AN UPPER BOUND ON THE ESSENTIAL DIMENSION OF A CENTRAL SIMPLE ALGEBRA

AUREL MEYER^{†} AND ZINOVY REICHSTEIN^{††}

Abstract. We prove a new upper bound on the essential p-dimension of the projective linear group PGL_{n}.

Contents

1. Introduction 1 2. G/H-crossed products 3 3. G-lattices 5 4. An upper bound 6 5. Proof Theorem 1.2 8 Acknowledgments 9 References 9

1. Introduction

Let k be a base field; all other fields will be assumed to be extensions of k.

Given a central simple algebra A over a field K one can ask whether A can be written as A = A_{0} ⊗_{K_{0}} K where A_{0} is a central simple algebra over some subfield K_{0} of K. In that situation we say that A descends to K_{0}. The essential dimension of A, denoted ed(A), is the minimal transcendence degree over k of a field K_{0} ⊂ K such that A descends to K_{0}. It can be thought of as “the minimal number of independent parameters” required to define A.

For a prime number p, the related notion of essential dimension at p of an algebra A/K is defined as ed(A; p) = min ed(A_{K′}), where K′/K runs over all finite field extensions of degree prime to p.

2000 Mathematics Subject Classification. 16K20, 20C10.

Key words and phrases. Essential dimension, central simple algebra, projective linear group, G-lattice.

† Aurel Meyer was partially supported by a University Graduate Fellowship at the University of British Columbia.

†† Z. Reichstein was partially supported by NSERC Discovery and Accelerator Supplement grants.

We also define

ed(PGL_{n}) := max { ed(A) } ,

and

ed(PGL_{n}; p) := max { ed(A; p) } ,

where the maximum is taken over all fields K/k and over all central simple K-algebras A of degree n. The appearance of PGL_{n} in the symbols ed(PGL_{n}) and ed(PGL_{n}; p) has to do with the fact that central simple algebras of degree n are in a natural bijective correspondence with PGL_{n}-torsors. In fact, one can define ed(G) and ed(G; p) for every algebraic k-group G in a similar manner, using G-torsors instead of central simple algebras; see [Re_{2}],

[RY] or [BF].

To the best of our knowledge, the problem of computing ed(PGL_{n}) was first raised by C. Procesi in the 1960s. Procesi and S. Amitsur constructed so-called universal division algebras UD(n) and showed that UD(n) has various generic properties among central simple algebras of degree n. In particular, their arguments can be used to show that

ed(UD(n)) ≥ ed(A) and ed(UD(n); p) ≥ ed(A; p)

for any prime integer p; cf. [LRRS, Remark 2.8]. Equivalently,

ed(UD(n)) = ed(PGL_{n}) and ed(UD(n); p) = ed(PGL_{n}; p).

Since the center of UD(n) has transcendence degree n^{2} + 1 over k, we conclude that ed(PGL_{n}) ≤ n^{2} +1. Procesi showed (using different terminology) that in fact,

ed(PGL_{n}) ≤ n^{2} ;

see [Pr, Theorem 2.1].

The problem of computing ed(PGL_{n}) was raised again by B. Kahn in the early 1990s. In particular, in 1992 Kahn asked the second author if ed(PGL_{n}) grows sublinearly in n, i.e., whether

ed(PGL_{n}) ≤ an + b

for some positive real numbers a and b. To the best of our knowledge, this question never appeared in print but it is implicit in [Ka, Section 2]. It remains open; the best known upper bound,

(1)  ed(PGL_{n}) ≤ \begin{cases}\frac{(n−1)(n−2)}{2}, for every odd n ≥ 5 and \\ n^{2} − 3n + 1, for every n ≥ 4 \end{cases}

(see [LR], [LRRS, Theorem 1.1], [Le, Proposition 1.6] and [FF]), is quadratic in n and the best known lower bound,

ed(PGL_{p^{r}}) ≥ ed(PGL_{p^{r}}; p) ≥ 2r ,

is logarithmic.

Note that if p^{s} is the largest power of p dividing n then one easily checks, using primary decomposition of central simple algebras, that ed(PGL_{n}; p) = ed(PGL_{p^{s}}; p). Thus for the purpose of computing ed(PGL_{n}; p) it suffices to consider the case where n = p^{s}. In this case we have showed that

ed(PGL_{p^{s}}; p) ≤ p^{2s−1} − p^{s} + 1

for any s ≥ 2; see [MR, Corollary 1.2]. The main result of this paper is the following stronger upper bound.

