe1), or equivalently Lemma 9.2. Let Xn := {(a1, . . . ,an) ∈ (Z/pZ)n |a1 + · · ·+an = 0 in Z/pZ }. Then every Pn-invariant generating subset of Xn has at least pe(n− pe) elements. In the statement of the lemma we allow e = 0, to facilitate the induction argu- ment. For the purpose of proving minimality in Proposition 9.1(d) we only need this lemma for e≥ 1. Proof. Once again, we consider the p-adic expansion (9.9) of n, with 0 ≤ e1 < e2 < ... < eu and 1 ≤ ni < p. We may assume that n is not a power of p, since otherwise the lemma is vacuous. We will argue by induction on e = e1. For the base case, let e1 = 0. Here the lemma is obvious: since Xn has rank n−1, every generating set (Pn-invariant or not) has to have at least n−1 elements. For the induction step, we may suppose e = e1 ≥ 1; in particular, n is divisible by p. Define Σ : Xn → Xn/p by sending (a1, ....,an) to (s1, ...,sn/p), where s j = a( j−1)p+1 + · · ·+a jp for j = 1, . . . ,n/p. Arguing as before we see that Σ(Λ)\ pXn/p is a (Ppe1−1)n1 × ·· ·× (Ppeu−1) nu -invariant generating subset of Xn/p and that every s ∈ Σ(Λ)\ pXn/p has at least p2 preimages in Λ. By the induction assumption, |Σ(Λ)\ pXn/p| ≥ pe−1( n p − pe−1) and thus |Λ| ≥ p2 · pe−1(n p − pe−1) = pe(n− pe) This completes the proof of Lemma 9.2 and thus of Proposition 9.1 and of Theo- rem 9.1. 84 10 Central Simple Algebras As noted there is an equivalence of the functors H1(∗,PGLn) and CSAn. In this chapter we work on the side of central simple algebras. For background material on central simple algebras we refer to [Pie82], [Row80, 3] or [Sal99]. We intro- duce some new functors which describe classes of simple algebras with additional structure, such as crossed product algebras or algebras split by a distinguished Galois extension. These will help us obtain information on ed(CSAn). The main result of this chapter is the following theorem (and its variants The- orems 10.2, 10.4). Theorem 10.1. Let A/K be a central simple algebra of degree n. Suppose A con- tains a field F, Galois over K and Gal(F/K) can be generated by r ≥ 1 elements. If [F : K] = n then we further assume that r ≥ 2. Then ed(A)≤ r n 2 [F : K] −n+1 . Note that we always have [F : K]≤ n. In the special case where equality holds, i.e., A is a crossed product in the usual sense, Theorem 10.1 reduces to [LRRS03, Corollary 3.10(a)]. Let us assume for the moment that the central simple algebra A/K in Theo- rem 10.1 has a separable maximal subfield L/K, containing the Galois extension F/K. We will denote the Galois closure of L over K by E and the associated Galois groups by G = Gal(E/K), H = Gal(E/L) and N = Gal(E/F), as in the diagram 10 Central Simple Algebras 85 below. E        H z       N        G ' % # !            A ?? ?? ?? ?? L F G/N px  F N K In the terminology of [LRRS03], A/K is a G/H-crossed product; cf. also [FSS94, Appendix]. Note that since E/K is the smallest Galois extension containing L/K, we have CoreG(H) = ⋂ g∈G Hg = {1} . (10.1) where Hg := gHg−1. We will assume that this condition is satisfied whenever we talk about G/H-crossed products. In general no (strictly) maximal subfield L needs to exist and it is more natural to allow étale algebras, in particular if the property of being a G/H-crossed prod- uct needs to be preserved under scalar extensions (as in Definition 10.2). However, for