ESSENTIAL DIMENSION AND ERROR-CORRECTING CODES

SHANE CERNELE AND ZINOVY REICHSTEIN, WITH AN APPENDIX BY ATHENA NGUYEN

To the memory of Robert Steinberg

Abstract. One of the important open problems in the theory of central simple algebras is to compute the essential dimension of GL_{n} /μ_{m}, i.e., the essential dimension of a generic division algebra of degree n and exponent dividing m. In this paper we study the essential dimension of groups of the form

G = (GL_{n_{1}} × ⋯ × GL_{n_{r}})/C ,

where C is a central subgroup of GL_{n_{1}} × ⋯ × GL_{n_{r}}. Equivalently, we are interested in the essential dimension of a generic r-tuple (A_{1}, …, A_{r}) of central simple algebras such that deg(A_{i}) = n_{i} and the Brauer classes of A_{1}, …, A_{r} satisfy a system of homogeneous linear equations in the Brauer group. The equations depend on the choice of C via the error-correcting code Code(C) which we naturally associate to C. We focus on the case where n_{1}, …, n_{r} are powers of the same prime. The upper and lower bounds on ed(G) we obtain are expressed in terms of coding-theoretic parameters of Code(C), such as its weight distribution. Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when r ≥ 3; in some cases we even compute the exact value. The Appendix by Athena Nguyen contains an explicit description of the Galois cohomology of groups of the form (GL_{n_{1}} × ⋯ × GL_{n_{r}})/C. This description and its corollaries are used throughout the paper.

1. Introduction

Let k be a base field. Unless otherwise specified, we will assume that every field appearing in this paper contains k and every homomorphism (i.e., inclusion) of fields restricts to the identity map on k.

We begin by recalling the definition of the essential dimension of a covariant functor F from the category of fields to the category of sets. Given a field K and an object α ∈ F(K), we will say that α descends to an intermediate field k ⊂ K_{0} ⊂ K if α lies in the image of the natural map F(K_{0}) → F(K). The essential dimension ed(α) of α is defined as the minimal value of trdeg_{k}(K_{0}) such that α descends to a subfield k ⊂ K_{0} ⊂ K. Given a prime integer p, the essential dimension ed_{p}(α) of α at p is defined as the minimal value of trdeg_{k}(K_{0}), where the minumum is taken over all finite field extensions L/K and all intermediate intermediate fields k ⊂ K_{0} ⊂ L, such that [L : K] is prime to p and α_{L} descends to K_{0}.

2010 Mathematics Subject Classification. Primary 20G15, 16K20, 16K50. Secondary 94B05. This paper is based on a portion of the first author’s Ph.D. thesis completed at the University of

British Columbia. Both authors gratefully acknowledge financial support from the University of British Columbia and the Natural Sciences and Engineering Research Council of Canada.

The essential dimension ed(F) (respectively, the essential dimension ed_{p}(F) at p) of the functor F is defined as the maximal value of ed(α) (respectively of ed_{p}(α)), where the maximum is taken over all feld extensions K/k and all objects α ∈ F(K).

Informally speaking, ed(α) is the minimal number of independent parameters required to define α, ed(F) is the minimal number of independent parameters required to define any object in F, and ed_{p}(α), ed_{p}(F) are relative versions of these notions at a prime p. These relative versions are somewhat less intuitive, but they tend to be more accessible and more amenable to computation than ed(α) and ed(F). Clearly ed(α) ⩾ ed_{p}(α) for each α, and ed(F) ⩾ ed_{p}(F). In most cases of interest ed(α) is finite for every α. On the other hand, ed(F) (and even ed_{p}(F)) can be infinite. For an introduction to the theory of essential dimension, we refer the reader to the surveys [BF03], [Re10], [Re12] and [M13].

To every algebraic group G one can associate the functor

F_{G} := H^{1}(∗, G): K ↦ {isomorphism classes of G-torsors over Spec(K)}.

If G is affine, then the essential dimension of this functor is known to be finite; it is usually denoted by ed(G), rather than ed(F_{G}). For many specific groups G, H^{1}(K, G) is in a natural bijective correspondence with the set of isomorphism classes of some algebraic objects defined over K. In such cases, ed(G) may be viewed as the minimal number of independent parameters required to define any object of this type. This number is often related to classical problems in algebra.

For example, in the case where G is the projective linear group PGL_{n}, the objects in question are central simple algebras. That is,

(1) H^{1}(K, PGL_{n}) = {isomorphism classes of cenral simple K-algebras of degree n}.

The problem of computing ed(PGL_{n}) is one of the important open problems in the theory of central simple algebras; see [ABGV11, Section 6]. This problem was first posed by C. Procesi, who showed (using different terminology) that

(2)  ed(PGL_{n}) ⩽ n^{2} ;

see [Pr67, Theorem 2.1]. Stronger (but still quadratic) upper bounds can be found in [LRRS03, Theorem 1.1] and [Le04, Theorem 1.6].

A more general but closely related problem is computing ed(GL_{n} /μ_{m}), where m and n are positive integers and m divides n. Note that

(3)

H^{1}(K, GL_{n} /μ_{m}) = {isomorphism classes of central simple K-algebras

of degree n and exponent dividing m}.

In particular, ed(PGL_{n}) = ed(GL_{n} /μ_{n}). The problem of computing ed(GL_{n} /μ_{m}) partially reduces to the case where m = p^{s} and n = p^{a} are powers of the same prime p and 1 ⩽ s ⩽ a.

From now on we will always assume throughout that char(k) ≠ p. The inequalities

(4)  p^{2a−2} + p^{a−s} ⩾ ed_{p}(GL_{p^{a}} /μ_{p^{s}}) ⩾ \begin{cases}(a − 1)2^{a−1} & if p = 2 and s = 1, \\ (a − 1)p^{a} + p^{a−s} & otherwise, \end{cases}

proved in [BM12] represent a striking improvement on the best previously known bounds. (Here a ⩾ 2.) Yet the gap between the lower and upper bounds in (4) remains wide.

The gap between the best known upper and lower bounds becomes even wider when ed_{p}(GL_{p^{a}} /μ_{p^{s}}) is replaced by ed(GL_{p^{a}} /μ_{p^{s}}).

These gaps in our understanding of ed(GL_{n} /μ_{m}) will not deter us from considering the vastly more general problem of computing the essential dimension of groups of the form

(5)  G := (GL_{n_{1}} × ⋯ × GL_{n_{r}})/C,

in the present paper. Here n_{1}, …, n_{r} ⩾ 2 are integers, and C ⊂ 𝔾_{m}^{r} is a central subgroup of GL_{n_{1}} × ⋯ × GL_{n_{r}}.

As usual, we will identify elements (m_{1}, …, m_{r}) of ℤ^{r} with characters x: 𝔾_{m}^{r} → 𝔾_{m}, where x: (τ_{1}, …, τ_{r}) ↦ τ_{1}^{m_{1}} … τ_{r}^{m_{r}}. The subgroup C ⊂ 𝔾_{m}^{r} is completely determined by the ℤ-module

(6)  X(𝔾_{m}^{r} /C) = {(m_{1}, …, m_{r}) ∈ ℤ^{r} | τ_{1}^{m_{1}} … τ_{r}^{m_{r}} = 1 ∀(τ_{1}, …, τ_{r}) ∈ C} ,

consisting of characters of 𝔾_{m}^{r} which vanish on C. The Galois cohomology of G is explicitly described in the appendix: by Theorem A.1, H^{1}(K, G) is naturally isomorphic to the set of isomorphism classes of r-tuples (A_{1}, …, A_{r}) of central simple K-algebras such that

deg(A_{i}) = n_{i} and A_{1}^{⊗m_{1}} ⊗ ⋯ ⊗ A_{r}^{⊗m_{r}} is split over K

for every (m_{1}, …, m_{r}) ∈ X(𝔾_{m}^{r} /C). (Note that in the special case where r = 1, we recover (1) and (3).) It follows from this description that the essential dimension of G does not change if C is replaced by C ∩ μ, where

(7)  μ := μ_{n_{1}} × ⋯ × μ_{n_{r}};

see Corollary A.2. Thus we will assume throughout that C ⊂ μ. Unless otherwise specified, we will also assume that n_{1} = p^{a_{1}}, …, n_{r} = p^{a_{r}} are powers of the same prime p. Here a_{1}, …, a_{r} ⩾ 1 are integers. Under these assumptions, instead of X(𝔾_{m}^{r} /C) ⊂ ℤ^{r} we will consider the subgroup of X(μ) = (ℤ /p^{a_{1}} ℤ) × ⋯ × (ℤ /p^{a_{r}} ℤ) given by

(8) Code(C) := X(μ/C) = {(m_{1}, …, m_{r}) ∈ X(μ) | τ_{1}^{m_{1}} … τ_{r}^{m_{r}} = 1 ∀(τ_{1}, …, τ_{r}) ∈ C} .

In other words, Code(C) consists of those characters of μ which vanish on C. The symbol “Code” indicates that we will view this group as an error-correcting code. In particular, we will define the Hamming weight w(y) of

y = (m_{1}, …, m_{r}) ∈ (ℤ /p^{a_{1}} ℤ) × ⋯ × (ℤ /p^{a_{r}} ℤ)

as follows. Write m_{i} := u_{i}p^{e_{i}} with u_{i} ∈ (ℤ / p^{a_{i}} ℤ)^{∗} and 0 ≤ e_{i} ≤ a_{i}. Then

w(y) := ∑_{i=1}^{r}(a_{i} − e_{i}) .

