University of California Los Angeles
Algebraic Tori and Essential Dimension
A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics
by
Anthony Michael Ruozzi
2012 c Copyright by Anthony Michael Ruozzi 2012 Abstract of the Dissertation Algebraic Tori and Essential Dimension
by
Anthony Michael Ruozzi Doctor of Philosophy in Mathematics University of California, Los Angeles, 2012 Professor Alexander Merkurjev, Chair
Interest in essential dimension problems has been growing over the past decade. In part, it is because the idea of essential dimension captures quite elegantly the problem of parametrizing a wide range of algebraic objects. But perhaps more, it is because the study of essential dimension requires most of the algebraic arsenal. What began as a problem in Galois co- homology and representation theory now has connections to versal torsors, stacks, motives, birational geometry, and invariant theory. This exposition will focus on just a small bit of this theory: algebraic tori and how they can be used to help us calculate the essential p-dimension for PGLn.
ii The dissertation of Anthony Michael Ruozzi is approved.
Amit Sahai
Paul Balmer
Richard Elman
Alexander Merkurjev, Committee Chair
University of California, Los Angeles 2012
iii Table of Contents
1 Introduction ...... 1
1.1 Basic Notations ...... 2
2 Groups and Algebras ...... 3
2.1 p-closed fields ...... 3
2.2 Central Simple Algebras ...... 4
2.2.1 Etale´ Algebras ...... 5
2.2.2 Division Algebras of Degree ps ...... 5
2.3 Algebraic Groups ...... 8
2.3.1 Groups of Multiplicative Type ...... 9
2.3.2 Galois Cohomology ...... 10
3 Essential Dimension ...... 11
3.1 Definitions ...... 12
3.2 Essential p-dimension ...... 13
3.3 Essential Dimension of Algebraic Groups ...... 14
3.3.1 Examples ...... 15
3.3.2 Essential Dimension of Groups of Multiplicative Type ...... 16
3.3.3 Essential Dimension of Algebraic Tori ...... 17
3.3.4 Essential Dimension of Finite Abelian Groups ...... 28
4 Essential Dimension of PGLn ...... 32
4.1 The Setup ...... 33
4.2 An Upper Bound ...... 34
iv 4.3 A Lower Bound and Some Consequences ...... 38
References ...... 39
v Vita
2003–2006 Teaching Assistant for MATH 151-153 Calculus at the Univer- sity of Chicago.
2006 B.S. (Mathematics), University of Chicago.
2007 M.A. (Mathematics), UCLA, Los Angeles, California.
2006–2009 Teaching Assistant, Mathematics Department, UCLA.
Fall 2009 NSF VIGRE instructorship, Mathematics Department, UCLA.
2010 Research Assistant, Mathematics Department, UCLA.
Fall 2011 NSF VIGRE instructorship, Mathematics Department, UCLA.
2011-present Teaching Assistant, Mathematics Department, UCLA.
Publications
Essential p-dimension of PGLn. Journal of Algebra 238 (2011), no. 1, 488-494.
vi CHAPTER 1
Introduction
Essential dimension is a relatively young idea, but its study has generated much interest. This dissertation will explore a small facet of the essential dimension problem and provide a framework for understanding the computation for algebraic tori. Any new-comer to the field would certainly stumble across with the oft-referenced paper by Berhuy and Favi on the subject, and it still contains much of the basic theory. Endeavoring to not repeat that theory here, I present a slightly different outlook that I hope novices and experts alike can appreciate.
The trajectory of this dissertation is as follows. In Chapter 2, I outline the basic knowl- edge that forms the foundation of all subsequent chapters. For the well-informed reader, this is mostly a refresher. The highlight is a slightly improved version of a theorem of Rowen and Saltman that is useful later for the computation of the essential p-dimension of PGLn.
In Chapter 3, I describe the essential dimension problem. The main focus is on groups of multiplicative type, and I present some new results that give a practical approach to the essential dimension problem for tori. This new method agrees with all previously know results. Indeed, I conjecture that this should be the general solution to the essential dimen- sion problem. However, with no proof in hand, I simply demonstrate how my theory works through many examples: norm one tori, most tori of dimension 2, and small order abelian groups.
In Chapter 4, I show how algebraic tori can be used to give some bounds on the essential p-dimension of PGLn. To date, this is the best known upper bound for the problem. The exposition mostly follows the paper I published on the subject with a notable addition: I show that the upper bound that I find actually agrees with the essential p-dimension of a
1 particular torus. I conclude with some general remarks on the known lower bounds for this problem and what conclusions can be drawn from both bounds together.
1.1 Basic Notations
k will be a fixed base field. The following is a list, in roughly chronological order, of notations that will be used in this text.
k An algebraic closure of k.
ksep A separable closure of k.
Mn(R) The algebra of n × n matrices over the ring R.
AF For a k-algebra A, A ⊗k F .
CAB The centralizer of the A ⊂ B. deg(A) The degree of the central simple algebra A. ind(A) The index of the central simple algebra A. ℘(x) ℘(x) = xp − x. R×n R × R × ... × R, the product of n copies of R.
Gm The multiplicative group scheme.
Sn The symmetric group on n elements.
Zn The cyclic group on n elements.
µn The group scheme of n-th roots of unity.
RF/k(G) The Weil restriction of the group scheme G.
trdegk(F ) The transcendence degree of the field F over k.
edk(F) The essential dimension of the functor F.
edp(F) The essential p-dimension of the functor F. ed(F; p) The essential p-dimension of the functor F.
A(V ) The affine space associated to the vector space V .
Γp A Sylow p-subgroup of the profinite group Γ. (1) RF/k(Gm) The torus of norm one elements for the field extension F/k. φ(n) The Euler phi function of n. 2 CHAPTER 2
Groups and Algebras
In this chapter, I outline the basic theory of central simple algebras, algebraic groups, and Galois cohomology as it will be used later in this work. Therefore, the reader well-informed of these ideas may skip this chapter with the notable exception of Section 2.2.2 where I prove a new result about the structure of degree ps division algebras after prime to p extension (Corollary 5). This theorem is the basis for the argument in Chapter 4 which computes some bounds on the essential p-dimension of PGLn.
2.1 p-closed fields
I make use of the following simplification employed in [LMMR09]. A field F/k is called p- closed if every finite extension of F has degree a power of p. After fixing a separable closure ksep of k, a p-closure of k can be constructed as follows. Let Γ = Gal(ksep/k) and choose Γp to be a Sylow p-subgroup. I remark that Γ is actually a profinite group, but this notion still makes sense [RZ00]. Then
Γp kp = [x ∈ k|x is purely inseparable over ksep]
is the desired p-closure of k. For a further discussion of the p-closure see [LMMR09].
It will be useful later to have the following notion as well. A natural transformation of functors F → G will be called p-surjective if for any K/k, there is a finite extension L/K of
degree prime to p such that F(L) G(L).
3 2.2 Central Simple Algebras
The following can found in any standard reference on central simple algebras c.f. [KMRT98]. A central simple algebra over k is a finite dimensional k-algebra A 6= 0 with center k such that there are no non-trivial two sided ideals. A central simple algebra will be called a division algebra if every non-zero element is invertible. The canonical examples of a central simple algebras are the matrix algebras Mn(k) for some n. It turns out that all other central simple algebras are “twisted forms” of these:
Theorem 1. (Wedderburn) The following are all equivalent:
1. A is a central simple algebra over k.
2. There is a field extension L/k and an integer s such that AL ' Ms(L).
3. Ak ' Ms(k) for k/k any algebraic closure of k.
4. There is a unique up to isomorphism division k-algebra D and an integer t such that
A ' Mt(D).
The fields in (2) will be referred to as splitting fields. A given algebra A can have many splitting fields. For example, any central simple algebra is split by a finite separable extension. Also, the function field of the generic point of the associated Severi-Brauer variety splits A [Ami82]. This is an example of a generic splitting field, the study of which is closely related to the concept of essential dimension.
Because the dimension of an algebra does not change after a field extension, (2) also
2 implies that every central simple algebra has dimkA = s . As a consequence, the integer s will be called the degree of A (denoted deg(A)). The degree of the division algebra D in (4) is called the index of A, ind(A).