our proves we may always assume that L and E are fields as the Lemma 10.1 below shows. Lemma 10.1. In the course of proving Theorem 10.1 we may assume without loss of generality that F is contained in a subfield L of A such that L/K is a separable extension of degree n = deg(A). Proof. Note that we are free to replace K by K(t), F by F(t) and A by A(t) = A⊗K K(t), where t is an independent variable. Indeed, edk A(t) = edk(A); see, e.g., [LRRS03, Lemma 2.7(a)]. Thus if the inequality of Theorem 10.1 is proved for A(t), it will also hold for A. The advantage of passing from A to A(t) is that K(t) if Hilbertian for any infinite field K; see, e.g., [FJ08, Proposition 13.2.1]. Thus after adjoining two variables, t1 and t2 as above, we may assume without loss of generality that K is Hilbertian. (Note that a subfield L ⊂ A of degree n over K may not exist without this assumption). 10 Central Simple Algebras 86 Let F ⊂ F ′ be maximal among separable field extension of F contained in A. We will look for L inside the centralizer CA(F ′). By the Double Centralizer Theorem, CA(F ′) is a central simple algebra with center F ′. The maximality of F ′ tells us that CA(F ′) contains no non-trivial field extensions of F ′. In particular, CA(F ′) = Mr(F ′), where r[F ′ : K] = n. On the other hand, since K is Hilbertian, so is its finite separable exten- sion F ′; cf. [FJ08, 12.2.3]. Consequently, F ′ admits a finite separable exten- sion L/F ′ of degree r. To construct L/F ′, start with the field extension Lr = F ′(t1, . . . , tr)[x]/( f (x)) of F ′(t1, . . . , tn) of degree r, where f (x) = xr + t1xr−1 + . . .+ tn−1x + tn is the general polynomial of degree r. Then specialize t1, . . . , tr in F ′, using the Hilbertian property, to obtain a field extension L/F ′ of degree r. Any such L/F ′ can be embedded into Mr(F ′) via the regular representation of L on L = (F ′)r; cf. [Pie82, Lemma 13.1a]. By the maximality of F ′, we conclude that L = F ′, i.e., r = 1 and [L : K] = n, as desired. Since [G : H] = [L : K] = deg(A) = n, and n [F : K] = [L : F ] = [N : H], we can restate Theorem 10.1 as follows. Theorem 10.2. Let A be a G/H-crossed product. Suppose H is contained in a normal subgroup N of G and G/N is generated by r elements. Furthermore, assume that either H 6= {1} or r ≥ 2. Then ed(A)≤ r[G : H] · [N : H]− [G : H]+1 . 10.1 Crossed Products, Splitting Groups and Fields Recall the following definitions, cf. [Pie82, 13.2], [TA85, 6.1]: Definition 10.1. An étale algebra E/K is said to split the algebra A/K, if E = L1×·· ·×Lr for some fields Li/K and ALi := A⊗K Li ≃ Mn(Li) is split for each i = 1, . . . ,r. Alternatively the Azumaya algebra A⊗K E ≃Mn(E) is split. A finite group G is called splitting group of A/K if there exists a Galois étale algebra E/K with Galois group Gal(E/K)⊂ G such that E splits A. In the diagram above, L and E are splitting fields (see [Pie82, 13.3]) and G a splitting group. This gives rise to the following functors Fields/k → Sets: Definition 10.2. Let K/k be a field. Define 10 Central Simple Algebras 87 (a) Suppose E/k is an étale algebra. Splitn,E/k(K)= {deg. n central simple K-algebras, split by EK = E⊗k K}/≃ (b) Let G⊂Sn a finite group and H ⊂ G of index n. CPG/H(K) = {G/H-crossed product algebras over K}/≃ (c) More generally, let G be any finite group. Splitn,G(K) = {Central