Our main results relate ed(G) to coding-theoretic invariants of Code(C), such as its weight distribution; cf. also Corollary A.3. For an introduction to error-correcting coding theory, see [MS77].

At this point we should warn the reader that our notions of error-correcting code and Hamming weight are somewhat unusual. In coding-theoretic literature (linear) codes are usually defined as linear subspaces of 𝔽_{q}^{n}, where 𝔽_{q} is the field of q elements. In this paper by a code we will mean an additive subgroup of (ℤ /p^{a_{1}} ℤ)×⋯×(ℤ /p^{a_{r}} ℤ). Nevertheless, in an important special case, where a_{1} = ⋯ = a_{r} = 1, our codes are linear codes of length r over 𝔽_{p} in the usual sense of error-correcting coding theory, and our definition of the Hamming weight coincides with the usual definition.

Theorem 1.1. Let p be a prime, G := (GL_{p^{a_{1}}} × ⋯×GL_{p^{ar}})/C, where C ⊂ μ_{p^{a_{1}}} ×⋯×μ_{p^{ar}} is a central subgroup, and y_{1}, …, y_{t} be a minimal basis for Code(C); see Definition 3.2. Then

(a) ed_{p}(G) ⩾ (∑_{i=1}^{t} p^{w(y_{i})}) − p^{2a_{1}} − ⋯ − p^{2a_{r}} + r − t,

(b) ed(G) ⩽ (∑_{i=1}^{t} p^{w(y_{i})}) − t + ed(\overline{G}) and ed_{p}(G) ⩽ (∑_{i=1}^{t} p^{w(y_{i})}) − t + ed_{p}(\overline{G}), where \overline{G} := PGL_{p^{a_{1}}} × ⋯ × PGL_{p^{ar}}.

Although the upper and lower bounds of Theorem 1.1 never meet, for many central

subgroups C ⊂ μ ⊂ G, the term ∑_{i=1}^{t} p^{w(y_{i})} is much larger than any of the other terms

appearing in the above inequalities and may be viewed as giving the asymptotic value of ed(G). (In particular, note that in view of (2),

(9)  ed_{p}(\overline{G}) ⩽ ed(\overline{G}) ⩽ ed(PGL_{p^{a_{1}}}) + ⋯ + ed(PGL_{p^{ar}}) ⩽ p^{2a_{1}} + ⋯ + p^{2a_{r}}.)

Under additional assumptions on C, we will determine ed(G) exactly; see Theorem 1.2.

The fact that we can determine ed(G) for many choices of C, either asymptotically or exactly, was rather surprising to us, given the wide gap between the best known upper and lower bounds on ed(G) in the simplest case, where r = 1; see (4). Our informal explanation of this surprising phenomenon is as follows. If Code(C) can be generated by

vectors y_{1}, …, y_{t} of small weight, then ∑_{i=1}^{t} p^{w(y_{i})} no longer dominates the other terms. In

particular, this always happens if r ⩽ 2. In such cases the value of ed(G) is controlled by the more subtle “lower order effects”, which are poorly understood.

To state our next result, we will need the following terminology. Suppose 2 ⩽ n_{1} ⩽ ⋯ ⩽ n_{t} and z = (z_{1}, …, z_{r}) ∈ (ℤ /n_{1} ℤ) × ⋯ × (ℤ /n_{r} ℤ), where z_{j_{1}}, …, z_{j_{s}} ≠ 0 for some 1 ⩽ j_{1} < ⋯ < j_{s} ⩽ r and z_{j} = 0 for any j ∉ {j_{1}, …, j_{r}}. We will say that z is balanced if

(i) n_{j_{s}} ⩽ \frac{1}{2}n_{j_{1}}n_{j_{2}} … n_{j_{s−1}} and

(ii) (n_{j_{1}}, …, n_{j_{s}}) ≠ (2, 2, 2, 2), (3, 3, 3) or (2, n, n) for any n ⩾ 2. Note that condition (i) can only hold if s ⩾ 3. In particular, (ℤ /n_{1} ℤ) × ⋯ × (ℤ /n_{r} ℤ) has no balanced elements if r ⩽ 2. In the sequel we will usually assume that n_{1}, …, n_{r} are powers of the same prime p. In this situation condition (ii) is vacuous, unless p = 2 or 3.

Theorem 1.2. Let p be a prime, G := (GL_{p^{a_{1}}} × ⋯ × GL_{p^{ar}})/C, where a_{r} ⩾ ⋯ ⩾ a_{1} ⩾ 1 are integers, and C is a subgroup of μ, as in (7). Assume that the base field k is of characteristic zero and Code(C) has a minimal basis y_{i} = (y_{i1}, …, y_{ir}), i = 1, …, t satisfying the following conditions:

(a) y_{ij} = −1, 0 or 1 in ℤ /p^{a_{j}} ℤ, for every i = 1, …, t and j = 1, …, r.

(b) For every j = 1, …, r there exists an i ∈ {1, …, t} such that y_{i} is balanced and y_{ij} ≠ 0.

Then ed(G) = ed_{p}(G) = (∑_{i=1}^{t} p^{w(y_{i})}) − p^{2a_{1}} − ⋯ − p^{2a_{r}} + r − t.

Specializing Theorem 1.2 to the case where Code(C) is generated by the single element (1, …, 1), we obtain the following.

Theorem 1.3. Let a_{r} ⩾ a_{r−1} ⩾ ⋯ ⩾ a_{1} ⩾ 1 be integers and F: Fields_{k} → Sets be the covariant functor given by

F(K) := \{ \begin{matrix}isomorphism classes of r-tuples (A_{1}, …, A_{r}) of central simple K-algebras \\ such that deg(A_{i}) = p^{a_{i}} ∀ i = 1, …, r, and A_{1} ⊗ … ⊗ A_{r} is split over K. \end{matrix} \}

(a) If a_{r} ⩾ a_{1} + ⋯ + a_{r−1}, then ed(F) = ed(PGL_{p^{a_{1}}} × ⋯ × PGL_{p^{a_{r−1}}}) and

ed_{p}(F) = ed_{p}(PGL_{p^{a_{1}}} × ⋯ × PGL_{p^{a_{r−1}}}) .

In particular, ed(F) ⩽ p^{2a_{1}} + ⋯ + p^{2a_{r−1}}.

(b) Assume that char(k) = 0, a_{r} < a_{1} + ⋯ + a_{r−1}, and (p^{a_{1}}, …, p^{a_{r}}) is not of the form (2, 2, 2, 2), (3, 3, 3) or (2, 2^{a}, 2^{a}), for any a ⩾ 1. Then

(10)  ed(F) = ed_{p}(F) = p^{a_{1}+···+a_{r}} − ∑_{i=1}^{r} p^{2a_{i}} + r − 1 .

(c) If (p^{a_{1}}, …, p^{a_{r}}) = (2, 2, 2), then ed(F) = ed_{2}(F) = 3.

Here part (c) treats the smallest of the exceptional cases in part (b). Note that in this case p = 2, r = 3 and a_{1} = a_{2} = a_{3} = 1. Thus p^{a_{1}+···+a_{r}} − ∑_{i=1}^{r} p^{2a_{i}} + r − 1 = −2, and formula (10) fails. The values of ed(F) and ed_{p}(F) in the other exceptional cases, where (p^{a_{1}}, …, p^{a_{r}}) = (2, 2, 2, 2), (3, 3, 3), or (2, 2^{a}, 2^{a}) for some a ⩾ 2, remain open.

The results of this paper naturally lead to combinatorial questions, which we believe to be of intependent interest but will not address here. For each code (i.e. subgroup) X ⊂ (ℤ /p^{a_{1}} ℤ) × ⋯ × (ℤ /p^{a_{r}} ℤ) of rank t, let (w_{1}, …, w_{t}) be the minimal profile of X with respect to the Hamming weight function, in the sense of Proposition 3.1. That is, w_{i} = w(y_{i}), where y_{1}, …, y_{t} is a minimal basis of X. Fixing p, a_{1} ⩽ ⋯ ⩽ a_{r} and t, and letting X range over all possible codes with these parameters,

• What is the lexicographically largest profile (w_{1}, …, w_{t})?

• What is the maximal value of w_{t}?

• What is the probability that w_{1} = ⋯ = w_{t}?

• What is the maximal value of p^{w_{1}} + ⋯ + p^{w_{t}}?

• What is the average value of p^{w_{1}} + ⋯ + p^{w_{t}}?

• What is the probability that w_{t} > 2a_{r}?

Note that the expression p^{w_{1}} + ⋯ + p^{w_{t}} appears in the formulas given in Theorem 1.1. For large p the condition that w_{t} > 2a_{r} makes p^{w_{1}} + ⋯ + p^{w_{t}} the dominant term in these formulas. To the best of our knowledge, questions of this type (focusing on the minimal profile of a code, rather than the minimal weight) have not been previously investigated by coding theorists even in the case, where a_{1} = ⋯ = a_{r} = 1.

The rest of this paper is structured as follows. In Section 2 we prove general bounds on the essential dimension of certain central extensions of algebraic groups. These bounds will serve as the starting point for the proofs of the main theorems. To make these bounds explicit for groups of the form (GL_{p^{a_{1}}} × ⋯×GL_{p^{ar}})/C we introduce and study the notion of a minimal basis in Section 3. Theorem 1.1, 1.2 and 1.3 are then proved in Sections 4, 5 and 6, respectively. The Appendix by Athena Nguyen contains an explicit description of the Galois cohomology of groups of the form (5). This description and its corollaries are used throughout the paper.