Theorem 1.1. Let n = p^{s} for some s ≥ 2. Then

ed(PGL_{n}; p) ≤ 2\frac{n^{2}}{p^{2}} − n + 1

A. S. Merkurjev [Me_{2}] recently showed that for s = 2 this bound is sharp, i.e., ed(PGL_{p^{2}}; p) = p^{2} +1. We conjecture that this bound is sharp for every s ≥ 2; this would imply, in particular, that ed(PGL_{n}) is not sublinear in n.

Our upper bound on ed(PGL_{n}; p) is a consequence of the following result. Here n is not assumed to be a prime power.

Theorem 1.2. Let A/K be a central simple algebra of degree n. Suppose A contains a field F, Galois over K and Gal(F/K) can be generated by r ≥ 1 elements. If [F : K] = n then we further assume that r ≥ 2. Then

ed(A) ≤ r \frac{n^{2}}{[F : K]} − n + 1

Note that we always have [F : K] ≤ n. In the special case where equality holds, i.e., A is a crossed product in the usual sense, Theorem 1.2 reduces to [LRRS, Corollary 3.10(a)].

To deduce Theorem 1.1 from Theorem 1.2, let n = p^{s} and A = UD(n). In [RS_{1}, 1.2], L. H. Rowen and D. J. Saltman showed that if s ≥ 2 then there is a finite field extension K′/K of degree prime to p, such that A′ := A⊗_{K} K′ contains a field F, Galois over K′ with Gal(F/K′) ≃ ℤ/p × ℤ/p. Thus, if s ≥ 2, Theorem 1.2 tells us that

ed(PGL_{n}; p) = ed(A; p) ≤ ed(A′) ≤ 2\frac{n^{2}}{p^{2}} − n + 1 .

This proves Theorem 1.1. □

The remainder of this paper will be devoted to proving Theorem 1.2. We reduce the problem to a question about G-lattices, using the same approach as in [LRRS, Sections 2–3], but our analysis is more delicate here, and the results (Theorems 1.2 and 4.1) are stronger.

2. G/H-crossed products

Lemma 2.1. In the course of proving Theorem 1.2 we may assume without loss of generality that F is contained in a subfield L of A such that L/K is a separable extension of degree n = deg(A).

Proof. Note that we are free to replace K by K(t), F by F (t) and A by A(t) = A ⊗_{K} K(t), where t is an independent variable. Indeed, ed_{k} A(t) = ed_{k}(A); see, e.g., [LRRS, Lemma 2.7(a)]. Thus if the inequality of Theorem 1.2 is proved for A(t), it will also hold for A.

The advantage of passing from A to A(t) is that K(t) if Hilbertian for any infinite field K; see, e.g., [FJ, Proposition 13.2.1]. Thus after adjoining two variables, t_{1} and t_{2} as above, we may assume without loss of generality that K is Hilbertian. (Note that a subfield L ⊂ A of degree n over K may not exist without this assumption.)

Let F ⊂ F′ be maximal among separable field extension of F contained in A. We will look for L inside the centralizer C_{A}(F′). By the Double Centralizer Theorem, C_{A}(F′) is a central simple algebra with center F′. The maximality of F′ tells us that C_{A}(F′) contains no non-trivial field extensions of F′. In particular, C_{A}(F′) = M_{r}(F′), where r[F′ : K] = n.

On the other hand, since K is Hilbertian, so is its finite separable extension F′; cf. [FJ, 12.2.3]. Consequently, F′ admits a finite separable extension L/F′ of degree r. (To construct L/F′, start with the field extension L_{r} = F′(t_{1}, …, t_{r})[x]/(f(x)) of F′(t_{1}, …, t_{n}) of degree r, where f(x) = x^{r} + t_{1}x^{r−1} + … + t_{n−1}x + t_{n} is the general polynomial of degree r. Then specialize t_{1}, …, t_{r} in F′, using the Hilbertian property, to obtain a field extension L/F′ of degree r.) Any such L/F′ can be embedded into M_{r}(F′) via the regular representation of L on L = (F′)^{r}; cf. [Pi, Lemma 13.1a]. By the maximality of F′, we conclude that L = F′, i.e., r = 1 and [L : K] = n, as desired. □

Let us now assume that our central simple algebra A/K has a separable maximal subfield L/K, as in Lemma 2.1. We will denote the Galois closure of L over K by E and the associated Galois groups by G = Gal(E/K), H = Gal(E/L) and N = Gal(E/F), as in the diagram below.