simple K-algebras of degree n, split by G}/≃ (d) G as before. Pairsn,G(K) = { (A,E) A central simple of degree n over KE/K G-Galois algebra, splitting A } /≃ Remark. Some similar functors are discussed in [Mer09]. If one starts say with a field extension L/K which not necessarily descends to a field extension over k, it is sometimes useful to consider the functor Splitn,L/K : Fields/K → Sets over the larger base field K. In everything that follows we assume that G⊂Sn, H ⊂G of index n. Let E/k be a G-Galois algebra, the functors are related as follows: Pairsn,G // // Splitn,G = CPG/H   // CSAn Splitn,E/k ? OO (10.2) Here the first map sends a pair (A,E) to A and the fact that Splitn,G = CPG/H is easily checked, using e.g. [Pie82, Cor. 13.3]. By [BF03, 1.9] and Proposi- tion 4.2 we have Lemma 10.2. ed(Pairsn,G)≥ ed(Splitn,G) = ed(CPG/H) The same inequality holds for essential p-dimension. 10 Central Simple Algebras 88 If F is a subfunctor of G and α ∈ F(K) ⊂ G(K), then we can calculate the essential dimension of α with respect to both of these functors (which we will indicate by superscripts) and we have edF(α)≥ edG(α), edF(α; p)≥ edG(α; p). (10.3) Now we can do the following: If we start with an arbitrary central simple K- algebra A ∈ CSAn(K) of degree n, let L be a maximal subfield (or étale algebra) and E a Galois closure and suppose G is the Galois group Gal(E/K). Thus A ∈ Splitn,G(K) and (A,E) ∈Pairsn,G(K). By the lemma above and the definition of essential dimension we have edCSA(A)≤ edSplitn,G(A)≤ ed(A,E)≤ ed(Pairsn,G). (10.4) Let T = Gmn /∆ be a maximal split torus in PGLn, see also Chapter 9. Now G ⊂ Sn acts on Gmn by permuting the elements in the product and this action descends to T . We can thus form the semidirect product T⋊G which is a subgroup of the normalizer N = T ⋊Sn and thus of PGLn. Lemma 10.3. There is an equivalence of functors Pairsn,G = H1(∗,T ⋊G). Proof. Let (Mn(K),Z[G]⊗K) be the split element in Pairsn,G(K). An automor- phism of the pair must preserve (Z[G]⊗K)H = Z[G/H]⊗K which is of degree n and we can think of a maximal commutative subalgebra of Mn(K). Ones checks that the automorphism group of this pair is exactly T ⋊G. We conclude with [KMRT98, 29.12]. Corollary 10.1. Assume A ∈CSAn(K) is split by a G-Galois algebra E/K. Then ed(A)≤ ed(T ⋊G). Example. Let G = Sn then T ⋊Sn = N is the normalizer studied in the previous chapter. We have Pairsn,Sn = H 1(∗,N) = N. where N was defined as the functor of pairs of an algebra with a distinguished maximal étale subalgebra. 10 Central Simple Algebras 89 Suppose we start again with an arbitrary central simple K-algebra A∈CSAn(K), L is a maximal subfield (or étale algebra) and E a Galois closure. If in addi- tion E/K is defined over k, i.e. there exists an étale algebra E0/k which be- comes isomorphic to E over K, then A ∈Splitn,E0/k(K) and we have ed CSA(A)≤ edSplitn,E0/k(A) ≤ ed(Splitn,E0/k). The condition that E/K is defined over k is of course a strong restriction. The functor Splitn,E/k becomes most interesting by the following observation, due to Merkurjev [Mer09], which relates it to a certain torus (for which we have Theorem 8.3 to compute the essential dimension): For any étale algebra E/k the Weil restriction RE/k(Gm) is a torus over k. Let T be the torus RE/K(Gm)/Gm where Gm embeds diagonally. Lemma 10.4 ([Mer09, (7)]). There is an equivalence of functors Splitn,E/k = H1(∗,T ) . 