2. Essential dimension and central extensions

Let T = 𝔾_{m}^{r} be a split k-torus of rank r, and

(11)  1 → T → G → \overline{G} → 1

be a central exact sequence of affine algebraic groups. This sequence gives rise to the exact sequence of pointed sets

H^{1}(K, G) → H^{1}(K, \overline{G}) →^{∂} H^{2}(K, T)

for any field extension K of the base field k. Any character x: T → 𝔾_{m}, induces a homomorphism x_{∗}: H^{2}(K, T) → H^{2}(K, 𝔾_{m}). We define ind^{x}(G, T) as the maximal index of x_{∗} ∘ ∂_{K}(E) ∈ H^{2}(K, T), where the maximum is taken over all field extensions K/k and over all E ∈ H^{1}(K, \overline{G}). In fact, this maximal value is always attained in the case where E = E_{vers} → Spec(K) is a versal G-torsor (for a suitable field K). That is,

(12)  ind^{x}(G, T) = ind(x_{∗} ∘ ∂_{K}(E_{vers}))

for every x ∈ X(T); see [M13, Theorem 6.1]. Finally, we set

(13)  ind(G, T) := min {∑_{i=1}^{r} ind^{x_{i}}(G, T) | x_{1}, …, x_{r} generate X(T)} .

Our starting point for the proof of the main theorems is the following proposition.

Proposition 2.1. Assume that the image of every E ∈ H^{1}(K, \overline{G}) under

∂: H^{1}(K, \overline{G}) → H^{2}(K, T)

is p-torsion for every field extension K/k. Then (a) ed_{p}(G) ⩾ ind(G, T) − dim(G),

(b) ed(G) ⩽ ind(G, T) + ed(\overline{G}) − r and ed_{p}(G) ⩽ ind(G, T) + ed_{p}(\overline{G}) − r.

These bounds are variants of results that have previously appeared in the literature. Part (a) is a generalization of [BRV12, Corollary 4.2] (where r is taken to be 1). In the case where T is μ_{p}^{r}, rather than 𝔾_{m}^{r}, a variant of part (a) is proved in [Re10, Theorem 4.1] (see also [M13, Theorem 6.2]) and a variant of part (b) in [M13, Corollaries 5.8 and 5.12].

Our proof of Proposition 2.1 proceeds along the same lines as these earlier proofs; it relies on the notions of essential and canonical dimension of a gerbe (for which we refer the reader to [BRV12] and [M13]), and the computation of the canonical dimension of a product of p-primary Brauer-Severi varieties in [KM08, Theorem 2.1]. In fact, the argument is easier for T = 𝔾_{m}^{r} than for μ_{p}^{r}. In the former case (which is of interest to us here) the essential dimension of a gerbe banded by T is readily expressible in terms of its canonical dimension (see formula (15) below), while an analogous formula for gerbes banded by μ_{p}^{r} requires a far greater effort to prove. (For r = 1, compare the proofs of parts (a) and (b) of [BRV12, Theorem 4.1]. For arbitrary r ⩾ 1, see [KM08, Theorem 3.1] or [M13, Theorem 5.11].)

Proof. If K/k is a field, and E ∈ H^{1}(K, G), i.e. E → Spec(K) is a \overline{G}-torsor, then the quotient stack [E/G] is a gerbe over Spec(K) banded by T. By [BRV12, Corollary 3.3] and [M13, Corollary 5.7], ed(G) ⩾ max_{K,E} ed([E/G]) − dim(\overline{G}) and similarly

ed_{p}(G) ⩾ max_{K,E} ed_{p}([E/G]) − dim(\overline{G}) ,

where the maximum is taken over all field extensions K/k and all E ∈ H^{1}(K, \overline{G}). On the other hand, by [Lö13, Example 3.4(i)]

ed(G) ⩽ ed(\overline{G}) + max_{K,E} ed([E/G]) and ed_{p}(G) ⩽ ed_{p}(\overline{G}) + max_{K,E} ed_{p}([E/G]) ;

see also [M13, Corollary 5.8]. Since dim(G) = dim(\overline{G}) + r, it remains to show that

(14)  max_{K,E} ed([E/G]) = max_{K,E} ed_{p}([E/G]) = ind(G, T) − r .

Choose a ℤ-basis x_{1}, …, x_{r} for the character group X(T) ≃ ℤ^{r} and let P := P_{1}×⋯×P_{r}, where P_{i} is the Brauer–Severi variety associated to (x_{i})_{∗} ∘ ∂(E) ∈ H^{2}(K, 𝔾_{m}). Since T is a special group (i.e., every T -torsor over every field K/k is split), the set [E/G](K) of isomorphism classes of K-points of [E/G] consists of exactly one element if P (K) ≠ ∅ and is empty otherwise. Thus

(15)  ed([E/G]) = cdim(P) and ed_{p}([E/G]) = cdim_{p}(P),

where cdim(P) denotes the canonical dimension of P. (The same argument is used in the proof of [BRV12, Theorem 4.1(a)] in the case, where r = 1.) Since we are assuming that ∂(E) is p-torsion, the index of each Brauer-Severi variety P_{i} is a power of p. Thus by [KM08, Theorem 2.1],

cdim(P) = cdim_{p}(P) = min {∑_{i=1}^{r} ind((x_{i})_{∗} ∘ ∂_{K}(E)) | x_{1}, …, x_{r} generate X(T)} − r;

see also [M13, Theorem 4.14]. Taking E := E_{vers} to be a versal \overline{G}-torsor, we obtain

cdim(P) = cdim_{p}(P) = min {∑_{i=1}^{r} ind^{x_{i}}(G, T)) | x_{1}, …, x_{r} generate X(T)} − r;

see (12). By the definition (13) of ind(G, T), the last formula can be rewritten as cdim(P) = cdim_{p}(P) = ind(G, T) − r. Combining these equalities with (15), we obtain (14). □

Remark 2.2. Our strategy for proving Theorem 1.1 will be to apply Proposition 2.1 to the exact sequence (11) with G = (GL_{p^{a_{1}}} × ⋯ × GL_{p^{ar}})/C, and T := 𝔾_{m}^{r} /C. The only remaining issue is to find an expression for ind(G, T) in terms of Code(C).

Usually, the term ind(G, T) is computed using the formula ind^{x}(G, T) = gcd dim(ρ), as ρ: G → GL(V) ranges over all finite-dimensional representations of G, such that τ ∈ T acts on V via scalar multiplication by x(τ). See, for example, [KM08, Theorem 4.4] or [M13, Theorem 6.1] or [LMMR13, Theorem 3.1]. We will not use this approach in the present paper. Instead, we will compute the values of ind^{x}(G, T) and ind(G, T) directly from the definition, using the description of the connecting map ∂: H^{1}(K, \overline{G}) → H^{2}(K, T) given by Theorem A.1; see the proof of Proposition 4.1 below.

3. Minimal bases

To carry out the program outlined in Remark 2.2 we will need the notion of a minimal basis. This section will be devoted to developing this notion.

The general setting is as follows. Let R be a local ring with maximal ideal I ⊂ R and A be a finitely generated R-module. We will refer to a generating set S ⊂ A as a basis if no proper subset of S generates A. In the sequel we will specialize R to ℤ /p^{a} ℤ and A to a submodule of (ℤ /p^{a_{1}} ℤ) × ⋯ × (ℤ /p^{a_{t}} ℤ), where a = max(a_{1}, …, a_{r}). However, in this section it will be convenient for us to work over an arbitrary local ring R.

Let π: A → A/IA be the natural projection. We will repeatedly appeal to Nakayama’s Lemma, which asserts that a subset S ⊂ A generates A as an R-module if and only if π(S) generates A/IA as an R/I-vector space; see [Lang02, Section X.4].

By a weight function on A we shall mean any function w : A → ℕ, where ℕ denotes the set of non-negative integers. We will fix w throughout and will sometimes refer to w(y) as the weight of y ∈ A. For each basis B = {y_{1}, …, y_{t}} of A, we will define the profile of B as

w(B) := (w(y_{1}), …, w(y_{t})) ∈ ℕ^{t} ,

where y_{1}, …, y_{t} are ordered so that w(y_{1}) ⩽ w(y_{2}) ⩽ ⋯ ⩽ w(y_{t}). Let Prof(A) ⊂ ℕ^{t} denote the set of profiles of bases of A.

Proposition 3.1. Prof(A) has a unique minimal element with respect to the partial order on ℕ^{t} given by (α_{1}, …, α_{t}) ⪯ (β_{1}, …, β_{t}) if α_{i} ⩽ β_{i} for every i = 1, …, t.

Note that since every descending chain in (Prof(A), ⪯) terminates, the unique minimal element is comparable to every element of Prof(A).

Proof. We argue by contradiction. Set t := dim(A/IA). Suppose X = {x_{1}, …, x_{t}} and Y = {y_{1}, …, y_{t}} are bases of A such that w(X) and w(Y) are distinct minimal elements of Prof(A). Let us order X and Y so that w(x_{1}) ⩽ ⋯ ⩽ w(x_{t}) and w(y_{1}) ⩽ ⋯ ⩽ w(y_{t}). Since w(X) ≠ w(Y), there exists an s between 0 and t − 1 such that

w(x_{i}) = w(y_{i}) for all i = 1, …, s

but w(x_{s+1}) ≠ w(y_{s+1}). After possibly interchanging X and Y, we may assume without loss of generality that w(x_{s+1}) < w(y_{s+1}).