[figure: E; H; N; G; A; L; F; K]

In the terminology of [LRRS], A/K is a G/H-crossed product; cf. also [FSS, Appendix]. Note that since E/K is the smallest Galois extension containing L/K, we have

(2)  Core_{G}(H) = ⋂_{g∈G} H^{g} = {1} .

where H^{g} := gHg^{−1}. We will assume that this condition is satisfied whenever we talk about G/H-crossed products.

Using the notation introduced above and remembering that [G : H] = [L : K] = deg(A) = n, and \frac{n}{[F : K]} = [L : F] = [N : H], we can restate

Theorem 1.2 as follows.

Theorem 2.2. Let A be a G/H-crossed product. Suppose H is contained in a normal subgroup N of G and G/N is generated by r elements. Furthermore, assume that either H ≠ {1} or r ≥ 2. Then

ed(A) ≤ r[G : H] · [N : H] − [G : H] + 1 .

3. G-lattices

In the sequel H ≤ G will be finite groups. Given g ∈ G we will write \overline{g} for the left coset gH of H. We will denote the identity element of G by 1.

Recall that a G-lattice M is a (left) ℤ[G]-module, which is free of finite rank over ℤ. In particular, any finite set X with a G-action gives rise to a Glattice ℤ[X]; G-lattices of this form are called permutation. For background material on G-lattices we refer the reader to [Lo].

Of particular interest to us will be the G-lattice ω(G/H), which is defined as the kernel of the natural augmentation map ℤ[G/H] → ℤ, sending n_{1}\overline{g_{1}} + ⋯ + n_{s}\overline{g_{s}} to n_{1} + ⋯ + n_{s}.

The starting point for our proof of Theorem 2.2 (and hence, of Theorem 1.2) will be the following result from [LRRS].

Theorem 3.1. ([LRRS, Theorem 3.5]) Let P be a permutation G-lattice and

0 → M → P → ω(G/H) → 0

be an exact sequence of G-lattices. If the G-action on M is faithful then

ed(A) ≤ rank(M) − n + 1

for any G/H-crossed product A. □

The condition that G acts faithfully on M is not automatic. However, the following lemma shows that it is satisfied for many natural choices of P.

Lemma 3.2. Let G ≠ {1} be a finite group H ≤ G be a subgroup of G, H_{1}, …, H_{r} be subgroups of H and

(3)  0 → M → ⊕_{i=1}^{r}ℤ[G/H_{i}] → ω(G/H) → 0

be an exact sequence of G-lattices. Assume that H does not contain any nontrivial normal subgroup of G (i.e., H satisfies condition (2) above). Then the G-action on M fails to be faithful if and only if s = 1 and H_{1} = H.

Here we are not specifying the map ⊕_{i=1}^{r}ℤ[G/H_{i}] → ω(G/H); the lemma holds for any exact sequence of the form (3). We also note that in the case where H_{1} = ⋯ = H_{r} = {1}, Lemma 3.2 reduces to [LRRS, Lemma 2.1].

Proof. To determine whether or not the G-action on M is faithful, we may replace M by M_{ℚ} := M ⊗ℚ. After tensoring with ℚ, the sequence (3) splits, and we have an isomorphism

(4)  ω(G/H)_{ℚ} ⊕ M_{ℚ} ≃ ⊕_{i=1}^{r}ℚ[G/H_{i}] .

Case 1: r ≥ 2. Then H_{r} is a subgroup of H, we have a natural surjective map ℚ[G/H_{r}] → ℚ[G/H]. Using complete irreducibility over ℚ once again, we see that ℚ[G/H] (and hence ω(G/H)) is a subrepresentation of ℚ[G/H_{r}]. Thus (4) tells us that ℚ[G/H_{r−1}] is a subrepresentation of M_{ℚ}. The kernel of the G-representation on ℚ[G/H_{r−1}] is a normal subgroup of G contained in H_{r−1} (and hence, in H); by our assumption on H, any such subgroup is trivial. This shows that G acts faithfully on ℚ[G/H_{r−1}] and hence, on M.

Case 2: Now assume r = 1. Our exact sequence now assumes the form

0 → M_{ℚ} → ℚ[G/H_{1}] → ω(G/H)_{ℚ} → 0 .