10.2 G-Lattices In the sequel H ≤G will be finite groups. Given g ∈G we will write g for the left coset gH of H. We will denote the identity element of G by 1. Here again we will reduce the task to a lattice theoretic problem. This is achieved directly via the theorem below from [LRRS03]. Any finite set X with a G-action gives rise to a permutation G-lattice Z[X ]. Of particular interest to us will be the G-lattice ω(G/H), which is defined as the kernel of the natural augmentation map Z[G/H] → Z, sending n1g1 + · · ·+ nsgs to n1 + · · ·+ns. Theorem 10.3. [LRRS03, Theorem 3.5] Let P be a permutation G-lattice and 0→M → P→ ω(G/H)→ 0 be an exact sequence of G-lattices. If the G-action on M is faithful then ed(A)≤ rk (P)−n+1 for any G/H-crossed product A. 10 Central Simple Algebras 90 The theorem is easy to deduce from the preparations made earlier here and we include a short proof. Proof. Let A ∈ CPG/H(K) be a G/H-crossed product of degree n over K and E a G-Galois algebra which is the Galois closure of a maximal étale subalgebra of A. By Corollary 10.1 and Corollary 3.3, ed(A)≤ ed(Pairsn,G) = ed(T ⋊G)≤ rk M = rk P−n+1. The condition that G acts faithfully on M is not automatic. However, the following lemma shows that it is satisfied for many natural choices of P. Lemma 10.5. Let G 6= {1} be a finite group H ≤G be a subgroup of G, H1, . . . ,Hr be subgroups of H and 0→M →⊕ri=1Z[G/Hi]→ ω(G/H)→ 0 (10.5) be an exact sequence of G-lattices. Assume that H does not contain any nontrivial normal subgroup of G (i.e., H satisfies condition (10.1) above). Then the G-action on M fails to be faithful if and only if s = 1 and H1 = H. Here we are not specifying the map ⊕ri=1Z[G/Hi] → ω(G/H); the lemma holds for any exact sequence of the form (10.5). We also note that in the case where H1 = · · ·= Hr = {1}, Lemma 10.5 reduces to [LRRS03, Lemma 2.1]. Proof. To determine whether or not the G-action on M is faithful, we may replace M by MQ := M⊗Q. After tensoring with Q, the sequence (10.5) splits, and we have an isomorphism ω(G/H)Q⊕MQ ≃⊕ri=1Q[G/Hi] . (10.6) Case 1: r ≥ 2. Then Hr is a subgroup of H, we have a natural surjective map Q[G/Hr] → Q[G/H]. Using complete irreducibility over Q once again, we see that Q[G/H] (and hence ω(G/H)) is a subrepresentation of Q[G/Hr]. Thus (10.6) tells us that Q[G/Hr−1] is a subrepresentation of MQ. The kernel of the G-representation on Q[G/Hr−1] is a normal subgroup of G contained in Hr−1 (and hence, in H); by our assumption on H, any such subgroup is trivial. This shows that G acts faithfully on Q[G/Hr−1] and hence, on M. 10 Central Simple Algebras 91 Case 2: Now assume r = 1. Our exact sequence now assumes the form 0→MQ→Q[G/H1]→ ω(G/H)Q→ 0 . If H = H1 then M ≃ Z, with trivial (and hence, non-faithful) G-action. Our goal is thus to show that if H1 ( H then the G-action on MQ is faithful. Denote byQ[1] the trivial representation (it will be clear from the context of which group). Observe that Q[G/H1] ≃ IndGH1Q[1] ≃ Ind G H IndHH1Q[1] ≃ Ind G HQ[H/H1] ≃ IndGH(ω(H/H1)Q⊕Q[1]) ≃ IndGH ω(H/H1)Q ⊕ Q[G/H] ≃ IndGH ω(H/H1)Q ⊕ ω(G/H)Q ⊕ Q[1] and we obtain MQ ≃ IndGH ω(H/H1)Q ⊕ Q[1] . If H1 ( H then the kernel of the G-representation IndGH ω(H/H1)Q is a normal subgroup of G contained in H1 (and hence, in H). By our assumption on H, this kernel is trivial. 