The following basic fact about central simple algebras will be used in the sequel, so it is stated here for reference.
Theorem 2. ([KMRT98][Theorem 1.5]) Let A be a central simple k-algebra and B ⊂ A a simple subalgebra with center F/k. Denote by CAB the set of elements of A which commute 4 with every element of B. It is also a simple subalgebra of A and
dim(A) = dim(B) · dim(CAB).
Moreover, if F = k, then CAB is a central simple algebra and A = B ⊗ CAB.
Two central simple algebras, A and B, will be called Brauer equivalent if DA and DB, the division algebras from (4), are isomorphic. The set of all central simple algebras under Brauer equivalence forms a group under tensor product. This group is of fundamental importance to the study of the cohomology of algebraic groups as we shall see later.
2.2.1 Etale´ Algebras
Let E be any (commutative, associative) finite dimensional k-algebra. E is an ´etale algebra if it satifies one of the following equivalent properties:
1. E ' F1 × ... × Fn where the Fi are finite separable field extensions of k.
2. Eksep = ksep × ... × ksep.
Etale´ algebras figure prominently in the theory of central algebras. If A is a central simple algebra/k, then by Theorem 1 (3), Ak ' Mn(k) for some n. In particular, there is a ×n subalgebra E ⊂ A such that Ek ' k is the subalgebra of diagonal matrices. By (2) above, E is an ´etale algebra. In fact, E is maximal: if any other ´etalesubalgebra of A contains E then it is equal to E. All maximal ´etalesubalgebras of A have dimension n = deg(A) [Alb39].
Etale´ algebras can be classified by Galois cohomology groups as we will see later.
2.2.2 Division Algebras of Degree ps
The goal of this section is to prove that any central simple algebra of degree ps for a prime p has a maximal ´etalealgebra of a special form after a suitable prime to p extension. Because of this, throughout this section, F/k will be a p-closure of k see Section 2.1. Let D/F be 5 a division algebra of degree n = ps. The objective is then to construct a maximal subfield of D whose normal closure has Galois group as small as possible. We begin with a classical result:
Proposition 3. If D contains a non-trivial separable subfield K/F then K contains a degree p cyclic Galois extension of F .
Proof. See [RS92][Prop 1.1].
The following theorem was proven without reference to a fixed subfield L1 in [RS92][Theorem 1.2]. The slight modification of this argument below illuminates the ba- sic strategy for the fundamental construction of this section.
Theorem 4. Let L1 ⊂ D be a degree p cyclic Galois extension/F . Then there is another
degree p cyclic Galois extension L2/F contained in D such that L1L2 := L1 ⊗F L2 ⊂ D is a bicyclic Galois extension.
Proof. Denote L1 = F (a) and fix a generator hσi = Gal(L1/F ). The Skolem-Noether
× −1 theorem gives an element y ∈ D such that yxy = σ(x) for all x ∈ L1. Let A be the
p degree p algebra generated over F by y and a with center F (y ), and set B = CDA. There are two possibilities:
B contains a non-trivial separable field extension K/F :
Applying the proposition to D and K, there is a degree p cyclic extension L2/F contained
in K. L1 ∩B = F by definition, so L1 ∩L2 = F . Thus, L1L2 is the desired bicyclic extension.
Every subfield of B is purely inseparable/F :
Since every division algebra has a maximal separable subfield, degB = 1. It then follows
p p that A = CDF (y ) and degA · [F (y ): F ] = degD. In particular, degA = p and so F (y) is a maximal purely inseparable subfield of D. This is equivalent to D being a cyclic p-algebra [Alb39][VII Theorem 25].
We will show by induction on s that any cyclic p-algebra D containing a degree p subfield
L1/F contains a bicyclic extension L1L2/F . First, suppose s = 2. Following the argument 6 in [Sal77] [Proposition 7], we have the degree p cyclic sub-algebra C = CDL1. By definition,
p p2 C ⊃ F (y ), so we can write C ' [x, y ) for some x ∈ L1. By [Sal77][Lemma6] (choosing
0 p a = 0), C = [a0, b0) with a0 ∈ F . Let ℘(x0) = x0 −x0 = a0 for x0 ∈ C and set L = L1(x0). If a and a0 are linearly independent elements of the Fp-vector space F/℘(F ) then L2 = F (x0) is the desired extension. Otherwise, a0 ∈ ℘(F ) which implies that C is split. Since degree p split algebras contain every degree p extension of F , we can choose for L2 any cyclic extension disjoint from L1. Such a field always exists if dimFp F/℘(F ) ≥ 2. Since D is a division algebra of degree ≥ 2, this condition holds by [Sal77][Theorem 3].
0 ps−1 Now assume that we have the result for s ≥ 3 and consider A = CDF (y ). It is a degree
s−1 p algebra containing F (y) and L1. Therefore, it is also cyclic and the induction hypothesis
0 ps−1 ps−1 implies that there is a cyclic degree p extension L2/F (y ) disjoint from L1(y ). Since ps−1 ps−1 0 F (y ) is purely inseparable, there is a cyclic degree p field L2/F such that L2(y ) = L2 which is still disjoint from L1.
s 2 Corollary 5. For A any central simple algebra/F of degree p ≥ p and K1 ⊂ A an ´etale subalgebra of degree p, there is a maximal ´etalesubalgebra K ⊂ A that can be written as
s−1 K = K1 ⊗F K2 for K2/F an ´etalealgebra of dimension p .
Proof. First, consider the case where A is a division algebra. By the theorem, A contains
two distinct degree p cyclic extensions K1 and L over F . Proceed by induction on s.
If s = 2 , then taking K2 = L we are done. Otherwise, assume we have the result for
s−1 0 any division algebra of degree p and any degree p subfield L . The centralizer CAL is a
s−1 division algebra over L of degree p . By definition it contains the degree p subfield K1L/L,
s−2 so by induction hypothesis, CAL has subfield K2/L disjoint from K1L of degree p .
Since K2 ∩ K1L = L and L is disjoint from K1, K2 ∩ K1 = F . It follows that K2/F is a
s−1 degree p extension disjoint from K1 and K1 ⊗ K2 ⊂ A.
Suppose A is not a division algebra. If A is split, the result is immediate. Otherwise, choose a division algebra D ∼ A. Then
deg(D) = pt ≤ ps = deg(A).
7 ×ps−1 If t = 1, then any maximal subfield K1 ⊂ D gives the desired ´etalealgebra K1 ⊗F ⊂ A, so suppose t ≥ 2. By the above argument, D has a subfield
L = L1L2 ' L1 ⊗ L2
t−1 ×ps−t with [L1 : F ] = p and [L2 : F ] = p . Set K = L an ´etalesub-algebra of A. Since K
s ×ps−t ×ps−t has dimension p , it is maximal. But K ' L1 ⊗ L2 , so L1 and L2 are ´etalealgebras of the desired degree.
2.3 Algebraic Groups
An affine algebraic group is a functor G : CommAlgk → Group that is representable by a Hopf algebra/k. That is, there is a commutative k-algebra A equipped with three homomor- phisms c : A → A ⊗ A
i : A → A
u : A → F
satisfying the usual commutative diagrams for the co-multiplication, co-inverse, and co-unit such that G(R) = Hom(A, R) for any commutative k-algebra R see [KMRT98] for more details.
Many of the most familiar groups are actually group schemes: finite (constant) groups, the
additive group scheme Ga, the multiplicative group scheme Gm, GLn, PGLn, etc. Moreover, group schemes have all of the usual properties of groups. There are well-defined notions of subgroup, normal subgroup, homomorphism, and quotient group that satisfy the usual properties. One important exception is that a surjective map between two group schemes need not be surjective on every R!
Group schemes behave quite nicely with respect to field extensions. In fact, given a field extension F/k and a group scheme G over k, we can produce the restriction over F simply by
setting GF (R) := Hom(AF ,R) where A is the Hopf algebra associated to G. Similarly, if H
8 is a group scheme over F , we define the corestriction (or more frequently Weil restriction) to
be the group scheme RF/k(H)(R) := H(R ⊗k F ). Restrictions are obviously group schemes, but the corestriction requires a bit of work to produce the corresponding Hopf algebra c.f. [KMRT98][Lemma 20.6].
