LATTICES AND PARAMETER REDUCTION IN DIVISION ALGEBRAS

M. LORENZ AND Z. REICHSTEIN

Abstract. Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose center F contains k. We shall say that D can be reduced to r parameters if we can write D ≃ D_{0} ⊗_{F_{0}} F, where D_{0} is a division algebra, the center F_{0} of D_{0} contains k and trdeg_{k}(F_{0}) = r.

We show that every division algebra of odd degree n ≥ 5 can be reduced to ≤ \frac{1}{2} (n − 1)(n − 2) parameters. Moreover, every crossed product division algebra of degree n ≥ 4 can be reduced to ≤ (⌊log_{2}(n)⌋ − 1)n + 1 parameters. Our proofs of these results rely on lattice-theoretic techniques.

Contents

1. Introduction 2 2. Preliminaries 3 2.1. G-varieties 3 2.2. (G, H)-sections and compressions 4 2.3. G-lattices 5 2.4. The symmetric and exterior squares 5 3. Groups of the form T_{n−1} {> \atop ◁} G and lattices 5 3.1. Notations 5 3.2. T_{n−1} {> \atop ◁} G-varieties and G-lattices 5 3.3. Compressions and G-lattices 6 3.4. Linearization and essential dimension 7 4. Proof of Theorem 1.1 7 4.1. Reduction to a lattice-theoretic problem 7 4.2. Solution of the lattice-theoretic problem 9 5. Proof of Theorem 1.2 11 5.1. General observations 11 5.2. Conclusion of the proof 14 6. Algebras of degree four 15 References 18

1991 Mathematics Subject Classification. 16K20, 14L30, 20C10, 20J06. Key words and phrases. division algebra, integral representation, lattice, algebraic transformation

group, essential dimension, crossed product, central simple algebra. M. Lorenz was supported in part by NSF grant DMS-9618521. Z. Reichstein was supported in part by NSF grant DMS-9801675.

1. Introduction

Throughout this paper k denotes a (fixed) algebraically closed base field of characteristic zero. Let K be a field containing k and let A be a finite-dimensional K-algebra. We would like to write A as A = A_{0} ⊗_{K_{0}} K for some K_{0}-algebra A_{0} over an intermediate field k ⊆ K_{0} ⊆ K with trdeg_{k}(K_{0}) as low as possible; the minimal value of trdeg_{k}(K_{0}) will be denoted by τ(A). Note that if trdeg_{k}(K_{0}) < trdeg_{k}(K) then passing from A to A_{0} may be viewed as “parameter reduction” in A.

We shall be particularly interested in the case where A = UD(n) is the universal division algebra of degree n and K is the center of UD(n) which we shall denote by Z(n). Recall that UD(n) is the subalgebra of M_{n}(k(x_{ij}, y_{ij})) generated (as a division algebra) by two generic n × n-matrices X = (x_{ij}) and Y = (y_{ij}), where x_{ij} and y_{ij} are 2n^{2} independent variables over k; see, e.g., [Pr, Section II.1] or [Row_{1}, Section 3.2]. We will denote τ(UD(n)) by d(n). It is easy to show that d(n) ≥ τ(A) for any central simple algebra A of degree n whose center contains k (see, e.g., [Re_{2}, Lemma 9.2]); in other words, every central simple algebra of degree n can be “reduced to at most d(n) parameters”. In the language of [Re_{2}], d(n) = ed(PGL_{n}), where ed denotes the essential dimension; see [Re_{2}, Lemma 9.2].

To the best of our knowledge, the earliest attempt to determine the value of d(n) is due to Procesi, who showed that d(n) ≤ n^{2}; see [Pr, Thm. 2.1]. Note that if UD(n) is cyclic then d(n) = 2, because we can take A_{0} to be a symbol algebra; cf. [Re_{2}, Lemma 9.4]. This is known to be the case for n = 2, 3 and 6. For other n the exact value of d(n) is not known. However, the following inequalities hold:

d(n) ≤ n^{2} − 2n ([Re_{2}, Proposition 4.5]) ,  (1.1)
d(n) ≤ d(nm) ≤ d(n) + d(m) if (n, m) = 1 ([Re_{2}, Section 9.4]) ,  (1.2)
d(n^{r}) ≥ 2r ([Re_{1}, Theorem 16.1]) ,  (1.3)
d(n) ≤ \frac{1}{2}(n − 1)(n − 2) + n if n is odd ([Row_{2}]; cf. [Re_{2}, Section 9.3]) . (1.4)

The last inequality is due to Rowen. In this paper we will sharpen it by showing that, in fact, d(n) ≤ \frac{1}{2}(n − 1)(n − 2) for every odd n ≥ 5. Moreover, in UD(n), reduction to this number of parameters can be arranged in a particularly nice fashion:

Theorem 1.1. Let n ≥ 5 be an odd integer, UD(n) be the universal division algebra of degree n and Z(n) be its center. Then there exists a subfield F of Z(n) and a division algebra D of degree n with center F such that

(a) UD(n) = D ⊗_{F} Z(n),

(b) trdeg_{k}(F) = \frac{1}{2}(n − 1)(n − 2) and

(c) Z(n) is a rational extension of F. In particular, d(n) = ed(PGL_{n}) ≤ \frac{1}{2}(n − 1)(n − 2).