10.3 An Upper Bound In this section we will prove the following upper bound on the essential dimension of a G/H-crossed product. We will say that g1, . . . ,gs ∈ G generate G over H if G = 〈g1, . . . ,gs,H〉. Theorem 10.4. Let A be a G/H-crossed product. Suppose that (i) g1, . . . ,gs ∈ G generate G over H, and (ii) if G is cyclic then H 6= {1}. Then ed(A)≤ ∑si=1[G : (H ∩Hgi)]− [G : H]+1. Remark. The index [G : (H∩Hgi)] appearing in the above formula can be rewritten as [G : H] · [H : (H ∩Hgi)] = [G : H] · [(H ·Hgi) : H] ; see, e.g., [Hup67, I 2.12]. Note H ·Hg := {hh′ |h ∈ H, h′ ∈ Hg} is a subset of G but may not be a subgroup, and [(H ·Hg) : H] is defined as |H ·H g| |H| . 10 Central Simple Algebras 92 If H is contained in a normal subgroup N of G then clearly H ·Hg lies in N, each [H ·Hg : H]≤ [N : H] and thus Theorem 10.4 yields ed(A)≤ s[G : H] · [N : H]− [G : H]+1 . This is a bit weaker than the inequality of Theorem 10.2, even though the two look very similar. The difference is that we have replaced r in the inequality of Theorem 10.2 by s, where G is generated by s elements over H and by r elements over N. A priori r can be smaller than s. Nevertheless in the next section we will deduce Theorem 10.2 from Theorem 10.4 by a more delicate argument along these lines. Our proof of Theorem 10.4 will rely on the following lemma. Lemma 10.6. Let V be a Z[G]-submodule of ω(G/H). Then GV := {g ∈ G |g−1 ∈V} is a subgroup of G containing H. Proof. The inclusion H ⊂ GV is obvious from the definition. To see that GV is closed under multiplication, suppose g,g′ ∈GV . That is, both g−1 and g′−1 lie in V . Then gg′−1 = g · (g′−1)+(g−1) also lies in V , i.e., gg′ ∈ GV , as desired. Proof of Theorem 10.4. We claim that the elements g1 − 1, . . . ,gs − 1 generate ω(G/H) as a Z[G]-module. Indeed, let V be the Z[G]-submodule of ω(G/H) generated by these elements. Lemma 10.6 and condition (i) tell us that V contains g−1 for every g ∈G. Trans- lating these elements by G, we see that V contains a−b for every a,b∈G. Hence, V = ω(G/H), as claimed. For i = 1, . . . ,s, let Si := {g ∈ G |g · (gi−1) = gi−1} be the stabilizer of gi−1 in G. We may assume here that gi is not in H, otherwise it could be removed since it is not needed to generate G over H. Then clearly 10 Central Simple Algebras 93 g ∈ Si iff ggi = gi and g = 1. From this one easily sees that Si = H∩Hgi . Thus we have an exact sequence 0→M →⊕si=1Z[G/Si] φ −→ ω(G/H)→ 0 where φ sends a generator of Z[G/Si] to gi− 1 ∈ ω(G/H). By Theorem 10.3 it remains to show that G acts faithfully on M. By Lemma 10.5 G fails to act faithfully on M if and only if r = 1 and S1 = H = Hg1 . But this possibility is ruled out by (ii). Indeed, assume that s = 1 and S1 = H = Hg1 . Then G = 〈g1,H〉 and H = Hg1 . Hence, H is normal in G. Condition (10.1) then tells us that H = {1}. Moreover, in this case G = 〈g1,H〉= 〈g1〉 is cyclic, contradicting (ii). 