Let π: A → A/IA be the natural projection, as above. By Nakayama’s Lemma, π(x_{1}), …, π(x_{s+1}) are R/I-linearly independent in A/IA. Choose t − s − 1 elements of Y, say y_{j_{s+2}}, …, y_{j_{t}} such that π(x_{1}), …, π(x_{s+1}), π(y_{j_{s+2}}), …, π(y_{j_{t}}) form an R/I-basis of A/IA. After permuting y_{j_{s+2}}, …, y_{j_{t}}, we may assume that w(y_{j_{s+2}}) ⩽ ⋯ ⩽ w(y_{j_{t}}).

Applying Nakayama’s lemma once again, we see that Z = {x_{1}, …, x_{s+1}, y_{j_{s+2}}, …, y_{j_{t}}} is a basis of A.

We claim that w(Z) ≺ w(Y), where the inequality is strict. Since we assumed that w(Y) is minimal in Prof(A), this claim leads to a contradiction, thus completing the proof of Proposition 3.1.

To prove the claim, let z_{1}, …, z_{t}, be the elements of Z, in increasing order of their weight: w(z_{1}) ⩽ w(z_{2}) ⩽ ⋯ ⩽ w(z_{t}). It suffices to show that w(z_{i}) ⩽ w(y_{i}) for every i = 1, …, t, and w(z_{s+1}) < w(y_{s+1}). Let us consider three cases.

(i) i ⩽ s. Since

w(x_{1}) = w(y_{1}) ⩽ w(x_{2}) = w(y_{2}) ⩽ ⋯ ⩽ w(x_{i}) = w(y_{i}) ,

Z has at least i elements whose weight is ⩽ w(y_{i}), namely x_{1}, …, x_{i}. Thus w(z_{i}) ⩽ w(y_{i}).

(ii) i = s + 1. Z has at least s + 1 elements, namely x_{1}, …, x_{s+1} whose weight is at most w(x_{s+1}). Hence, w(z_{s+1}) ⩽ w(x_{s+1}) < w(y_{s+1}), as desired.

(iii) i > s + 1. Recall that both y_{1}, …, y_{t} and y_{j_{s+2}}, …, y_{j_{t}} are arranged in weightincreasing order. For any i ⩾ s + 2 there are at least t − i + 1 elements of Y whose weight is ⩾ w(y_{j_{i}}), namely y_{j_{i}}, y_{j_{i+1}}, …, y_{j_{t}}. Thus

w(y_{j_{i}}) ⩽ w(y_{i})

for any i = s+2, …, t. Consequently, Z has at least i elements of weight ⩽ w(y_{i}), namely x_{1}, …, x_{s+1}, y_{j_{s+2}}, …, y_{j_{i}}. Hence, w(z_{i}) ⩽ w(y_{i}), as desired.

This completes the proof of the claim and hence of Proposition 3.1. □

Definition 3.2. We will say that a basis y_{1}, …, y_{t} of A is minimal if its profile is the minimal element of Prof(A), as in Proposition 3.1. Note that a minimal basis in A is usually not unique; however any two minimal bases have the same profile in ℕ^{t}.

Remark 3.3. We can construct a minimal basis of A using the following “greedy algorithm”. Select y_{1} ∈ A of minimal weight, subject to the condition that π(y_{1}) ≠ 0. Next select y_{2} of minimal weight, subject to the condition that π(y_{1}) and π(y_{2}) are R/I-linear independent in A/IA. Then select y_{3} of minimal weight, subject to the condition that π(y_{1}), π(y_{2}) and π(y_{3}) are R/I-linear independent in A/IA. Continue recursively. After t = dim_{R/I}(A/IA) steps, we obtain a minimal basis y_{1}, …, y_{t} for A.

Example 3.4. Set R := 𝔽_{p}, I := (0), G a finite p-group, D := Z(G)[p] the subgroup of p-torsion elements of the center Z(G), and A := X(D) the group of characters of D. For x ∈ A, define w(x) to be the minimal dimension of a representation G → GL(V_{x}), such that D acts on V_{x} via scalar multiplication by x. If {x_{1}, …, x_{t}} is a minimal basis of A, then V_{x_{1}} ⊕ ⋯ ⊕ V_{x_{t}} is a faithful representation of G of minimal dimension; see [KM08, Remark 4.7].

4. Conclusion of the proof of Theorem 1.1

Recall that we are interested in the essential dimension of the group

G = (GL_{p^{a_{1}}} × ⋯ × GL_{p^{ar}})/C ,

where C is a subgroup of μ := μ_{p^{a_{1}}} × ⋯ × μ_{p^{ar}}. We will think of the group of characters X(𝔾_{m}^{r}) as ℤ^{r} by identifying the character x(τ_{1}, …, τ_{r}) = τ_{1}^{m_{1}} … τ_{r}^{m_{r}} with (m_{1}, …, m_{r}) ∈ ℤ^{r}. Characters of T := 𝔾_{m}^{r} /C are identified in this manner with the r-tuples (m_{1}, …, m_{r}) ∈ ℤ^{r} such that τ_{1}^{m_{1}} … τ_{r}^{m_{r}} = 1 for every (τ_{1}, …, τ_{r}) ∈ C. The relationship among these character groups is illustrated by the following diagram

[figure: X(𝔾_{m}^{r} /C); X(𝔾_{m}^{r}); π; ℤ × ⋯ × ℤ (r times); π; Code(C); X(μ/C); X(μ); (ℤ /p^{a_{1}} ℤ) × ⋯ × (ℤ /p^{a_{r}} ℤ).]

Here Code(C) is as in (8) and π is the natural projection, given by restricting a character from 𝔾_{m}^{r} to μ.

Our proof of Theorem 1.1 will be based on the strategy outlined in Remark 2.2. In view of Proposition 2.1 it suffices to establish the following:

Proposition 4.1. Consider the central exact sequence

(16)  1 → T → G → \overline{G} → 1 ,

where G = (GL_{p^{a_{1}}} × ⋯×GL_{p^{ar}})/C, C is a subgroup of μ := μ_{p^{a_{1}}} ×⋯×μ_{p^{ar}}, T := 𝔾_{m}^{r} /C and \overline{G} := PGL_{p^{a_{1}}} × ⋯ × PGL_{p^{ar}}.

(a) If x ∈ X(T) and y = π(x) ∈ Code(C) then ind^{x}(G, T) = p^{w(y)}.

(b) ind(G, T) = p^{w(z_{1})} + ⋯ + p^{w(z_{t})} + r − t, where z_{1}, …, z_{t} is a minimal basis of Code(C).

Proof of Proposition 4.1(a). Consider the connecting map ∂: H^{1}(K, \overline{G}) → H^{2}(K, T) associated to the central exact sequence (16). Given a character x: T → 𝔾_{m}, x(τ_{1}, …, τ_{r}) = τ_{1}^{m_{1}} … τ_{r}^{m_{r}}, ind^{x}(G, T) is, by definition, the maximal value of ind(x_{∗}∂(E)), as K ranges over all fields containing k and E ranges over H^{1}(K, \overline{G}). In this case, \overline{G} = PGL_{p^{a_{1}}} × ⋯× PGL_{p^{ar}}, and thus H^{1}(K, \overline{G}) is the set of r-tuples (A_{1}, …, A_{r}) of central simple algebras, where the degree of A_{i} is p^{a_{i}}. The group H^{2}(K, 𝔾_{m}) is naturally identified with the Brauer group Br(K), and the map x_{∗}∂ takes an r-tuple (A_{1}, …, A_{r}), as above, to the Brauer class of A := A_{1}^{⊗m_{1}} ⊗ ⋯ ⊗ A_{r}^{⊗m_{r}}.

Since deg(A_{i}) = p^{a_{i}}, the Brauer class of A depends only on

y = π(x) = (m_{1} mod p^{a_{1}} , …, m_{r} mod p^{a_{r}}) ∈ (ℤ /p^{a_{1}} ℤ) × ⋯ × (ℤ /p^{a_{r}} ℤ)) .

Moreover, if m_{i} ≡ u_{i}p^{e_{i}} (mod p^{a_{i}}), where u_{i} is prime to p and 0 ⩽ e_{i} ⩽ a_{i} then ind(A_{i}^{⊗m_{i}}) ⩽ p^{a_{i}−e_{i}}. Now recall that w(y) is defined as ∑_{i=1}^{r}(a_{i} − e_{i}). Thus

ind(A) ⩽ ∏_{i=1}^{r} ind(A_{i}^{⊗m_{i}}) ⩽ ∏_{i=1}^{r} p^{a_{i}−e_{i}} = p^{w(y)} .

To prove the opposite inequality, we set A_{i} to be the symbol algebra (α_{i}, β_{i})_{p^{a_{i}}}, over the field K = k(ζ)(α_{1}, …, α_{r}, β_{1}, …, β_{r}), where ζ is a primitive root of unity of degree p^{max(a_{1},...,a_{r})} and α_{1}, …, α_{r}, β_{1}, …, β_{r} are 2r independent variables over k. Writing m_{i} = u_{i}p^{e_{i}}, as above, we see that A_{i}^{⊗m_{i}} is Brauer equivalent to B_{i} := (α_{i}, β_{i}^{u_{i}})_{p^{a_{i}−e_{i}}} over K. An easy valuation-theoretic argument shows that B := B_{1} ⊗_{K} ⋯ ⊗_{K} B_{t} is a division algebra.

(In particular, the norm form of B is a Pfister polynomial and hence, is anisotropic; see [Re99, Theorem 3.2 and Proposition 3.4].) Thus

ind(A) = ind(B) = ind(B_{1}) · . . . · ind(B_{t}) = p^{(a_{1}−e_{1})+···+(a_{t}−e_{t})} = p^{w(y)} ,

as desired. We conclude that ind^{x}(G, T) ⩾ ind(A) = p^{w(y)}, thus completing the proof of Proposition 4.1(a). □

Our proof of Proposition 4.1(b) will rely on the following elementary lemma.