In the course of the proof of Theorem 1.1 we will obtain an explicit description of the center F of D: F ≃ k(⋀^{2} A_{n−1})^{S^{n}}, where S_{n} denotes the symmetric group on n symbols and

A_{n−1} = {(a_{1}, …, a_{n}) ∈ ℤ^{n} | a_{1} + ⋯ + a_{n} = 0} ,  (1.5)

with the natural S_{n}-action. Our argument relies on the results of [LL], where the symmetric square Sym^{2} A_{n−1} is shown to be stably permutation for n odd; see Proposition 4.5 below.

For our next result, recall that if A is a central simple algebra of degree n with center F and L is a subfield of A then L is called strictly maximal if F ⊂ L and [L : F] = n.

Theorem 1.2. Let A be a finite-dimensional central simple algebra of degree n with center F, L be a strictly maximal subfield of A, L^{norm} be the normal closure of L over F, and G = Gal(L^{norm}/F). Suppose G is generated by r elements together with H = Gal(L^{norm}/L). If either r ≥ 2 or H ≠ {1} then τ(A) ≤ r|G| − n + 1.

If A be a central simple algebra of degree n then the upper bounds we have for τ(A) (or, equivalently, for d(n)), are all quadratic in n; see (1.1), (1.4) and Theorem 1.1. However, if we assume that A is a crossed product, Theorem 1.2 yields an asymptotically better bound:

Corollary 1.3. Suppose a group G of order n can be generated by r ≥ 2 elements. Then τ(A) ≤ (r − 1)n + 1 for any G-crossed product central simple algebra A. In particular, τ(A) ≤ (⌊log_{2}(n)⌋ − 1)n + 1, for any crossed product central simple algebra of degree n ≥ 4. Here, as usual, ⌊x⌋ denotes the largest integer ≤ x.

Recall that A is called a G-crossed product if it contains a strictly maximal subfield L, such that L/F is a Galois extension and Gal(L/F) = G; cf. [Row_{1}, Definition 3.1.23]. Thus the first assertion of the corollary is an immediate consequence of Theorem 1.2. The second assertion follows from the first, because any group of order n can be generated by r ≤ log_{2}(n) elements. (Indeed, |<G_{0}, g>| ≥ 2|G_{0}| for any subgroup G_{0} of G and any g ∈ G \ G_{0}.) Note also that ⌊log_{2}(n)⌋ ≥ 2 for any n ≥ 4.

The case of central simple algebras of degree 4 is of special interest since, by a theorem of Albert, every such algebra is a ℤ/2ℤ × ℤ/2ℤ-crossed product; see e.g., [Row_{1}, Theorem 3.2.28]. Thus Corollary 1.3 says that d(4) = τ(UD(4)) ≤ 5. On the other hand, d(4) ≥ 4 by (1.3). This proves:

Corollary 1.4. d(4) = ed(PGL_{4}) = 4 or 5.

The rest of this paper is structured as follows. In Section 2 we discuss preliminary material from invariant theory and the theory of G-lattices. In Section 3 we explain how G-lattices can be used to give an upper bound on essential dimensions of certain groups. We prove Theorem 1.1 in Section 4 and Theorem 1.2 in Section 5. In Section 6 we show that the methods of this paper cannot be used to decide whether the exact value of d(4) equals 4 or 5.

2. Preliminaries

2.1. G-varieties. A G-variety X is an algebraic variety with a (regular) action of an algebraic group G. If G acts freely (i.e., with trivial stabilizers) on a dense open subset of X, then X is called a generically free G-variety.

A dominant rational map π: X −→ Y is called the rational quotient map if k(Y) = k(X)^{G} and π^{∗}: k(Y) ↪ k(X) is the natural inclusion k(X)^{G} ↪ k(X). We will usually denote the rational quotient Y by X/G; note that X/G is only defined up to birational equivalence. By a theorem of Rosenlicht a rational quotient map separates points in general position in X; see [Ros_{1}, Theorem 2] and [Ros_{2}] (also cf. [PV, Theorem 2.3]). In other words, there exists a dense open subset U of X such that f is regular on U and x, y ∈ U lie in the same G-orbit iff f(x) = f(y).

We will not generally assume that X is irreducible; however, we will always require X to be primitive. This means that G transitively permutes the irreducible components of X; equivalently, X/G is irreducible, i.e., k(X)^{G} is a field; cf. [Re_{2}, Section 2.2]. Note that an irreducible G-variety is always primitive, and, if G is connected, a primitive variety is necessarily irreducible. Thus the notion of a primitive variety is only of interest if the group G is disconnected.