If H = H_{1} then M ≃ ℤ, with trivial (and hence, non-faithful) G-action.

Our goal is thus to show that if H_{1} ⊊ H then the G-action on M_{ℚ} is faithful. Denote by ℚ[1] the trivial representation (it will be clear from the context of which group). Observe that

ℚ[G/H_{1}] ≃ Ind_{H_{1}}^{G} ℚ[1] ≃ Ind_{H}^{G} Ind_{H_{1}}^{H} ℚ[1] ≃ Ind_{H}^{G} ℚ[H/H_{1}]
≃ Ind_{H}^{G}(ω(H/H_{1})_{ℚ} ⊕ ℚ[1])
≃ Ind_{H}^{G} ω(H/H_{1})_{ℚ} ⊕ ℚ[G/H]
≃ Ind_{H}^{G} ω(H/H_{1})_{ℚ} ⊕ ω(G/H)_{ℚ} ⊕ ℚ[1]

and we obtain

M_{ℚ} ≃ Ind_{H}^{G} ω(H/H_{1})_{ℚ} ⊕ ℚ[1] .

If H_{1} ⊊ H then the kernel of the G-representation Ind_{H}^{G} ω(H/H_{1})_{ℚ} is a normal subgroup of G contained in H_{1} (and hence, in H). By our assumption on H, this kernel is trivial. □

4. An upper bound

In this section we will prove the following upper bound on the essential dimension of a G/H-crossed product.

We will say that g_{1}, …, g_{s} ∈ G generate G over H if G = ⟨g_{1}, …, g_{s}, H⟩.

Theorem 4.1. Let A be a G/H-crossed product. Suppose that (i) g_{1}, …, g_{s} ∈ G generate G over H, and (ii) if G is cyclic then H ≠ {1}.

Then ed(A) ≤ ∑_{i=1}^{s}[G : (H ∩ H^{g_{i}})] − [G : H] + 1.

Remark 4.2. The index [G : (H ∩ H^{g_{i}})] appearing in the above formula can be rewritten as

[G : H] · [H : (H ∩ H^{g_{i}})] = [G : H] · [(H · H^{g_{i}}) : H] ;

see, e.g., [Ro, 1.3.11(i)]. Note H · H^{g} := {hh′ | h ∈ H, h′ ∈ H^{g}} is a subset

of G but may not be a subgroup, and [(H · H^{g}) : H] is defined as \frac{|H · H^{g}|}{|H|}.

If H is contained in a normal subgroup N of G then clearly H · H^{g} lies in N, each [H · H^{g} : H] ≤ [N : H] and thus Theorem 4.1 yields

ed(A) ≤ s[G : H] · [N : H] − [G : H] + 1 .

This is a bit weaker than the inequality of Theorem 2.2, even though the two look very similar. The difference is that we have replaced r in the inequality of Theorem 2.2 by s, where G is generated by s elements over H and by r elements over N. A priori r can be smaller than s. Nevertheless in the next section we will deduce Theorem 2.2 from Theorem 4.1 by a more delicate argument along these lines.

Our proof of Theorem 4.1 will rely on the following lemma.

Lemma 4.3. Let V be a ℤ[G]-submodule of ω(G/H). Then

G_{V} := \{g ∈ G | \overline{g} − \overline{1} ∈ V \}

is a subgroup of G containing H.

Proof. The inclusion H ⊂ G_{V} is obvious from the definition.

To see that G_{V} is closed under multiplication, suppose g, g′ ∈ G_{V}. That is, both \overline{g} − \overline{1} and \overline{g′} − \overline{1} lie in V. Then

\overline{gg′} − \overline{1} = g · (\overline{g′} − \overline{1}) + (\overline{g} − \overline{1})

also lies in V, i.e., gg′ ∈ G_{V}, as desired. □

Proof of Theorem 4.1. We claim that the elements \overline{g_{1}}−\overline{1}, …, \overline{g_{s}}−\overline{1} generate ω(G/H) as a ℤ[G]-module.

Indeed, let V be the ℤ[G]-submodule of ω(G/H) generated by these elements. Lemma 4.3 and condition (i) tell us that V contains \overline{g} − \overline{1} for every g ∈ G. Translating these elements by G, we see that V contains \overline{a} − \overline{b} for every a, b ∈ G. Hence, V = ω(G/H), as claimed.