Lemma 4.2. Let p be a prime, M be a finite abelian p-group, and f: ℤ^{n} → M be a surjective ℤ-module homomorphism for some n ⩾ 1. Then for every basis y_{1}, …, y_{t} of M, there exists a ℤ-basis x_{1}, …, x_{n} of ℤ^{n} and an integer c prime to p, such that f(x_{1}) = cy_{1}, f(x_{2}) = y_{2}, …, f(x_{t}) = y_{t} and f(x_{t+1}) = ⋯ = f(x_{n}) = 0.

Proof. By [Lang02, Theorem III.7.8] there exists a basis e_{1}, …, e_{n} of ℤ^{n} such that Ker(f) is generated by p^{d_{i}}e_{i} for some integers d_{1}, …, d_{t} ⩾ 0. Since M has rank t, we may assume without loss of generality that d_{1}, …, d_{t} ⩾ 1 and d_{t+1} = ⋯ = d_{n} = 0. That is, we may identify M with (ℤ /p^{d_{1}} ℤ) × ⋯ × (ℤ /p^{d_{t}} ℤ) and assume that

f(r_{1}, …, r_{n}) = ( r_{1} mod p^{d_{1}}, …, r_{t} mod p^{d_{t}}) ∀(r_{1}, …, r_{n}) ∈ ℤ^{n} .

It now suffices to lift cy_{1}, …, y_{t} ∈ M to a basis x_{1}, …, x_{t} of ℤ^{t}, for a suitable integer c, prime to p. Indeed, if we manage to do this, then we will obtain a basis of ℤ^{n} of the desired form by appending

x_{t+1} := e_{t+1}, …, x_{n} := e_{n} ∈ Ker(f)

to x_{1}, …, x_{t}. Thus we may assume that n = t.

Now observe that f: ℤ^{n} → M, factors as ℤ^{n} → (ℤ /p^{d} ℤ)^{n} → M, where d := max(d_{1}, …, d_{t}). Lift each y_{i} ∈ M to some y′_{i} ∈ (ℤ /p^{d} ℤ)^{n}. By Nakayama’s Lemma y′_{1}, …, y′_{n} form a ℤ /p^{d} ℤ-basis of (ℤ /p^{d} ℤ)^{n}. It now suffices to lift cy′_{1}, y′_{2}, …, y′_{n} to a basis of ℤ^{n}, for a suitable integer c, prime to p. In other words, we may assume without loss of generality that M = (ℤ /p^{d} ℤ)^{n}, and f: ℤ^{n} → (ℤ /p^{d} ℤ)^{n} is the natural projection.

Now suppose y_{i} = (y_{i1}, …, y_{in}) for some y_{ij} ∈ ℤ /p^{d} ℤ. Since y_{1}, …, y_{m} form a basis of (ℤ /p^{d} ℤ)^{n}, the matrix A = (y_{ij}) is invertible, i.e., A ∈ GL_{n}(ℤ /p^{d} ℤ). After rescaling y_{1} by c := det(A)^{−1} in ℤ /p^{d} ℤ, we may assume that det(A) = 1. The lemma now follows from the surjectivity of the natural projection SL_{t}(ℤ) → SL_{t}(ℤ /p^{d} ℤ); see [Sh71, Lemma 1.38]. □

Proof of Proposition 4.1(b). By definition, ind(G, T) is the minimal value of ind^{x_{1}}(G, T)+ ⋯ + ind^{x_{r}}(G, T), as x_{1}, …, x_{r} range over the bases of X(T) ⊂ ℤ^{r}. By part (a), we can rewrite this as

ind(G, T) = min{p^{w(π(x_{1}))} + ⋯ + p^{w(π(x_{r}))} | x_{1}, …, x_{r} is a ℤ-basis of X(T)}.

Here, as before, π(x_{i}) ∈ Code(C) is the restriction of x_{i} from T = 𝔾_{m}^{r} /C to μ/C.

Let z_{1}, …, z_{t} ∈ Code(C) be a minimal basis, as in the statement of the proposition. We will prove part (b) by showing that

(i) p^{w(π(x_{1}))} + ⋯ + p^{w(π(x_{r}))} ⩾ p^{w(z_{1})} + ⋯ + p^{w(z_{t})} + r − t for every ℤ-basis x_{1}, …, x_{r} of X(T), and

(ii) there exists a particular ℤ-basis x_{1}, …, x_{r} of X(T) such that p^{w(π(x_{1}))} + ⋯ + p^{w(π(x_{r}))} = p^{w(z_{1})} + ⋯ + p^{w(z_{t})} + r − t.

To prove (i), note that if x_{1}, …, x_{r} form a ℤ-basis of X(T), then π(x_{1}), …, π(x_{r}) form a generating set for Code(C). By Nakayama’s Lemma every generating set for Code(C) contains a basis. After renumbering x_{1}, …, x_{r} we may assume that π(x_{1}), …, π(x_{t}) is a basis of Code(C) and w(π(x_{1})) ⩽ ⋯ ⩽ w(π(x_{t})). By Proposition 3.1, w(z_{i}) ⩽ w(π(x_{i})) for every i = 1, …, t. Thus

p^{w(π(x_{1}))}+⋯+p^{w(π(x_{r}))} ⩾ p^{w(π(x_{1}))}+⋯+p^{w(π(x_{t}))}+\underbrace{p^{0} + ⋯ + p^{0}}_{r − t times} ⩾ p^{w(z_{1})}+⋯+p^{w(z_{t})}+r−t .

To prove (ii), recall that by Lemma 4.2 there exists an integer c, prime to p, and a ℤ-basis x_{1}, …, x_{r} of X(T) such that π(x_{1}) = cz_{1}, π(x_{2}) = z_{2}, …, π(x_{t}) = z_{t}, and π(x_{t+1}) = ⋯ = π(x_{r}) = 0. Since c is prime to p, w(cz_{1}) = w(z_{1}). Thus for this particular choice of x_{1}, …, x_{r}, we have

p^{w(π(x_{1}))}+⋯+p^{w(π(x_{r}))} = p^{w(cz_{1})}+p^{w(z_{2})} ⋯+p^{w(z_{t})}+\underbrace{p^{0} + ⋯ + p^{0}}_{r − t times} = p^{w(z_{1})}+⋯+p^{w(z_{t})}+r−t .

as desired. □

5. Proof of Theorem 1.2

Consider the action of a linear algebraic group Γ on an absolutely irreducible algebraic variety X defined over k. We say that a subgroup S ⊂ Γ is a stabilizer in general position for this action if there exists a dense open subset U ⊂ X such that the scheme-theoretic stabilizer Stab_{Γ}(x) is conjugate to S over \overline{k} for every x ∈ U(\overline{k}). Here, as usual, \overline{k} denotes the algebraic closure of k. In the sequel we will not specify U and will simply say that Stab_{Γ}(x) is conjugate to S for x ∈ X(\overline{k}) in general position. Note that a stabilizer in general position S for a Γ-action on X does not always exist, and when it does, it is usually not unique. However, over \overline{k}, S is unique up to conjugacy.

For the rest of this section we will always assume that char(k) = 0. A theorem of R. W. Richardson [Ri72] tells us that under this assumption every linear action of a reductive group Γ on a vector space V has a stabilizer S ⊂ Γ in general position. Note that in Richardson’s paper [Ri72], k is assumed to be algebraically closed. Thus a priori the subgroup S and the open subset U ⊂ V, where all stabilizers are conjugate to S, are only defined over \overline{k}. However, after intersecting U with its Galois translates, we may assume that U is defined over k. Moreover, we may take S := Stab_{G}(x) for some k-point x ∈ U(k) and thus assume that S is defined over k. For a detailed discussion of stabilizers in general position over an algebraically closed field of characteristic zero, see [PV94, Section 7].

We will say that a Γ-action on X is generically free if the trivial subgroup S = {1_{Γ}} ⊂ Γ is stabilizer in general position for this action.

Lemma 5.1. Let Γ be a reductive linear algebraic group and ρ: Γ → GL(V) be a finitedimensional representation. If Stab_{Γ}(v) is central in Γ for v ∈ V in general position, then the induced action of Γ/ Ker(ρ) on V is generically free.

Proof. Let S ⊂ Γ be the stabilizer in general position for the Γ-action on V. Clearly Ker(ρ) ⊂ S. We claim that, in fact, Ker(ρ) = S; the lemma easily follows from this claim.

To prove the opposite inclusion, S ⊂ Ker(ρ), note that under the assumption of the lemma, S is central in Γ. Let U ⊂ V be a dense open subset such that the stabilizer of every v ∈ U(\overline{k}) is conjugate to S. Since S is central, Stab_{Γ}(v) is, in fact, equal to S. In other words, S stabilizes every point in U and thus every point in V. That is, S ⊂ Ker(ρ), as claimed. □

Our interest in generically free actions in this section has to do with the following fact: if there exists a generically free linear representation G → GL(V) then

(17)  ed(G) ⩽ dim(V) − dim(G) ;

see, e.g., [Re10, (2.3)] or [M13, Proposition 3.13]. This inequality will play a key role in our proof of Theorem 1.2.