2.3.1 Groups of Multiplicative Type
The primary concern of this work will be with algebraic tori, so I will explain this theory in more detail. For any abelian group H (written multiplicatively), the group ring khHi has the structure of a Hopf algebra given by the choices c(h) = h ⊗ h, i(h) = h−1, and u(h) = 1. Therefore, the corresponding representable functor is a group scheme. Such groups are called diagonalizable. As an example, Gm is the diagonalizable group associated to the abelian group Z. A group of multiplicative type is any group scheme that becomes diagonalizable over a separable closure ksep. This family of group schemes includes the algebraic tori: group schemes T such that Tksep ' Gm × ... × Gm and the roots of unity: µn. Part of their value in the theory of algebraic groups is the following equivalence of categories.
Theorem 6. [KMRT98][Proposition 20.17] There is an equivalence of categories between the category of group schemes of multiplicative type/k and the category of abelian groups with continuous Gal(ksep/k)-action.
The equivalence is given by sending a group of multiplicative type to its characters,
∗ G = Hom(Gksep , Gm,ksep ). And conversely, sending an abelian group A to the functor ∗ A(R) = Hom(A, (R ⊗k ksep) ). We will see later how this correspondence can be fruitful in many calculations. As a final notation, a torus will be called quasi-split if the associated Γ-module is a permutation module.
9 2.3.2 Galois Cohomology
Again, following [KMRT98], I will give a brief survey of Galois cohomology. Let Γ be a profinite group. A Γ-group is a group M with Γ acting by group homomorphisms. We can define cohomology (pointed) sets as follows
H0(Γ,M) = {m ∈ M|γm = m for all γ ∈ Γ}.
This is simply the subgroup of elements of M fixed by the Γ-action. The first cohomology groups is defined in two steps. First, we consider 1-cocycles:
1 Z (Γ,M) = {continuous maps α :Γ → M|α(γ1γ2) = α(γ1)γ1α(γ2) for γi ∈ Γ}.
Two 1-cocyles α and β are equivalent if there is an element m ∈ M such that β(γ) = mα(γ)γm−1 for all γ ∈ Γ. H1(Γ,M) is defined to be Z1(Γ,M)/ ∼.
The most important case for our purposes will be when Γ = Gal(ksep) and M = G(ksep) for some algebraic group G. In this case, we will abbreviate H1(Γ,M) = H1(k, G). If N is smooth then an exact sequence of algebraic groups
1 −−−→ N −−−→ G −−−→ H −−−→ 1
induces an exact sequence of pointed sets
Γ Γ Γ 1 1 1 1 → N(ksep) → G(ksep) → H(ksep) → H (k, N) → H (k, G) → H (k, H).
If N is abelian, then this sequence may be continued to the right. Moreover, H1(k, G) is functorial in both k and G.
For many algebraic groups, the first Galois cohomology group classifies algebras [KMRT98][Section 29.B]. That is, the elements of the pointed set H1(k, G) are in bijective correspondence with isomorphism classes of a certain type of algebra/k. Some important
examples are for Sn, PGLn, and On which classify, n-dimensional ´etale algebras, central simple algebras of degree n, and n-dimensional quadratic forms respectively. This fact will be quite important for us in later chapters.
10 CHAPTER 3
Essential Dimension
The essential dimension of an algebraic group G is a measure of the “least number of param- eters” needed to define a generic G-torsor. Since G-torsors classify special classes of algebras, as shown in Chapter 2, this allows us to study the essential dimension via our knowledge of algebras. Theoretically, one would like to go the other way as well: can we use the essential dimension to study the structure of algebras? So far, we have no results in this direction.
The study of essential dimension began in 1997 with J. Buhler and Z. Reichstein’s study of Galois G-algebras [BR97] for a finite group G. In the most familiar case, this is just a choice of Galois extension over a field F with Galois group G. Let α be a generator of this extension over F . It has a minimal polynomial with coefficients in F . The number of algebraically independent coefficients in this equation gives a measure of the complexity of this field extension. Of course, this choice of generator is not canonical, so perhaps with a different choice of generator β, we can get a minimal polynomial with a smaller number of coefficients. The least number of algebraically independent coefficients in a minimal polynomial of a field generator is exactly the essential dimension of this Galois extension.
Later, A. Merkurjev reformulated the essential dimension problem functorially, c.f. [BF03]. In this way, essential dimension is a general notion that encompasses the usual dimensions for schemes and vector spaces as well as the “dimension”of Galois cohomology functors. This latter case is perhaps the most studied example with applications to inverse Galois theory and the theory of central simple algebras.
The essential dimension problem is also inherently geometric. P. Brosnan, Z. Reichstein, and A. Vistoli [BRV07] have discussed a generalization to algebraic stacks, and because Galois cohomology groups classify the torsors of a algebraic group, the essential dimension
11 problem also admits an interpretation in terms of versal torsors c.f. [Rei10, Section 2].
In this chapter, I will introduce the theory of essential dimension and survey some of the known results. My focus is on the essential dimension of algebraic tori (or more generally groups of multiplicative type) because they play a central role in most of our bounds. In fact, most of the known results for abelian groups can be summarized by a single idea which I call a permutation representation. Even for non-abelian groups, though, tori are crucial to most of our constructions but you will have to wait until Chapter 4 for more on this.
3.1 Definitions
We wish to extend the idea of dimension to a broader class of algebraic objects. This can be accomplished using the following observation. Let Fields/k be the category of field extensions of k with inclusion maps. Any scheme X over k can be considered as functor
X : Fields/k → Set
defined as X(F ) = Hom(Spec k, X). The dimension of this scheme can be computed as
dim(X) = sup trdegkk(x) x∈X where k(x) is the residue field at x.
This definition extends to an arbitrary functor F : Fields/k → Set. An element a ∈ F(K) is said to be defined over k ⊂ K0 ⊂ K (and that K0 is a field of definition of a) if it is in the image of the map F(K0) → F(K). The essential dimension of a, de- noted edk(a), is the least transcendence degree/k of a field of definition for a. The essential dimension of F is defined as the possibly infinite value
edk(F) = sup{edk(a)}
where the supremum is taken over all a ∈ F(K) and all field extensions K/k.
Proposition 7. Let F and G be functors as above. Then we have the following useful properties: 12 (1) For any field extension L/k, edk(F) ≥ edL(F).
(2) Given a surjective natural transformation φ : F G then edk(F) ≥ edk(G).
Proof. (1) Choose a field extension K/L and an element a ∈ F(K). By definition, there is an
field K0 and an element b ∈ F(K0) mapping to a such that edk(a) = trdegk(K0) ≤ edk(F).
Consider the composite field extension K0L. By construction, trdegL(K0L) ≤ edk(F) and
b maps to a under the composite F(K0) → F(K0L) → F(L). Therefore a is defined over
K0L and
edL(a) ≤ trdegL(K0L) ≤ edk(F).
(2) Choose a field extension K/k and an element a ∈ G(K) such that
edk(a) = edk(G) = trdegk(K). Since φ is surjective, choose an element c ∈ F(K) map-
ping to a. By definition, there is a subfield k ⊂ K0 ⊂ K such that c is defined over K0 and edk(c) = trdegk(K0) ≤ edk(F). By commutativity of the diagram
F(K) → G(K) ↓ ↓
F(K0) → G(K0) it follows that a is also defined over K0, and so
edk(F) ≥ edk(c) = trdegk(K0) ≥ edk(a) = edk(G).
Remark 8. If F ,→ G it does not follow that edk(F) ≤ edk(G). In general, we need stronger conditions to make this true c.f. [BF03][Prop 1.8].