For i = 1, …, s, let

S_{i} := \{g ∈ G | g · (\overline{g_{i}} − \overline{1}) = \overline{g_{i}} − \overline{1}\}

be the stabilizer of \overline{g_{i}} − \overline{1} in G. We may assume here that g_{i} is not in H, otherwise it could be removed since it is not needed to generate G over H. Then clearly g ∈ S_{i} iff \overline{gg_{i}} = \overline{g_{i}} and \overline{g} = \overline{1}. From this one easily sees that S_{i} = H ∩ H^{g_{i}}. Thus we have an exact sequence

0 → M → ⊕_{i=1}^{s}ℤ[G/S_{i}] ⟶^{ϕ} ω(G/H) → 0

where ϕ sends a generator of ℤ[G/S_{i}] to \overline{g_{i}} − \overline{1} ∈ ω(G/H). By Theorem 3.1 it remains to show that G acts faithfully on M.

By Lemma 3.2 G fails to act faithfully on M if and only if r = 1 and S_{1} = H = H^{g_{1}}. But this possibility is ruled out by (ii). Indeed, assume that s = 1 and S_{1} = H = H^{g_{1}}. Then G = ⟨g_{1}, H⟩ and H = H^{g_{1}}. Hence, H is normal in G. Condition (2) then tells us that H = {1}. Moreover, in this case G = ⟨g_{1}, H⟩ = ⟨g_{1}⟩ is cyclic, contradicting (ii). □

5. Proof Theorem 1.2

As we saw above, it suffices to prove Theorem 2.2.

Let t_{1}, …, t_{r} ∈ G/N be a set of generators for G/N. Choose g_{1}, …, g_{r} ∈ G representing t_{1}, …, t_{r}. and let H′ := ⟨H, H^{g_{1}}, …, H^{g_{r}}⟩. Since H ≤ N and N is normal in G, H′ ≤ N. The group H′ depends on the choice of g_{1}, …, g_{r} ∈ G, so that g_{i}N = t_{i}. Fix t_{1}, …, t_{r} and choose g_{1}, …, g_{r} ∈ G representing them, so that H′ has the largest possible order or equivalently the smallest possible index in N. Denote this minimal possible value of [N : H′] by m. In particular

(5)  m = [N : H′] ≤ [N : (H^{g_{i}g} · H)]

for any i = 1, …, r and any g ∈ N. Here [N : (H^{g_{i}g} · H)] = \frac{|N|}{|H^{g_{i}g} · H|}, as

in Remark 4.2.

Choose a set of representatives 1 = n_{1}, n_{2}, …, n_{m} ∈ N for the distinct left cosets of H′ in N. We claim that the elements

{g_{i}n_{j} | i = 1, …, r; j = 1, …, m}

generate G over H. Indeed, let G_{0} be the subgroup of G generated by these elements and H. Since n_{1} = 1, G_{0} contains g_{1}, …, g_{r}. Hence, G_{0} contains H′. Moreover, G_{0} contains n_{j} = g_{1}^{−1}(g_{1}n_{j}) for every j; hence, G_{0} contains all of N. Finally, since t_{1} = g_{1}N, …, t_{r} = g_{r}N generate G/N, we conclude that G_{0} contains all of G. This proves the claim.

We now apply Theorem 4.1 to the elements {g_{i}n_{j}}. Substituting

[G : H] · [H : (H · H^{g_{i}n_{j}})] for [G : (H ∩ H^{g_{i}n_{j}})],

as in Remark 4.2, we obtain

ed(A) ≤  ∑_{i=1}^{r} ∑_{j=1}^{m}[G : (H ∩ H^{g_{i}n_{j}})] − [G : H] + 1

=  [G : H] · ∑_{i=1}^{r} ∑_{j=1}^{m}[(H · H^{g_{i}n_{j}}) : H] − [G : H] + 1

=  [G : H] · ∑_{i=1}^{r} ∑_{j=1}^{m} \frac{[N : H]}{[N : (H · H^{g_{i}n_{j}})]} − [G : H] + 1

≤  (by (5)) [G : H] · ∑_{i=1}^{r} ∑_{j=1}^{m} \frac{[N : H]}{m} − [G : H] + 1

=  r[G : H] · [N : H] − [G : H] + 1

as desired. This completes the proof of Theorem 2.2 and thus of Theorem 1.2. □

Acknowledgments

The authors are grateful to B. A. Sethuraman, D. J. Saltman, and B. Totaro for helpful comments.

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