Now set Γ := GL_{n_{1}} × ⋯ × GL_{n_{r}} and Γ′ := SL_{n_{1}} × ⋯ × SL_{n_{r}}. Let V_{i} be the natural n_{i}-dimensional representation, V_{i}^{−1} be the dual representation, and V_{i}^{0} be the trivial 1dimensional representation of GL_{n_{i}}. For ϵ = (ϵ_{1}, …, ϵ_{r}), where each ϵ_{i} is −1, 0 or 1, we define ρ_{ϵ} to be the natural representation of Γ on the tensor product

(18)  V_{ϵ} = V_{1}^{ϵ_{1}} ⊗ ⋯ ⊗ V_{r}^{ϵ_{r}} .

Lemma 5.2. Suppose 2 ⩽ n_{1} ⩽ … ⩽ n_{r} ⩽ \frac{1}{2}n_{1} … n_{r−1}, and

(n_{1}, …, n_{r}) ≠ (2, 2, 2, 2), (3, 3, 3), or (2, n, n), for any n ⩾ 2.

If ϵ = (ϵ_{1}, …, ϵ_{r}) ∈ {±1}^{r}, then the induced action of Γ/ Ker(ρ_{ϵ}) on V_{ϵ} is generically free.

Proof. By Lemma 5.1 it suffices to prove the following claim: the stabilizer Stab_{Γ}(v) is central in Γ for v ∈ V_{ϵ} in general position. To prove this claim, we may assume without loss of generality that k is algebraically closed.

We first reduce to the case where ϵ = (1, …, 1). Suppose the claim is true in this case, and let (ϵ_{1}, …, ϵ_{r}) ∈ {±1}^{r}. By choosing bases of V_{1}, …, V_{r} we can identify V_{i} with V_{i}^{ϵ_{i}} (we can take the identity map if ϵ_{i} = 1). Define an automorphism:

σ : Γ → Γ
(g_{1}, …, g_{r}) ↦ (g_{1}^{∗}, …, g_{r}^{∗})

where

g_{i}^{∗} = \begin{cases}g_{i} & if ϵ_{i} = 1; \\ (g_{i}^{−1})^{T} & if ϵ_{i} = −1. \end{cases}

Now ρ_{(ϵ_{1},...,ϵ_{r})} is isomorphic to the representation ρ_{(1,...,1)} ∘ σ. Since the center of Γ is invariant under σ, we see that the claim holds for ρ_{ϵ} as well.

From now on we will assume ϵ = (1, …, 1). By [P87, Theorem 2],

Γ/Z(Γ) = PGL_{n_{1}} × ⋯ × PGL_{n_{r}} = Γ′/Z(Γ′)

acts generically freely on the projective space ℙ(V_{ϵ}) = V_{ϵ}/Z(Γ). In other words for v ∈ V_{ϵ} in general position the stabilizer in Γ of the associated projective point [v] ∈ ℙ(V_{ϵ}) is trivial. Hence, the stabilizer of v is contained in Z(Γ); see the exact sequence in [RV07, Lemma 3.1]. This completes the proof of the claim and thus of Lemma 5.2. □

We are now ready to proceed with the proof of Theorem 1.2. We begin by specializing n_{i} to p^{a_{i}} for every i = 1, …, r, so that Γ becomes GL_{p^{a_{1}}} × ⋯ × GL_{p^{ar}}. Let

y_{1}, …, y_{t} ∈ (ℤ /p^{a_{1}} ℤ) × ⋯ × (ℤ /p^{a_{r}} ℤ)

be a basis of Code(C) satisfying the conditions of Theorem 1.2. Lift each y_{i} = (y_{i1}, …, y_{ir}) to x_{i} := (x_{i1}, …, x_{ir}) ∈ ℤ^{r} by setting x_{ij} := −1, 0 or 1, depending on whether y_{ij} is −1, 0 or 1 in ℤ /p^{a_{j}} ℤ. (If p^{a_{j}} = 2, then we define each x_{ij} to be 0 or 1.) By Nakayama’s Lemma the images of y_{1}, …, y_{t} are 𝔽_{p}-linearly independent in Code(C)/p Code(C). Thus the integer vectors x_{1}, …, x_{t} are ℤ-linearly independent. (Note that, unlike in the situation of Lemma 4.2, here it will not matter to us whether x_{1}, …, x_{t} can be completed to a ℤ-basis of ℤ^{r}.) We view each x_{i} as a character 𝔾_{m}^{r} → 𝔾_{m} and set

C̃ := Ker(x_{1}) ∩ ⋯ ∩ Ker(x_{t}) ⊂ 𝔾_{m}^{r} .

Since x_{1}, …, x_{t} are linearly independent,

(19)  dim(C̃) = r − t.

Set G := Γ/C and G̃ := Γ/C̃. By our construction, C̃ ∩ μ = C. Corollary A.2 now tells us that ed_{p}(G) ⩽ ed(G) = ed(G̃). By Theorem 1.1(a)

ed(G) ⩾ ed_{p}(G) ⩾ (∑_{i=1}^{t} p^{w(y_{i})}) − p^{2a_{1}} − ⋯ − p^{2a_{r}} + r − t.

It thus suffices to show that ed(G̃) ⩽ (∑_{i=1}^{t} p^{w(y_{i})})−p^{2a_{1}} −⋯−p^{2a_{r}} +r−t or equivalently,

ed(G̃) ⩽ (∑_{i=1}^{t} p^{w(y_{i})}) − dim(G̃) ;

see (19). By (17), in order to prove the last inequality it is enough to construct a generically free linear representation of G̃ of dimension ∑_{i=1}^{t} p^{w(y_{i})}. Such a representation is furnished by the lemma below.

Recall that x_{i} = (x_{i1}, …, x_{ir}) ∈ ℤ^{r}, where each x_{ij} = −1, 0 or 1, and ρ_{x_{i}} is the natural representation of Γ := GL_{p^{a_{1}}} × ⋯ × GL_{p^{ar}} on V_{x_{i}} := V_{1}^{x_{i1}} ⊗ ⋯ ⊗ V_{r}^{x_{ir}}, as in (18), with dim(V_{i}) = n_{i} = p^{a_{i}}.

Lemma 5.3. Let V = V_{x_{1}} ⊕ ⋯ ⊕ V_{x_{t}} and ρ := ρ_{x_{1}} ⊕ ⋯ ⊕ ρ_{x_{t}}: Γ → GL(V). Then (a) dim(V) = p^{w(y_{1})} + ⋯ + p^{w(y_{t})},

(b) Ker(ρ) = C̃, and

(c) the induced action of G̃ = Γ/C̃ on V is generically free.

Proof. For each i = 1, … t, we have

dim(V_{x_{i}}) = ∏_{x_{ij}≠0} p^{a_{j}} = ∏_{y_{ij}≠0} p^{a_{j}} = p^{∑_{y_{ij} ≠0} a_{j}} .

Since each y_{ij} = −1, 0 or 1, ∑_{y_{ij}≠0} a_{j} = w(y_{i}). Thus dim(V_{x_{i}}) = p^{w(y_{i})}, and part (a) follows.

Now choose v_{i} ∈ V_{x_{i}} in general position and set v := (v_{1}, …, v_{r}). We claim that Stab_{Γ}(v) is central in Γ.

Suppose for a moment that this claim is established. Since the center Z(Γ) = 𝔾_{m}^{r} acts on V_{x_{i}} via scalar multiplication by the character x_{i}: 𝔾_{m}^{r} → 𝔾_{m}, we see that

Ker(ρ) = Ker(ρ_{| 𝔾_{m}^{r}}) = Ker(x_{1}) ∩ ⋯ ∩ Ker(x_{t}) = C̃ ,

and part (b) follows. Moreover, by Lemma 5.1, the induced action of Γ/ Ker(ρ) on V is generically free. By part (b), Ker(ρ) = C̃ and part (c) follows as well.

It remains to prove the claim. Choose v_{i} ∈ V_{x_{i}} in general position and assume that g = (g_{1}, …, g_{r}) stabilizes v := (v_{1}, …, v_{t}) in V for some g_{j} ∈ GL_{p^{a_{j}}}. Our goal is to show that g_{j} is, in fact, central in GL_{p^{a_{j}}} for each j = 1, …, r.

Let us fix j and focus on proving that g_{j} is central for this particular j. By assumption (b) of Theorem 1.2, there exists an i = 1, …, t such that y_{i} is balanced and y_{ij} ≠ 0. Let us assume that y_{ij_{1}}, …, y_{ij_{s}} = ±1 and y_{ih} = 0 for every h ∉ {j_{1}, …, j_{r}} and consequently, x_{ij_{1}}, …, x_{ij_{s}} = ±1 and x_{ih} = 0 for every h ∉ {j_{1}, …, j_{r}}. By our assumption, j ∈ {j_{1}, …, j_{s}}.

The representation ρ_{x_{i}} of Γ = GL_{p^{a_{1}}} × ⋯ × GL_{p^{ar}} on

V_{x_{i}} := V^{x_{i1}} ⊗ ⋯ ⊗ V^{x_{it}} = V^{x_{ij_{1}}} ⊗ ⋯ ⊗ V^{x_{ijs}}

factors through the projection Γ → GL_{p^{a_{j_{1}}}} × ⋯ × GL_{p^{a_{js}}}. Thus if g = (g_{1}, …, g_{r}) stabilizes v = (v_{1}, …, v_{t}) ∈ V then, in particular, g stabilizes v_{i} and thus (g_{j_{1}}, …, g_{j_{s}}) stabilizes v_{i}.