3.2 Essential p-dimension
The computation of the essential dimension of a functor can in general be quite challenging. It is often useful in the computation of lower bounds to replace it with a weaker notion: essential p-dimension. The basic idea is to allow a bit more flexibility in our fields of definition. 13 Let F be a functor as in the previous section. An element a ∈ F(K) is p-defined over
0 0 K0/k if there is a finite prime to p extension K ⊃ K0 such that the image of a in K is in the image of an element from K0. Again, define edp(a) = mintrdegk(K0) where the minimum is
taken over all fields of p-definition. Similarly, set edp(F) = sup{edp(a)}.
It is clear that edp satisfies the same properties as listed in the previous section. Moreover,
edp(F) ≤ ed(F) for any prime p. Sometimes, in order to avoid a preponderance of indices, I
will use the notation edk(−; p) for the essential p-dimension, especially when the base field is not understood.
The language of p-closures can help us to simplify the discussion of essential p-dimension (see Section 2.1). For limit-preserving functors, the essential p-dimension of any ele- ment can be computed over a p-closure [LMMR09][Lemma 3.3], so we can always as- sume that our field F is p-closed when doing such a computation. Moreover, if F/k is
p-closed and F(F ) G(F ) for two functors F and G, then the map is p-surjective and edk(F; p) ≥ edk(G; p) [LMMR09][Prop 3.4].
3.3 Essential Dimension of Algebraic Groups
For algebraic groups, we have the Galois cohomology functors as defined in Section 2.3.2. It is standard to abbreviate ed(H1(−,G)) by ed(G). Since these functors classify algebras (or more generally torsors), computing the essential dimension of algebraic groups is a very interesting problem. Before computing some examples, I would like to first give some connections of this theory with representations.
Let G be an algebraic group. G is said to act generically freely on a scheme X if there is a G-stable open subscheme U ⊂ X on which G acts freely. A generically free representation is a G-generically free action on the affine space A(V ) where V is a finite dimensional vector space. Such representations give convenient upper bounds for the essential dimension problem.
14 Theorem 9. [BF03][Proposition 4.11] If V is a generically free representation of G then ed(G) ≤ dim(V ) − dim(G).
This follows by finding a G-stable open U ⊂ A(V ) for which the quotient U/G exists (in the category of schemes) and then constructing a surjective morphism from U/G(−) to H1(−,G). Of course, in many cases, this bound will not be tight, but U → U/G is always a versal G-torsor. Therefore, we can study compressions of this to compute the essential dimension c.f. [BF03][Chapter 6] or [DR11].
3.3.1 Examples
Computing the essential dimension is a difficult problem even for finite abelian groups as we will see later in Section 3.3.4. Below are some know results to contrast between essential dimension and essential p-dimension.
Sn: This is one of the first groups for which the essential dimension was bounded. However, for n > 7, the precise computation is unknown. First, ed(S2) = ed(S3) = 1 and ed(S3) = 2 [BF03][Chapter 7]. For n ≥ 5, n b c ≤ ed(S ) ≤ n − 3. 2 n
This lower bound is not tight since Duncan showed that ed(S7) = 4 [Dun10]. The essential p-dimension problem is completely solved however [MR09][Corollary 4.2]: n ed (S ) = b c. p n p p-groups: A similar situation occurs for Zp. Later we will see that the best known upper bound for ed(Zp) is φ(p − 1). However, after a prime to p extension the groups Zp and µp become isomorphic. Therefore, we can compute edp(Zp) by looking at the Kummer sequence
1 −−−→ µp −−−→ Gm −−−→ Gm −−−→ 1.
Using Hilbert Theorem 90 and the long exact sequence in cohomology shows that
1 Gm(F ) H (F, µp). Therefore, edp(Zp) = ed(µp) = 1. Clearly, these values are sub- stantially different if the base field is k = Q. However, if k contains the p-th roots of unity, we have the following powerful result. 15 Theorem 10. [KM08][Theorem 4.1] If k is a field of char 6= p containing a primitive p-th
root of unity and G is a finite p-group, then ed(G) = edp(G) = min(dim(V )) where the minimum is taken over all faithful (generically free) representations V of G.
3.3.2 Essential Dimension of Groups of Multiplicative Type
Throughout this section, let T be any group of multiplicative type and p a prime different from char(k). Let Γ be the Galois group Gal(L/k) for some Galois splitting field L of T and
Γp a Sylow p-subgroup of Γ. A map of Γp modules M → P will be called a p-presentation if P is a permutation module and the cokernel is finite of order prime to p. As an extension of the Karpenko-Merkurjev result [KM08], we have the following general theorem which completely solves the essential p-dimension problem for groups of multiplicative type.
Theorem 11 ([LMMR09]Corollary 5.1). If T is split by a Galois extension of p-power order, then
edp(T ) = min(φ) − dim(T )
∗ where the minimum is taken over all p-presentations φ of T (viewed as Γp-module).
In general, the calculation of the essential dimension of groups of multiplicative type is mostly unknown. In the rest of this section, I will give a method for constructing upper bounds and use it to reproduce all known upper bounds for the essential dimension problem. I begin with some notation.
Definition 12. A G-module Q is a permutation representation of T if there is a map φ : Q → T ∗, a permutation module P , and another map Q → P such that the following diagram commutes: Q → P & ↓ T ∗ Such a representation will be called faithful if φ is a surjection.
16 By defintion, a faithful permutation representation of T yields a short exact sequence
0 −−−→ M −−−→ Q −−−→ T ∗ −−−→ 0
Consider the exact sequence in cohomology for the corresponding tori:
1 −−−→ T (K) −−−→ Qˆ(K) −−−→ Mˆ (K) −−−→ H1(K, Xˆ) −−−→ H1(K, Qˆ)
Because T → Mˆ factors through a quasi-split torus and quasi-trivial tori have no non-trivial ˆ 1 K-torsors, it follows that M(K) H (K,T ), and so ed(T ) ≤ rank(M). I will show that this bound is a very good one in the following sections. As a consequence, I propose the following conjecture whose demonstration either positively or negatively would be a good start to understanding the general problem:
Conjecture 13. For a group of multiplicative type T ,
ed(T ) = min(rank(Q)) − dim(T )
where the minimum is taken over all faithful permutation representations of T .
3.3.3 Essential Dimension of Algebraic Tori
To begin, I show that Theorem 11 can be re-written in terms of permutation representations.
Theorem 14. Let X = T ∗ be the Γ-lattice corresponding to the torus T which splits over a Galois extension of degree pn with Galois group G. Then,
edp(T ) = min(rank(Q)) − dim(T ) where the minimum is taken over all faithful permutation representations of X.
Proof. Following [LMMR09], we know that this essential p-dimension is given by rank(P )−rank(X) where P is a permutation module of minimal rank with a map φ : P → X such that coker(φ) is finite of order prime to p. Let I = Im(φ) and M = ker(φ). We then have an exact sequence:
0 −−−→ M −−−→ P −−−→ I −−−→ 0 17 By definition, I(p) := Z(p) ⊗ I = X(p), and since p is the only prime dividing the order of G, I and X are in the same genus. By an amazing result [CR81][Theorem 31.28], this implies that if we take any faithful G-lattice, say Z[G], there is another G-lattice J in the same genus as Z[G] such that I ⊕ Z[G] ' X ⊕ J.
Consider now the exact sequence
0 −−−→ M −−−→ P ⊕ Z[G] −−−→ I ⊕ Z[G] −−−→ 0
Using the above isomorphism, we have two maps i : X,→ I ⊕ Z[G] and p : I ⊕ Z[G] X
where p ◦ i = idX . Pulling back the above sequence along i gives a diagram 0 0 y y f 0 −−−→ M −−−→ Q −−−→ X −−−→ 0 g y iy h 0 −−−→ M −−−→ P ⊕ Z[G] −−−→ I ⊕ Z[G] −−−→ 0 Then f = p ◦ h ◦ g factors through a permutation module, so Q is a faithful permutation representation and the essential p-dimension is bounded above by rank(M). This is the exact value by assumption, so M has minimal rank coinciding with the essential dimension as desired.
Corollary 15. Let G be a group of multiplicative type which splits over a Galois extension of degree pn with Galois group H that fits into an exact sequence
1 −−−→ T −−−→ G −−−→ F −−−→ 1 where T is a maximal subtorus and F is a p-group. Then,
edp(T ) = min(rank(Q)) − dim(T ) where the minimum is taken over all faithful permutation representations of T ∗.