10.4 Proofs of Theorems 10.1 and 10.2 We will prove Theorem 10.2 which is equivalent to Theorem 10.1. Let t1, . . . , tr ∈ G/N be a set of generators for G/N. Choose g1, . . . ,gr ∈ G representing t1, . . . , tr. and let H ′ := 〈H,Hg1, . . . ,Hgr〉. Since H ≤ N and N is normal in G, H ′ ≤ N. The group H ′ depends on the choice of g1, . . . ,gr ∈ G, so that giN = ti. Fix t1, . . . , tr and choose g1, . . . ,gr ∈ G representing them, so that H ′ has the largest possible order or equivalently the smallest possible index in N. Denote this minimal possible value of [N : H ′] by m. In particular m = [N : H ′]≤ [N : (Hgig ·H)] (10.7) for any i = 1, . . . ,r and any g ∈ N. Here [N : (Hgig ·H)] = |N| |Hgig ·H| , as in Re- mark 10.3. Choose a set of representatives 1 = n1,n2, . . . ,nm ∈N for the distinct left cosets of H ′ in N. We claim that the elements {gin j | i = 1, . . . ,r; j = 1, . . . ,m} generate G over H. Indeed, let G0 be the subgroup of G generated by these elements and H. Since n1 = 1, G0 contains g1, . . . ,gr. Hence, G0 contains H ′. Moreover, G0 con- tains n j = g−11 (g1n j) for every j; hence, G0 contains all of N. Finally, since t1 = g1N, . . . , tr = grN generate G/N, we conclude that G0 contains all of G. This proves the claim. 10 Central Simple Algebras 94 We now apply Theorem 10.4 to the elements {gin j}. Substituting [G : H] · [H : (H ·Hgin j)] for [G : (H ∩Hgin j)], as in Remark 10.3, we obtain ed(A)≤ r ∑ i=1 m ∑ j=1 [G : (H ∩Hgin j)]− [G : H]+1 = [G : H] · r ∑ i=1 m ∑ j=1 [(H ·Hgin j) : H]− [G : H]+1 = [G : H] · r ∑ i=1 m ∑ j=1 [N : H] [N : (H ·Hgin j)] − [G : H]+1 ≤ (by (10.7)) [G : H] · r ∑ i=1 m ∑ j=1 [N : H] m − [G : H]+1 = r[G : H] · [N : H]− [G : H]+1 as desired. This completes the proof of Theorem 10.2 and thus of Theo- rem 10.1. 95 11 The Projective Linear Group PGLn Recall that the essential dimension of the projective linear group PGLn is given as ed(PGLn) = ed(H1(∗,PGLn)) = ed(TorsPGLn) = ed(CSAn) where TorsPGLn denotes the functor of PGLn-torsors and CSAn the functor of central simple algebras of degree n, cf. Section 1.1 and Chapter 10. By definition we also have ed(PGLn) = ed(CSAn) = max degA=n ed(A) , the maximum essential dimension that a degree n central simple algebra (over any field K/k) can have. The problem of computing ed(PGLn) was first raised by C. Procesi in the 1960s. Procesi and S. Amitsur constructed so-called universal division algebras UD(n) which have various generic properties among central simple algebras of degree n (see [Pie82, 20.8] or [Row80, 3.2] for the definition of UD(n)). In par- ticular, UD(n) has the rational specialization property: Let Z ⊃ k be the center of UD(n) and A any simple K algebra of degree n with K ⊃ k. Then there exists a field R, both containing Z and K such that R is rational (i.e. purely transcendental) over K and UD(n)⊗Z R≃ A⊗K R. It follows from this that ed(UD(n))≥ ed(A), see [LRRS03, 2.4]. Since A is arbi- trary, UD(n) is an instance of an algebra with maximal essential dimension, ed(PGLn) = ed(UD(n)). (Alternatively, one shows that the PGLn-torsor corresponding to UD(n) is ver- sal). Procesi also showed that the center Z of UD(n) has transcendence degree trdegk Z = n2 +1, thus giving the upper bound ed(PGLn) = ed(UD(n))≤ n2 +1. In fact Procesi himself improved this bound and showed (using different termi- nology) that ed(PGLn)≤ n2, [Pro67, 2.1]. The problem of computing ed(PGLn) 11 The Projective Linear Group PGLn 96 was raised again by B. Kahn in the early 1990s who asked (implicitly in [Kah00, Section 2]) if ed(PGLn) grows sublinearly in n, i.e., whether ed(PGLn)≤ an+b for some positive real numbers a and b. There has been much work to improve