Since y_{i} is assumed to be balanced, the conditions of Lemma 5.2 for the action of GL_{n_{j_{1}}} × ⋯ × GL_{n_{js}} on V_{x_{i}} = V^{x_{j_{1}}} ⊗ ⋯ ⊗ V^{x_{js}} are satisfied. (Recall that here n_{i} = p^{a_{i}}.) Since (g_{j_{1}}, …, g_{j_{s}}) stabilizes v_{i} ∈ V_{x_{i}} in general position, Lemma 5.2 tells us that g_{j_{1}}, …, g_{j_{s}} are central in GL_{n_{j_{1}}}, … GL_{n_{js}}, respectively. In particular, g_{j} is central in GL_{n_{j}}, as desired. This completes the proof of Lemma 5.3 and thus of Theorem 1.2. □

6. Proof of Theorem 1.3

Consider the central subgroups C̃ and C of Γ = GL_{p^{a_{1}}} × ⋯ × GL_{p^{ar}} given by

C̃ = {(τ_{1}, …, τ_{r}) ∈ 𝔾_{m}^{r} | τ_{1} … τ_{r} = 1} and C = {(τ_{1}, …, τ_{r}) ∈ μ | τ_{1} … τ_{r} = 1}.

Set G := Γ/C and G̃ := Γ/C̃. Note that C = C̃∩μ. Thus Theorem A.1 and Corollary A.2 tell us that the functors H^{1}(−, G̃) and H^{1}(−, G) are both isomorphic to the functor F defined in the statement of Theorem 1.3. In particular, ed(G̃) = ed(G) = ed(F) and ed_{p}(G̃) = ed_{p}(G) = ed_{p}(F). We are now ready to proceed with the proof of Theorem 1.3.

(a) If A_{1} ⊗ ⋯ ⊗ A_{r} is split over K, then A_{r} can be recovered from A_{1}, …, A_{r−1} as the unique central simple K-algebra of degree p^{a_{r}} which is Brauer-equivalent to

(A_{1} ⊗ ⋯ ⊗ A_{r−1})^{op} .

(Here B^{op} denotes the opposite algebra of B.) In other words, the morphism of functors

(20)  F → H^{1}(−, PGL_{p^{a_{1}}}) × ⋯ × H^{1}(−, PGL_{p^{a_{r−1}}})

given by (A_{1}, …, A_{r−1}, A_{r}) → (A_{1}, …, A_{r−1}) is injective. We claim that if a_{r} ⩾ a_{1} +⋯+ a_{r−1} (which is our assumption in part (a)), then this morphism if also surjective. Indeed,

deg(A_{1} ⊗ ⋯ ⊗ A_{r−1}) = p^{a_{1}+···+a_{r−1}}

for any choice of central simple K-algebras A_{1}, …, A_{r−1} such that deg(A_{i}) = p^{a_{i}}. Hence, for any such choice there exists a central simple algebra of degree p^{a_{r}} which is Brauerequivalent to (A_{1} ⊗ ⋯ ⊗ A_{r−1})^{op}. This proves the claim.

We conclude that if a_{r} ⩾ a_{1} + ⋯ + a_{r−1} then (20) is an isomorphism and thus ed(G̃) = ed(G) = ed(F) = ed(PGL_{p^{a_{1}}} × ⋯ × PGL_{p^{a_{r−1}}}) and

ed_{p}(G̃) = ed_{p}(G) = ed_{p}(F) = ed_{p}(PGL_{p^{a_{1}}} × ⋯ × PGL_{p^{a_{r−1}}}) .

The inequality ed(F) ⩽ p^{2a_{1}} + ⋯ + p^{2a_{r−1}} now follows from (9).

(b) Now suppose a_{r} < a_{1} + ⋯ + a_{r−1}. In this case Code(C) has a minimal basis consisting of the single element (1, …, 1) ∈ (ℤ /p^{a_{1}} ℤ) × ⋯ × (ℤ /p^{a_{r}} ℤ). Moreover,

p^{a_{r}} ⩽ \frac{1}{2}p^{a_{1}} … p^{a_{r−1}} and consequently, Theorem 1.2 applies. It tells us that if the r-tuple (p^{a_{1}}, …, p^{a_{r}}) is not of the form (2, 2, 2, 2), (3, 3, 3) or (2, 2^{a}, 2^{a}), then

ed(F) = ed_{p}(F) = ed(G̃) = ed_{p}(G̃) = ed(G) = ed_{p}(G) = p^{a_{1}+···+a_{r}} − ∑_{i=1}^{r} p^{2a_{i}} + r − 1 ,

as claimed.

(c) In the case where (p^{a_{1}}, …, p^{a_{r}}) = (2, 2, 2), F(K) is the set of isomorphism classes of triples (A_{1}, A_{2}, A_{3}) of quaternion K-algebras, such that A_{1} ⊗ A_{2} ⊗ A_{3} is split over K. We will show that (i) ed(F) ⩽ 3 and (ii) ed_{2}(F) ⩾ 3.

To prove (i), recall that by a theorem of Albert [Lam05, Theorem III.4.8], the condition that A_{1} ⊗ A_{2} ⊗ A_{3} is split over K implies that A_{1} and A_{2} are linked over K. That is, there exist a, b, c ∈ K^{∗} such that A_{1} ≃ (a, b) and A_{2} ≃ (a, c) over K. Hence, the triple (A_{1}, A_{2}, A_{3}) ∈ F(K) descends to the triple (B_{1}, B_{2}, B_{3}) ∈ F(K_{0}), where K_{0} = k(a, b, c), B_{1} = (a, b), B_{2} = (a, c) and B_{3} = (a, bc) over K_{0}. Since trdeg(K_{0}/k) ⩽ 3, assertion (i) follows.

To prove (ii), consider the morphism of functors f: F → H^{1}(−, SO_{4}) given by

f: (A_{1}, A_{2}, A_{3}) ↦ α ,

where α is a 4-dimensional quadratic form such that

α ⊕ ℍ ⊕ ℍ ≅ N(A_{1}) ⊕ (−N(A_{2})) .

Here ℍ denotes the 2-dimensional hyperbolic form ⟨1, −1⟩, N(A_{1}) denotes the norm form of A_{1}, and −N(A_{2}) denotes the opposite norm form of A_{2}, i.e., the unique 4-dimensional form such that N(A_{2}) ⊕ (−N(A_{2})) is hyperbolic. Since N(A_{1}) and N(A_{2}) are forms of discriminant 1, so is α (this will also be apparent from the explicit computations below). Thus we may view α as an element of the Galois cohomology set H^{1}(K, SO_{4}), which classifies 4-dimensional quadratic forms of discriminant 1 over K, up to isomorphism. Note also that by the Witt Cancellation Theorem, α is unique up to isomorphism. We conclude that the morphism of functors f is well defined.

Equivalently, using the definition of the Albert form given in [Lam05, p. 69], α is the unique 4-dimensional quadratic form such that α ⊕ ℍ ≅ q, where q is the 6-dimensional Albert form of A_{1} and A_{2}. Here the Albert form of A_{1} and A_{2} is isotropic, and hence, can be written as α ⊕ ℍ, because A_{1} and A_{2} are linked; once again, see [Lam05, Theorem III.4.8].

Suppose A_{1} = (a, b), A_{2} = (a, c), and A_{3} = (a, bc), as above. Then

N(A_{1}) = ⟨⟨−a, −b⟩⟩ = ⟨1, −a, −b, ab⟩ ,

and similarly N(A_{2}) = ⟨1, −a, −c, ac⟩; see, e.g., [Lam05, Corollary III.2.2]. Thus

N(A_{1}) ⊕ (−N(A_{2})) = ⟨1, −1, −a, a, −b, c, ab, −ac⟩ ≃ ⟨−b, c, ab, −ac⟩ ⊕ ℍ ⊕ ℍ

and we obtain an explicit formula for α = f(A_{1}, A_{2}, A_{3}): α ≅ ⟨−b, c, ab, −ac⟩.

It is easy to see that any 4-dimensional quadratic form of discriminant 1 over K can be written as ⟨−b, c, ab, −ac⟩ for some a, b, c ∈ K^{∗}. In other words, the morphism of functors f: F → H^{1}(−, SO_{4}) is surjective. Consequently,

ed_{2}(F) ≥ ed_{2}(H^{1}(−, SO_{4})) = ed_{2}(SO_{4}) ;

see, e.g., [BF03, Lemma 1.9] or [Re10, Lemma 2.2]. On the other hand, ed_{2}(SO_{4}) = 3; see [RY00, Theorem 8.1(2) & Remark 8.2] or [Re10, Corollary 3.6(a)]. Thus

ed_{2}(F) ⩾ ed_{2}(SO_{4}) = 3 .

This completes the proof of (ii) and thus of part (c) and of Theorem 1.3. □

Appendix: Galois Cohomology of central quotients of products of general linear groups

by Athena Nguyen ^{1}

In this appendix we will study the Galois cohomology of algebraic groups of the form

G := Γ/C,

where Γ := GL_{n_{1}} × ⋯ × GL_{n_{r}} and C ⊂ Z(Γ) = 𝔾_{m}^{r} is a central subgroup. Here n_{1}, …, n_{r} ⩾ 1 are integers, not necessarily prime powers. Let \overline{G} := G/Z(G) = PGL_{n_{1}} × ⋯× PGL_{n_{r}} = Γ/Z(Γ). Recall that for any field K/k, H^{1}(K, PGL_{n}) is naturally identified with the set of isomorphism classes of central simple K-algebras of degree n, and

H^{1}(K, \overline{G}) = H^{1}(K, PGL_{n_{1}}) × ⋯ × H^{1}(K, PGL_{n_{r}})

with the set of r-tuples (A_{1}, …, A_{r}) of central simple K-algebras such that deg(A_{i}) = n_{i}. Denote by ∂_{K}^{i} the coboundary map H^{1}(K, PGL_{n_{i}}) → H^{2}(K, 𝔾_{m}) induced by the short exact sequence

1 → 𝔾_{m} → GL_{n_{i}} → PGL_{n_{i}} → 1.