18 Proof. Clearly, permutation representations give upper bounds, so we only need prove the lower bound. By Theorem 11, we know that the essential p-dimension is given by dim(P ) − dim(G) where P is a quasi-split torus of minimal rank with a map φ : G → P such that N = Ker(φ) is finite of order prime to p. Let J = Im(φ) and set I to be the image of T in J. We have a diagram: 1 y N y 1 −−−→ T −−−→ G −−−→ F −−−→ 1 y J y 1 The cokernel of the composition from T to J and the cokernel of the composition from N to F agree, so since N and F are relatively prime, N → F is the trivial map. We get the exact sequence 1 −−−→ I −−−→ J −−−→ F −−−→ 1.
Similarly, both kernels must be the same, so it follows that Ker(T → P ) = N yielding the exact sequence 1 −−−→ N −−−→ T −−−→ I −−−→ 1.
19 The inclusions of I and J into P induce a diagram (1): 1 y 1 F y y 1 −−−→ I −−−→ P −−−→ S −−−→ 1 y y 1 −−−→ J −−−→ P −−−→ R −−−→ 1 y y F 1 y 1 where F is the kernel of S → R by the Snake Lemma.
⊕r ∗ Let Z[H] N be a faithful representation of N. If we take the exact sequence of ⊕r ∗ characters corresponding to the quotient of T by I above and pullback along Z[H] N we get 0 0 y y M M y y 0 −−−→ I∗ −−−→ X −−−→ Z[H]⊕r −−−→ 0 y y 0 −−−→ I∗ −−−→ T ∗ −−−→ N ∗ −−−→ 0 y y 0 0 Since Z[H]⊕r is projective, the middle sequence splits. To see that X ' M ⊕ T ∗ as well, it suffices by [CR81][Corollary 31.3(i)] to first localize at p. Luckily, N has order prime to p, so it vanishes in the localization and the splitting is trivial. Note, however, that if we do the same with the sequence:
1 −−−→ N −−−→ G −−−→ J −−−→ 1, 20 the fact that G is not a lattice means that we can’t get a splitting in general. However, we can combine these two pullback diagrams to get a commutative diagram (2):
0 −−−→ F ∗ −−−→ J ∗ ⊕ Z[H]⊕r −−−→ I∗ ⊕ Z[H]⊕r −−−→ 0 y y 0 −−−→ F ∗ −−−→ G∗ −−−→ T ∗ −−−→ 0.
Now, I argue for T ∗ as in the theorem. By pulling back along the inclusion of T ∗ into
I∗ ⊕ Z[H]⊕r we get 0 0 y y 0 −−−→ S∗ −−−→ Q∗ −−−→ T ∗ −−−→ 0 y iy 0 −−−→ S∗ −−−→ P ∗ ⊕ Z[H]⊕r −−−→ I∗ ⊕ Z[H]⊕r −−−→ 0 and since i splits, the first row is a permutation representation of T as before. Note that S is the torus given in diagram (1) above. This diagram can be extended to a commutative diagram (with non-exact columns) using (2): 0 −−−→ S∗ −−−→ Q∗ −−−→ T ∗ −−−→ 0 y iy 0 −−−→ S∗ −−−→ P ∗ ⊕ Z[H]⊕r −−−→ I∗ ⊕ Z[H]⊕r −−−→ 0 j y y y 0 −−−→ F ∗ −−−→ G∗ −−−→ T ∗ −−−→ 0. This defines a map Q∗ → G∗ that factors through a permutation module. The left hand map S∗ → F ∗ is exactly the one that follows from (1) and the Snake Lemma, so it is a surjection. On the right, j ◦ i is the identity. Therefore, by the 5-Lemma, the Q∗ → G∗ is also surjective.
Finally,
edp(G) = dim(P ) − dim(G) = dim(S)
= dim(Q) − dim(T ) = dim(Q) − dim(G) as desired. 21 Corollary 16. A torus T has ed(T ) = 0 if and only if T ∗ is an invertible Γ-module.
Proof. If T ∗ is invertible then H1(K,T ) = 0 for all field extensions K/F , so clearly it has essential dimension 0.
Conversely, ed(T ) ≥ edp(T ) for all p, so if ed(T ) = 0, then all essential p-dimensions are
likewise 0. By the theorem, for any p there is a faithful permutation resolution of Γp-modules:
0 −−−→ M −−−→ Q −−−→ T ∗ −−−→ 0
where rank(M) = 0. But Q is a free Z-module which implies that M = 0. Therefore, ∗ ∗ there is a Γp-isomorphism of T that factors through a permutation module. That is, T is
∗ an invertible Γp-module. Since this is true for any p, T is also an invertible Γ-module as desired.
I remark that this proof can be done without the aid of Theorem 11, but this would take us too far afield. I now want to compute some other more examples to show how difficult the essential dimension problem can be in general.
3.3.3.1 Examples
(1) RL/F(Gm): Let L/F be any finite separable extension, then the norm homomorphism gives an exact sequence: (1) 1 −−−→ RL/F (Gm) −−−→ RL/F (Gm) −−−→ Gm −−−→ 1
Since the middle group has no non-trivial F -torsors by Shapiro’s Lemma and Hilbert Theo- rem 90, there is, for any field extension K/F a surjection
1 (1) Gm(K) H (K, RL/F (Gm)).
(1) Therefore, edF (RL/F (Gm)) ≤ 1. A torus has essential dimension 0 if and only if it is a direct summand of a quasi-split torus by Corollary 16, so this torus must have essential dimension 1. 22 One Dimensional Tori:
Up to choice of a degree 2 field extension K/F , there are only two one dimensional tori: (1) Gm and RK/F (Gm). These have essential dimension 0 and 1 respectively. Two Dimensional Tori:
Two dimensional tori were classified in [Vos65]. Every two dimensional torus is split by a Galois extension whose Galois group is a subgroup of either D8 or D12. The essential dimension of the D8-tori can be computed directly with the aid of Theorem 11, so let us consider tori split by Galois extensions with degree divisible by six. There are three possible groups: Z6, S3, and D12.
There is only one class of torus for Z6: the norm one tori for a degree six cyclic extension. All such tori have essential dimension 1 by the previous example.
There are two class of tori split by an S3-extension. The first is a norm one torus for a non-normal degree 3 extension, so we can compute its essential dimension as above. Instead, consider its dual, the torus T
1 −−−→ Gm −−−→ RF/k(Gm) −−−→ T −−−→ 1 where F/k is a non-normal degree 3 extension. Then taking an algebraic closure, it is easy to see that T is split by a field extension L/k with Gal(L/k) = S3. This induces, in the
usual way, a sequence of S3-modules:
∗ 0 −−−→ T −−−→ Z[S3/Z2] −−−→ Z −−−→ 0
To get an upper bound, it suffices to find a good permutation representation. Let
S3 = Z3 o Z2 = {σ} o {τ}. That is, the group generated by σ and τ with the relation στ = τσ2. It follows that T ∗ is generated by the elementσ ¯ − 1¯ where the bar denotes the
coset in S3/{τ}. This means we have an exact sequence
∗ 0 −−−→ M −−−→ Z[S3] −−−→ T −−−→ 0
where the second map sends the generator 1 toσ ¯ − 1.¯ This shows that ed(T ) ≤ 4, but we can actually do better. It is easy to see that M = hσ2 + τ, σ2 + σ + 1i. Therefore the natural 23 map Z[S3] → Z[S3/hτi] gives a diagram: 0 0 y y 0 −−−→ N −−−→ Q −−−→ T ∗ −−−→ 0 y y ∗ 0 −−−→ M −−−→ Z[S3] −−−→ T −−−→ 0 y y
Z[S3/hτi] Z[S3/hτi] y y 0 0 By commutativity, Q → T ∗ factors via a permutation module, so ed(T ) ≤ rank(Q) − rank(T ∗) = 1. To get a lower bound, we can compute the essen- tial 3-dimension, but after a degree 2 extension T becomes isomorphic to a norm 1 torus.