the bounds on ed(PGLn) but still very little is known about the exact values of ed(PGLn) (assume chark 6 |n): Table 11.1: Essential dimension of PGLn. n ed(PGLn) reference 2,3,6 2 [Rei00, 9.4] 4 5 [Ros08, 12.4], [Mer10, 1.2] The best known upper bound, ed(PGLn)≤ { (n−1)(n−2) 2 , for every odd n≥ 5 and n2−3n+1, for every n≥ 4 (11.1) (see [LR00], [LRRS03, Theorem 1.1], [Lem04, Proposition 1.6] and [FF08]), is quadratic in n. Recently Merkurjev [Mer09] made a substantial improvement on the lower bound for n = pr a power of a prime: Theorem 11.1. ed(PGLpr)≥ ed(PGLpr ; p)≥ (r−1)pr +1 . The proof of this result relies on our Theorem 8.3. Essential p-dimension of PGLn is a bit more accessible. One first notes that if pr is the largest power of p dividing n then, using primary decomposition of central simple algebras, ed(PGLn; p) = ed(PGLpr ; p), cf. [RY00, 8.5]. Thus for the purpose of computing ed(PGLn; p) it suffices to consider the case where n = pr. Table 11.2 lists the powers of p for which ed(PGLpr ; p) is known (in all cases chark 6= p). For higher powers, a first new upper bound on ed(PGLpr ; p) follows from the computations on the essential dimension of the normalizer of a maximal torus, 11 The Projective Linear Group PGLn 97 Table 11.2: Essential p-dimension of PGLn. n ed(PGLn; p) reference p 2 [RY00, 8.5] p2 p2 +1 [Mer10, 1.2], [MR09a, remark; Cor. 1.2] see (9.5). With a more careful analysis and the results in Chapter 10 we will now establish the following new upper bound 1: Theorem 11.2. Let n = pr for some r ≥ 2. Then ed(PGLn; p)≤ 2 n2 p2 −n+1 . Proof. We will deduce Theorem 11.2 from Theorem 10.1. Let n = pr and A = UD(n). In [RS92, 1.2], L. H. Rowen and D. J. Saltman showed that if r ≥ 2 then there is a finite field extension K′/K of degree prime to p, such that A′ := A⊗K K′ contains a field F , Galois over K′ with Gal(F/K′) ≃ Z/p×Z/p. Thus, if r ≥ 2, Theorem 10.1 tells us that ed(PGLn; p) = ed(A; p)≤ ed(A′)≤ 2 n2 p2 −n+1 . Let us examine some of the ingredients that went into this proof: Let G⊂Sn be a finite group and T a maximal split torus in PGLn. Recall the diagram of functors (10.2) H1(∗,T ⋊G) = Pairsn,G։Splitn,G →֒ CSAn Let G now be a splitting group (see Definition 10.1) of the universal division algebra UD(n) over k. Amitsur [Ami91] showed the following Theorem 11.3. Suppose G is a splitting group of UD(n). Then G is a splitting group for any other central simple algebra A of degree n. 1We just learnt that Ruozzi [Ru10] showed that after a prime to p extension, UD(n) has a Sp×Spr−1 splitting group. Using our methods he sharpened the bound to ed(PGLn; p)≤ n 2 p2 +1. 11 The Projective Linear Group PGLn 98 This follows basically from the rational specialization property. In combina- tion with Lemma 10.2 we obtain Corollary 11.1. Suppose G is a splitting group of UD(n). Then CSAn = Splitn,G. Thus ed(PGLn) = ed(CSAn) = ed(Splitn,G)≤ ed(T ⋊G) The same holds for essential p-dimension. In fact one can show that if G is a splitting group of UD(n) then the map H1(K,T ⋊G)→ H1(K,PGLn) is surjective for any field K/k. We end with the following question: Does there exist a splitting group G of UD(n) such that ed(PGLn) = ed(T ⋊G)? The splitting groups of UD(n) were studied in [TA85], but they appear to be somewhat mysterious. 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