This map sends a central simple algebra A_{i} to its Brauer class [A_{i}] in H^{2}(K, 𝔾_{m}) = Br(K).

1This appendix is based on a portion of the author’s Master’s thesis completed at the University of British Columbia. The author gratefully acknowledges the financial support from the University of British Columbia and the Natural Sciences and Engineering Research Council of Canada.

Of particular interest to us will be

X(𝔾_{m}^{r} /C) = {(m_{1}, …, m_{r}) ∈ ℤ^{r} | τ_{1}^{m_{1}} … τ_{r}^{m_{r}} = 1 ∀(τ_{1}, …, τ_{r}) ∈ 𝔾_{m}^{r}} ,

as in (6). We are now ready to state the main result of this appendix.

Theorem A.1. Let π: G → \overline{G} := PGL_{n_{1}} × ⋯ × PGL_{n_{r}} be the natural projection and π_{∗}: H^{1}(K, G) → H^{1}(K, \overline{G}) be the induced map in cohomology. Here K/k is a field extension. Then

(a) π_{∗}: H^{1}(K, G) → H^{1}(K, \overline{G}) is injective for every field K/k.

(b) π_{∗} identifies H^{1}(K, G) with the set of isomorphism classes of r-tuples (A_{1}, …, A_{r}) of central simple K-algebras such that deg(A_{i}) = n_{i} and A_{1}^{⊗m_{1}} ⊗ ⋯ ⊗ A_{r}^{⊗m_{r}} is split over K for every (m_{1}, …, m_{r}) ∈ X(𝔾_{m}^{r} /C).

Proof. Throughout, we will identify H^{2}(K, 𝔾_{m}^{r}) with H^{2}(K, 𝔾_{m})^{r} and X(𝔾_{m}^{r}) with ℤ^{n}. A character x = (m_{1}, …, m_{r}) ∈ ℤ^{n}, i.e., a character x: 𝔾_{m}^{r} → 𝔾_{m} given by (τ_{1}, …, τ_{r}) → τ_{1}^{m_{1}} … τ_{r}^{m_{r}}, induces a map x_{∗}: H^{2}(K, 𝔾_{m})^{r} → H^{2}(K, 𝔾_{m}) in cohomology given by

(21)  x_{∗}(α_{1}, …, α_{r}) = α_{1}^{m_{1}} · . . . · α_{r}^{m_{r}} .

Let us now consider the diagram

[figure: 1; 𝔾_{m}^{r}; η; Γ; ∏_{i=1}^{r} PGL_{n_{i}}; 1; 1; 𝔾_{m}^{r} /C; G; π; _{∏_{i=1}^{r} PGL_{n_{i}}}; 1.]

Since H^{1}(K, 𝔾_{m}^{r} /C) = {1} by Hilbert’s Theorem 90, we obtain the following diagram in cohomology with exact rows:

[figure: H^{1}(K, ∏_{i=1}^{r} PGL_{n_{i}}); (∂_{K}^{1},...,∂_{K}^{r}); H^{2}(K, 𝔾_{m}^{r}); η_{∗}; 1; H^{1}(K, G); π_{∗}; H^{1}(K, ∏_{i=1}^{r} PGL_{n_{i}}); ∂_{K}; H^{2}(K, 𝔾_{m}^{r} /C)]

(a) It follows from [S97, I.5, Proposition 42] that π_{∗} is injective.

(b) Thus, π_{∗} identifies H^{1}(K, G) with the set of r-tuples (A_{1}, …, A_{r}), where A_{i} ∈ H^{1}(K, PGL_{n_{i}}) is a central simple algebra of degree n_{i}, and (∂_{K}^{1}(A_{1}), …, ∂_{K}^{r}(A_{r})) ∈ Ker(η_{∗}). Recall that ∂_{K}^{i} sends a central simple algebra A_{i} to its Brauer class [A_{i}] ∈ H^{2}(K, 𝔾_{m}). In the sequel we will use additive notation for the abelian group H^{2}(K, 𝔾_{m}) = Br(K).

Consider an r-tuple α := ([A_{1}], …, [A_{r}]) ∈ H^{2}(K, 𝔾_{m}^{r}). Since 𝔾_{m}^{r} /C is diagonalizable, η_{∗}(α) = 0 if and only if x_{∗}(η_{∗}(α)) = 0 for all x ∈ X(𝔾_{m}^{r} /C). If x = (m_{1}, …, m_{r}) ∈ X(𝔾_{m}^{r} /C), then x_{∗}∘η_{∗} = (m_{1}, ...m_{r}) ∈ X(G_{m}^{r}). By (21), x_{∗}(η_{∗}(α)) = [A_{1}^{⊗m_{1}} ⊗⋯⊗A_{r}^{⊗m_{r}}], and part (b) follows. □

Corollary A.2. Let Γ := GL_{n_{1}} × ⋯ × GL_{n_{r}}, C_{1}, C_{2} be k-subgroups of Z(Γ) = 𝔾_{m}^{r}, G_{1} = Γ/C_{1} and G_{2} = Γ/C_{2}. Denote the central subgroup μ_{n_{1}} × ⋯ × μ_{n_{r}} of Γ by μ.

If C_{1} ∩ μ = C_{2} ∩ μ then the Galois cohomology functors H^{1}(−, G_{1}) and H^{1}(−, G_{2}) are isomorphic.

Proof. By Theorem A.1, H^{1}(K, G_{i}) is naturally identified with the set of r-tuples (A_{1}, …, A_{r}) of central simple algebras such that deg(A_{i}) = n_{i} and

A_{1}^{⊗m_{1}} ⊗ ⋯ ⊗ A_{r}^{⊗m_{r}} is split over K for every (m_{1}, …, m_{r}) ∈ X(𝔾_{m} /C_{i}).

Note that since A_{i}^{⊗n_{i}} is split for every i, this condition depends only on the image of (m_{1}, …, m_{r}) under the natural projection

π: X(𝔾_{m}^{r}) = ℤ^{r} → (ℤ /n_{1} ℤ) × ⋯ × (ℤ /n_{r} ℤ) = X(μ) .

Our assumption that C_{1} ∩ μ = C_{2} ∩ μ is equivalent to X(𝔾_{m}^{r} /C_{1}) and X(𝔾_{m}^{r} /C_{2}) having the same image under π, and the corollary follows. □

In order to state the second corollary of Theorem A.1, we will need the following definition. By a code we shall mean a subgroup of X(μ) = (ℤ /n_{1} ℤ) × ⋯ × (ℤ /n_{r} ℤ). Given a subgroup C ⊂ μ, we define the code Code(C) := X(μ/C), as in (8).

We will say that two codes are called equivalent if one can be obtained from the other by repeatedly performing the following elementary operations:

(1) Permuting entries i and j in every vector of the code, for any i, j with n_{i} = n_{j}.

(2) Multiplying the i^{th} entry in every vector of the code by an integer c prime to n_{i}.

Corollary A.3. Suppose C_{1} and C_{2} are subgroups of μ := μ_{n_{1}} × ⋯ × μ_{n_{r}}, G_{1} = Γ/C_{1} and G_{2} := Γ/C_{2}. If Code(C_{1}) and Code(C_{2}) are equivalent, then

(a) the Galois cohomology functors H^{1}(−, G_{1}) and H^{1}(−, G_{2}) are isomorphic, and

(b) in particular, ed(G_{1}) = ed(G_{2}) and ed_{p}(G_{1}) = ed_{p}(G_{2}) for every prime p.

Proof. (a) It suffices to show that H^{1}(−, G_{1}) and H^{1}(−, G_{2}) are isomorphic if C_{2} is obtained from C_{1} by an elementary operation.

(1) Suppose n_{i} = n_{j} for some i, j = 1, …, r, and Code(C_{2}) is obtained from Code(C_{1}) by permuting entries i and j in every vector. In this case C_{2} = α(C_{1}), where α is the automorphism of Γ = GL_{n_{1}} × ⋯ × GL_{n_{r}} which swaps the ith and the jth components. Then α induces an isomorphism between G_{1} = Γ/C_{1} and G_{2} = Γ/C_{2}, and thus an isomorphism between H^{1}(−, G_{1}) and H^{1}(−, G_{2}).

(2) Now suppose that Code(C_{1}) is obtained from Code(C_{2}) by multiplying the i^{th} entry in every vector by some c ∈ (ℤ /n_{i} ℤ)^{∗}. The description of H^{1}(K, G/μ) given by Theorem A.1 now tells us that

H^{1}(K, G_{1}) → H^{1}(K, G_{2})
(A_{1}, …, A_{r}) ↦ (A_{1}, …, A_{i−1}, [A_{i}^{⊗c}]_{n_{i}}, A_{i+1}, …, A_{r})

is an isomorphism. Here, by [A_{i}^{⊗c}]_{n_{i}} we mean the unique central simple K-algebra of degree n_{i} which is Brauer equivalent to A_{i}^{⊗c}.

(b) follows from (a), because ed(G) and ed_{p}(G) are defined entirely in terms of the Galois cohomology functor H^{1}(−, G). □

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Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2

  E-mail address: shane.cernele@gmail.com, reichst@math.ubc.ca, athena@math.ubc.ca