Thus, ed3(T ) = 1 ≤ ed(T ) as desired.
There are also two classes of tori split by a D12-extension. Let T be
1 −−−→ T −−−→ RKL/k(Gm) −−−→ RK/k(Gm) × RL/k(Gm) −−−→ Gm −−−→ 1
where L/k is a degree 3 non-normal extension and K/k is a degree 2 extension. The other
class of D12-tori is the torus S dual to this one. Write D12 = S3 × S2 = hσi o hτi × hρi. It follows from the above that we get an exact sequence
∗ 0 −−−→ M −−−→ Z[D12/hτi] −−−→ T −−−→ 0 where M = hσ¯2 +σ ¯ + 1¯, ρ¯ + 1¯i. Therefore, ed(T ) ≤ 4. Again, we can do better. The
24 augmentation map P := Z[D12/hτi] → Z yields a commutative diagram 0 0 y y 0 −−−→ N −−−→ Q −−−→ T ∗ −−−→ 0 y y 0 −−−→ M −−−→ P −−−→ T ∗ −−−→ 0 y y Z Z y y 0 0 and which shows, as in the S3 case, that ed(T ) ≤ 3.
Now, we move on to lower bounds. Again, we use the essential p-dimension. Af- ter a degree 4 extension, T becomes isomorphic to the norm one torus for a de- gree 3 cyclic extension which implies that ed3(T ) = 1. After base extension by L,
TL ' RKL/L(Gm)/Gm × RKL/L(Gm)/Gm, and it is easy to see that such a group has es- sential dimension 2. Thus, ed2(T ) = 2 ≤ ed(T ) ≤ 3. This is the smallest algebraic torus for which the essential dimension cannot be computed. The essential dimension of its dual, S, has a similar bounds.
3.3.3.2 Norm Tori and Valuation Descent
I want to extend the first example of the previous section to a slightly broader class of algebraic groups. These are the algebraic tori that are Weil restrictions of norm one tori:
1 −−−→ T −−−→ RL/k(Gm) −−−→ RL0/k(Gm) −−−→ 1 where k ⊂ L0 ⊂ L are both finite separable field extensions and the second map is induced by the norm homomorphism. In the sequel, I fix the following notations
× × N : K 7→ K /(N((L ⊗k K) )) where L/k is a degree d field extension with norm N.
0 × × N : K 7→ (L ⊗k K) /(N((L ⊗k K) )) 25 where L/L0 is degree d0 extension with norm N and L0/k is a a degree d extension.
By Hilbert Theorem 90, N' H1(−,T ) where T is the torus above. Therefore, the essential dimension of this functor is the essential dimension of the group T . Of course, N is just the cohomology of a norm one torus, as computed in the examples of the previous section.
×n Proposition 17. edk(N = n)
Proof.
n There is an obvious surjection from the functor Gm to N , so the essential dimension is bounded above by n. I will now construct an element that has essential dimension n to demonstrate the lower bound.
Choose the field extension K = k(t1, ..., tn) where the ti are independent variables. Con- sider the element t = (t1, ..., tn) ∈ N (K). This element is defined over a field of transcendence degree n, namely K. Suppose that t can be defined over a subfield k ⊂ K0 ⊂ K. That is,
0 0 there are elements si ∈ K such that s = (s1, ...sn) maps to t under the map N (K ) → N (K).
Write si = tiN(xi) for xi ∈ L(t1, ..., tn). To show that the essential dimension of t is n, it
0 then suffices to show that trdegk(K ) ≥ n. The argument is valuation theoretic.
Let νi be the discrete valuation corresponding to the indeterminate ti and denote by
n ν : K → Z the composite valuation of all the νi, 1 ≤ i ≤ n. I claim that the restriction, 0 0 0 0 0 ν , of this valuation to K continues to have rank n. For each i, ν (si) = ν (ti) + ν (N(xi)).
0 n Because it is the restriction of ν, ν (ti) = ei ∈ Z , the standard generating elements. The argument reduces to computing the valuation of the norm.
Let ω1, ..., ωd be a basis for the degree d extension L(t1, ..., tn)/k(t1, ..., tn). In this basis
N(y1ω1 + ... + ydωd) is a degree d homogeneous polynomial in the yi with coefficients in k. By definition, it is the determinant of the multiplication endomorphism associated to an
× element of L(t1, ..., tn) . This means in particular that the norm is not zero on any unit so
d that each term yi appears with non-zero coefficient.
× Take x = y1ω1 + ... + ydωd ∈ L(t1, ..., tn) . Since the norm is multiplicative and any element of k(t1, ..., tn) can be written as a quotient of polynomials, we may assume that 26 0 x ∈ L[t1, ...tn]. Consequently, N(x) is also of this form. To compute ν (N(x)), we begin
with multiples of t1 thinking of the norm as a polynomial in t1. If t1 divides N(x), then t1
must appear as a term in at least one yi. Write yi = ait1 + bi where ai ∈ L[t1, ..., tn] and
bi ∈ L[t2, ..., tn]. Then substituting into the norm, we see that every term is divisible by t1 except for the term N(b1ω1 +...+bnωn). Since t1 divides the entire norm and every term but this one, we must have that N(b1ω1 + ... + bnωn) = 0. This can only be the case if bi = 0 for
all i. Thus, every yi must be a multiple of t1, and since the norm is homogeneous of degree
d, the multiple of t1 dividing the norm must be divisible by d.
If the norm is not divisible by t1 then we set t1 = 0 and look at multiples of t2. Considering this as a polynomial in the last n−1 variables, we repeat the process: either it is divisible by t2 or not. In the former case, this power must again be a multiple of d by the same argument as above. Continuing the computation inductively, we see that ν0(N(x)) ∈ dZn.
0 Combining with the original computation gives, ν (ti) = ei(mod d). In particular, the rank of ν0 modulo d is n and therefore also generally, as claimed. By [ZS75][VI Theorem 3
0 0 Corollary 1], we have n = rank(ν ) ≤ trdegk(K ) demonstrating the lower bound of n for the essential dimension.
Remark 18. Since the argument given in the proposition depended only on the homogeneity
of the norm, it remains true even if K is chosen to be F (t1, ..., tn) for an algebraic extension F/k. This will allow us to conclude the same result for base extensions of this functor.
×n Proposition 19. edk(N ) = nd
Proof.
n ×n We again have a surjection RL0/k(Gm) → N , which shows that the essential dimension is bounded above by nd. Consider the functor N defined over L0 instead of k. To avoid confusion we will denote it F. Using Weil restriction, we can study this functor over k:
0 0 × 0 × RL0/k(F)(K) = F(K ⊗k L ) = (K ⊗k L ) /N((K ⊗k L ⊗L0 L) )
0 × × = (K ⊗k L ) /N((K ⊗k L) ) = N (K) 27 That is, N is a Weil restriction of the functor N . We will use this identification and our previous results to give the desired lower bound. By Proposition 7(1), it suffices to check ×n ˜ that edF (N ) is at least nd for any F/k. Define L to be any finite separable extension of L0 that splits L0. Then we base extend N and compute the essential dimension:
×n ×n ×nd (N ) = (R 0 (F )) ' (F) . L˜ L /k L˜ L˜
Now, we conclude by the variant of the previous proposition mentioned in the remark.
3.3.4 Essential Dimension of Finite Abelian Groups
Consider first the case of a cyclic q-group for q = pr a prime power. Let p−1 if p is odd Z G = 1 if p = 2 and r = 1 Z2r−2 × Z2 if p = 2 and r 6= 1
be the automorphism group of Zq. The character module corresponding to the finite constant
group scheme Zq is Z/qZ with G acting by automorphisms. Following [Led02], we can construct a faithful permutation resolution to get the following upper bound.
r−1 Theorem 20. edQ(Zpr ) ≤ φ(p − 1)p .
Proof. First consider the case where p is odd and let X denote the set of pr-th roots of unity and fix a primitive pr-th root ζ. G = hσi acts by permutations on this set, so we get an exact sequence
f 0 −−−→ N −−−→ Z[X] −−−→ Z/qZ −−−→ 0 where f(ζn) = n.
Let r−1 r−1 Y Y Y g(x) = Φ(p−1)pi and h(x) = Φmpi i=0 i=0 m|(p−1) m r−1 If p = 2, it is well-known that Zq has a faithful representation of degree q/2 = 2 . It is given by the map Z[G] Z/qZ which sends 1 to 1. Note that for r = 1 this is just Z with the trivial action. This completes the proof. Consider now the group Zn = Z n1 nr . It follows from the primary decomposition p1 ...pr theorem that X ed (Zn) ≤ ed (Z ni ). Q Q pi i This is the best known bound for a general cyclic group. My conjecture stated at the end of the previous section would imply that this is an equality. More generally, if a field F/Q mi contains the pi -th roots of unity where mi ≤ ni. Set si = ni − mi. There is an obvious exact sequence which gives a permutation representation. L 0 −−−→ M −−−→ [Z si ] −−−→ Zn −−−→ 0 Z pi P Here rank(M) = [F (µ ni ): F ]. pi For any pair of distinct primes p and q, the sum of the augmentation maps gives a map P = Z[Zpa ] ⊕ Z[Zpb ] → Z such that the following diagram commutes: 0 0 y y 0 −−−→ N −−−→ Q −−−→ Zpaqb −−−→ 0 y y 0 −−−→ M −−−→ P −−−→ Zpaqb −−−→ 0 y y Z Z y y 0 0 where the equality in the last rows follows since p and q are relatively prime. 29 Applying this same idea r − 1 times to our original exact sequence produces a resolution 0 −−−→ N −−−→ Q −−−→ Zn −−−→ 0 P where Q → Zn factors through a permutation module and rank(N) = [F (µ ni ): F ]−(r−1) pi which is exactly the result computed in [WW11]: ni mi Theorem 21. If n = pi and F is a field containing the pi -th roots of unity then X edF (Zn) ≤ [F (µ ni ): F ] − (r − 1) pi i I want to conclude this section by computing the known values of the essential dimension for small abelian groups. 3.3.4.1 Examples Z2, Z3: By Theorem 20, edQ(Z2), edQ(Z3) ≤ φ(1) = 1. Since the only finite group with essential dimension 0 is 1, edQ(Z2) = edQ(Z3) = 1. Z4: Let k/Q be any field extension. Again by Theorem 20, edk(Z4) ≤ 2. A finite group can have essential dimension 1 over k only if it is a subgroup of PGL2(k) [BF03][Lemma 7.2]. For Z4, this happens precisely when −1 is a square in k. Therefore, 1 if − 1 is a square in k. edk(Z4) = 2 otherwise. Z5: edQ(Z5) ≤ φ(4) = 2, so again by [BF03][Lemma 7.2], it suffices to check whether or not Z5 is a subgroup of PGL2(Q). It is not which implies that edQ(Z5) = 2. Z6: edQ(Z6) = edQ(Z2 × Z3) ≤ edQ(Z2) + edQ(Z3) = 2. Moreover, the essential dimension of an abelian group is always bounded below by its rank [BF03][Proposition 3.7]. Therefore edQ(Z6) = 2. 30 Z7: edQ(Z7) ≤ φ(6) = 2, and as in the Z5 case, edQ(Z7) = 2. Z8: 2 edQ(Z8) ≤ φ(2)2 = 4. We compute the essential 2-dimension to get a lower bound. By Theorem 11, it suffices to produce a 2-faithful presentation P Z8 of minimal rank. If rank(P ) < 4, then since P cannot be fixed by the automorphism group, P = Z ⊕ Z[Aut(Z8)/Z2] or P = Z[Aut(Z8)/Z2], but it is easy to see that a generator of Z8 is not fixed by any subgroup of Aut(Z8) = Z2 × Z2. Therefore, rank(P ) ≥ 4, and so edQ(Z8) = 4. Z9: edQ(Z9) ≤ φ(2)3 = 3. Again, using Theorem 11 the essential 3-dimension can be computed. After a degree 2 extension, Aut(Z9) = Z3, and it is clear that no generator of Z9 is fixed by the Z3-action. Thus, arguing as in the Z8 case, the rank of any 3-faithful presentation must be at least 3 which implies that edQ(Z9) = 3. Z10, Z14: By the above remarks, edQ(Z10), edQ(Z14) ≤ φ(2) + φ(4) = φ(2) + φ(7) = 3. This is actually equal to the precise value as computed in [JLY02][Theorem 8.2.11]. Z11: This is the first unknown prime power case. By Theorem 20, edQ(Z11) ≤ φ(10) = 4. I am not aware of any non-trivial lower bound for this problem. Z12: This is the first unknown composite case. It has bounds 2 ≤ edQ(Z12) ≤ 3 given by ed(Z4) from below and by ed(Z4) + ed(Z3) from above. 31 CHAPTER 4 Essential Dimension of PGLn In this chapter, I will compute some bounds on the essential p-dimension of P GLn. Through- out, I will use the following notation 1 Algn(F ) := [iso classes of CSAs over F of degree n] = H (F, PGLn) General upper bounds for the essential dimension of Algn have been computed (n−1)(n−2) 2 if n ≥ 5 odd. edF (Algn) ≤ 2 n − 3n + 1 n ≥ 4,F = Fsep, char(F ) = 0. For references to the above results as well as an overview of what is known see [AAV10]. Unfortunately, these upper bounds differ substantially from the best known lower bounds given by the essential p-dimension. Moreover, even computing the essential p-dimension is a difficult and mostly unsolved problem. There is one nice break however. Because we can apply primary decomposition to central simple algebras, the computation of ed(Algn(−); p) s reduces to a computation of ed(Algps (−); p) where p is the largest power of p dividing n. For s = 1, over a prime to p extension, every non-trivial degree p central simple algebra is isomorphic to a cyclic algebra (see Proposition 3). Since such an algebra is described by a field element and a cyclic extension, ed(Algp; p) ≤ 1 + edp(Zp) = 2. In fact, this 0 is the precise value since, after base extending to F = Falg, there can be no non-trivial division algebras over F 0(x) by Tsen’s theorem. For s = 2, Merkurjev [Mer10a] showed that 2 ed(Algp2 ; p) = p + 1, but the proof is far more involved (see the discussion at the end of this chapter). In 2009, Meyer and Reichstein gave an upper bound for the case s ≥ 2 [MR11][Theorem 32 1.1]: 2s−2 s ed(Algps ; p) ≤ 2p − p + 1 The goal of this chapter is to show that even though this bound agrees with the actual value in the case s = 2, it can be improved for larger values of s: 2s−2 Theorem 22. [Ruo11] ed(Algps ; p) ≤ p + 1 for s ≥ 2. 4.1 The Setup Let φ : S → T be a map of split algebraic tori defined over F , and suppose that we have a finite group G acting on S and T by automorphisms so that φ is G-equivariant. Then by the usual anti-equivalence of categories (Theorem 6), there is an induced map of G-modules T ∗ → S∗ where T ∗ denotes the character group of the torus T . n Let T be a split maximal torus in PGLn(F ). T can be viewed a quotient of S = Gm by the diagonal elements. If we are given a G as above, then letting I denote the character module of T with G-action, we obtain an exact sequence of G-modules: 0 → I → Z[X] → Z → 0 where X is a G-set of n elements. Construct any G-module resolution of I 0 → M → P → I → 0 where P is a permutation module. Now, fixing bases and using the anti-equivalence of categories, this sequence corresponds to an exact sequence of split algebraic tori/F with G acting by automorphisms 1 → T → U → S → 1 Let B be a basis of P permuted by G and let V be the vector space k|B| with the corresponding action by G. Then U = Spec k⊗P embeds as the diagonal in GL(V ). Thus we have a faithful representation T,→ GL(V ). 33 Since T acts on V via this representation, G acts on V by automorphisms, and the above representation is G-equivariant, this extends to a representation T o G → GL(V ). I will construct an upper bound for the essential dimension using the following useful result: Theorem 23. If G acts faithfully on M in the resolution of I constructed above, then ed(T o G) ≤ rank(P ) − rank(I) = rank(M). Proof. [MR09][Lemma 3.3] It suffices by Section 3.3 to exhibit a generically free T o G- representation of dimension rank(M). I claim that V is the desired representation. First, U is a dense open subvariety in V . In particular, T o G acts generically freely on V if and only if it does so on U. Moreover, T,→ U, so T acts faithfully (and thus generically freely) on U by definition. For the T o G action to be generically free, it suffices to show now that G acts faithfully on U/T , but this is exactly the torus corresponding to M. Thus, V is a generically free representation and ed(T o G) ≤ dim(V ) − dim(T ) = rank(P ) − rank(I) = rank(M). One final result about the essential p-dimension of algebraic groups will be useful in the next section: Proposition 24. Let H = Sylp(G). Then ed(T o G; p) = ed(T o H; p). Proof. This is just a special case of [MR09][Lemma 4.1]. 4.2 An Upper Bound Let A be a central simple algebra/F of degree ps with s ≥ 2. By Corollary 2.5, there exists a maximal ´etalesubalgebra K = K1 ⊗ K2 where K1 is ´etaleof degree p and K2 is ´etaleof degree ps−1. 1 The main idea is to consider the functor H (−,Sn) which classifies n-dimensional ´etale algebras up to isomorphism [KMRT98][Section 29.9]. Let G = Sp ×Sps−1 . We have the usual 34 isomorphism 1 1 1 H (F,Sp) × H (F,Sps−1 ) → H (F,G). 1 Converting this to the language of algebras, H (F,G) can be identified as pairs (L1,L2) of s−1 ´etalealgebras of dimensions p and p , respectively. If Sp acts on a set Y of p elements and s−1 Sps−1 acts on a set Z of p elements, then G acts on the product Y ×Z, where Sp permutes the first factor and Sps−1 permutes the second. This yields an inclusion 1 1 H (F,G) → H (F,Sps ) which sends the pair (L1,L2) to the ´etalealgebra L1 ⊗ L2. Using these identifications, the maximal ´etalealgebra K ⊂ A can be viewed as an element of H1(F,G). Consider the split torus T for G as above and X a G-set of ps elements. For its cohomology group, we have a disjoint union of fibers 1 a 1 Γ H (F,T o G) = H (F,Tγ)/Gγ γ∈H1(F,G) sep where Tγ denotes T with action twisted by the cocycle γ and Γ = Gal(F /F ) [KMRT s 28.C]. Tγ is also a torus with character module I. In particular, if X corresponds to a p - dimensional ´etalealgebra N represented by γ then the usual anti-equivalence of categories gives an exact sequence 1 → Gm → RN/F(Gm,N) → Tγ → 1. Passing to cohomology and applying Hilbert theorem 90 gives 1 2 2 1 → H (F,Tγ) → H (F, Gm) → H (F,RN/F (Gm)) 1 Γ Γ showing that H (F,Tγ) ' Br(N/F ). Now, Gγ acts on Gm trivially, so given g ∈ Gγ , we get a diagram: 1 2 1 −−−→ H (F,Tγ) −−−→ H (F, Gm) g y 1 2 1 −−−→ H (F,Tγ) −−−→ H (F, Gm) 35 Γ and the commutativity implies that the action of g must be trivial for all g ∈ Gγ . Combining these observations with the above, 1 a H (F,T o G) = Br(N/F ), N 1 and thus we can define a map φG(F ):H (F,T oG) → Algps (F ) which sends [B] ∈ Br(N/F ) s to the unique (up to isomorphism) C ∼ B with degree C = p . Since any A ∈ Algps is split over L = L1 ⊗ L2, it is in the image of this map. I have proven: Proposition 25. φG is p-surjective. Let Gs = Sylp(Sp × Sps−1 ) = Sylp(Sp) × Sylp(Sps−1 ) := Σ1 × Σs−1. Using Proposition 24 and the property of p-surjective maps (see Section 2.1), edk(T o Gs; p) = edk(T o (Sp × Sps−1 ); p) ≥ edk(Algps (−); p). 4.2.0.2 The Computation The computation of the upper bound is now reduced to that of the essential dimension of T o Gs. By the above remarks, this requires finding a faithful Gs-module, M, in a resolution of I of smallest rank. In what follows, we denote Gs = Σs−1 × Z/p = Σs−1 × hσi, where as above Σs = Sylp(Sps ). s First observe that Gs acts by permutation on a set Xs of p elements. Rewriting s−1 Xs = Ys × Zs for Ys a set of p elements and Zs a set of p elements, this action can be descrbibed as an action of Z/p on Ys and an action of Σs−1 on Zs. In particular, the action is transitive, so if Hs = Stab(y, z) for some (y, z) ∈ Xs, then Xs ' Gs/Hs as Gs-sets. s−1 Let τ ∈ Σs−1 be any p cycle. Then Gs is generated by σ, τ, and the elements of Hs. 0 −1 I is then generated by σx − x and τx − x; [MR11][proof of Theorem 4.1]. Set Hs = τHτ . It is the subgroup of Gs that acts trivially on the element τx − x. This allows us to define a surjective morphism 0 Z[Gs/Hs] ⊕ Z[Gs/Hs] → I 36 by sending a generator of the first summand to σx − x and a generator of the second to τx − x. Because 0 0 s s−2 [Gs : Hs] = [Gs : Hs][Hs : Hs] = p · p this map has kernel M with 0 rank(M) = rank(Z[Gs/Hs]) + rank(Z[Gs/Hs]) − rank(I) = ps + ps+(s−2) − (ps − 1) = p2s−2 + 1. To use Theorem 23, It remains to show that the Gs-action on M is faithful (cf. [MR Lemma 3.2] for a more general argument). Faithfulness can be checked over Q, so we have the split exact sequences: 0 → I ⊗ Q → Q[Gs/Hs] → Q → 0 0 0 → N → Q[Gs/Hs] → Q[Gs/Hs] → 0 0 0 → M ⊗ Q → Q[Gs/Hs] ⊕ Q[Gs/Hs] → I ⊗ Q → 0 Combining these together, 0 (M ⊗ Q) ⊕ (I ⊗ Q) ' Q[Gs/Hs] ⊕ Q[Gs/Hs] ' Q[Gs/Hs] ⊕ N ⊕ Q[Gs/Hs] ' Q[Gs/Hs] ⊕ N ⊕ Q ⊕ (I ⊗ Q). Therefore, Q[Gs/Hs] is a direct summand of M ⊗ Q, so it suffices to check that the Gs action on Q[Gs/Hs] is faithful. However, if the coset gHs is fixed by every element in Gs, then g ∈ T g H g−1. A quick induction argument shows that for all s ≥ 2 this group is gs∈Gs s s s trivial. 37 4.3 A Lower Bound and Some Consequences Merkurjev [Mer10b] has given a lower bound for the the essential p-dimension problem: s edp(Algps ) ≥ (s − 1)p + 1 for s ≥ 2. Taken together with the upper bound of the previous section, it follows that 2 edp(Algp2 ) = p + 1. However, for larger values of s the bounds are quite different. I also remark that the argument used in Merkurjev’s paper is a valuation descent argument which reduces the problem inductively to the computation of the essential p-dimension of a torus. One other interesting consequence of these bounds is that for s = 3 and p = 2 they also coincide to give ed2(Alg8) = 17. In this case, Merkurjev’s lower bound is based on a caluculation for (Z/2Z)3 crossed product algebras, so the equality implies that the essential dimension can be calculated using crossed products. However, Amitsur showed that there are non-crossed product algebras of degree 8 [Ami72]. 38 References [AAV10] Skip Garibaldi Asher Auel, Eric Brussel and Uzi Vishne, Open problems in cen- tral simple algebras, preprint, 2010. [Alb39] A. A. Albert, Structure of algebras, Amer. Math. Soc., 1939. [Ami72] Shimshon Amitsur, On central division algebras, Israel J. Math. 12 (1972), 408– 420. MR MR0318216 (47 #6763) [Ami82] S. 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