If N is a normal subgroup of G then the G-action on X induces a (rational) G/Naction on X/N; moreover, one can choose a model Y of X/N such that the G/N-action on Y is regular; see [PV, Proposition 2.6], [Re_{2}, Remark 2.6].

Lemma 2.1. Let N be a normal subgroup of G and let X be a G-variety. Then X is generically free as a G-variety if and only if

(a) X is generically free as an N-variety and

(b) X/N is generically free as a G/N-variety.

Proof. Assume (a) and (b) hold. Choose x in X in general position and suppose g ∈ Stab(x). Then (b) implies that g ∈ N and (a) says that g = 1. This shows that the G-action on X is generically free. The converse in proved in a similar manner.

2.2. (G, H)-sections and compressions. Let X be a G-variety and let π: X −→ X/G be the rational quotient map. Furthermore, let H be a closed subgroup of G. An H-invariant subvariety S of X is called a (G, H)-section if the following conditions are satisfied.

(i) π(S) contains an open dense subset of X/G and

(ii) if x is a point in general position in S then gx ∈ S if and only if g ∈ H.

Recall that a G-compression X −→ Y is a dominant rational map of generically free G-varieties.

Lemma 2.2. Suppose S is a (G, H)-section of X. Then

(a) k(S)^{H} = k(X)^{G} and

(b) any H-compression S −→ S′ lifts to a G-compression X −→ X′, where S′ is a (G, H)-section of X′.

Proof. (a) See [PV, Section 2.8], [Po, 1.7.2] or [Re_{2}, Lemma 2.11].

(b) Note that X and G∗_{H} S are birationally equivalent as G-varieties, where G∗_{H} S is defined as G×S/H for the H-action given by h(g, s) = (gh^{−1}, hs); see [Po, Theorem 1.7.5] or [Re_{2}, Lemma 2.14]. Now we set X′ = G ∗_{H} S′ and extend f to a rational map X −→ X′ by f(g, s) ⟶ (g, f(s)). It is easy to see that this map has the desired properties; cf. the proof of [Re_{2}, Lemma 4.1].

2.3. G-lattices. Let G be a finite group. A G-lattice is a (left) module over the integral group ring ℤ[G] that is free of finite rank as a ℤ-module. A G-lattice M is called

• faithful if the structure map G ⟶ Aut_{ℤ}(M) is injective, and

• a permutation lattice if M has a ℤ-basis that is permuted by G. For any G-lattice M, the G-action on M extends canonically to actions of G on the group algebra k[M] and on the field of fractions k(M) of k[M]. The operative fact concerning the G-fields (i.e., fields with G-action) of the form k(M) for our purposes is the following result of Masuda [M]; cf. also [L, proof of Prop. (1.5)].

Proposition 2.3. Let 0 ⟶ M ⟶ E ⟶ P ⟶ 0 be an exact sequence of G-lattices with M faithful and P permutation. Then, as G-fields, k(E) ≃ k(M)(t_{1}, …, t_{r}), where the elements t_{i} are G-invariant and transcendental over k(M) and r = rank P.

2.4. The symmetric and exterior squares. Let M be a G-lattice. By definition, the symmetric square Sym^{2} M is the quotient of M^{⊗2} = M ⊗ M modulo the subgroup generated by the elements m ⊗ m′ − m′ ⊗ m for m, m′ ∈ M. Similarly, the exterior square ⋀^{2} M is the quotient of M^{⊗2} modulo the subgroup generated by the elements m ⊗ m, as m ranges over M. The action of G on M^{⊗2} restricts down to Sym^{2} M and ⋀^{2} M, making each a G-lattice. The G-lattice ⋀^{2} M can be identified with the sublattice of antisymmetric tensors in M^{⊗2}, that is,

⋀^{2} M ≃ A′_{2}(M) = {x ∈ M^{⊗2} | x^{τ} = −x}

where τ : M^{⊗2} ⟶ M^{⊗2} is the switch (m ⊗ m′)^{τ} = m′ ⊗ m; see [Bou, Exerc. 8 on p. A III.190]. Furthermore, A′_{2}(M) is exactly the kernel of the canonical map M^{⊗2} ↠ Sym^{2} M. Hence, we have an exact sequence of G-lattices

0 ⟶ ⋀^{2} M ⟶ M^{⊗2} ⟶ Sym^{2} M ⟶ 0 .  (2.1)

3. Groups of the form T_{n−1} {> \atop ◁} G and lattices

3.1. Notations. In this section we shall focus on the following situation. Let T_{n−1} = (𝔾_{m})^{n}/Δ be the (diagonal) maximal torus of PGL_{n}; here Δ ≃ 𝔾_{m} diagonally embedded in (𝔾_{m})^{n}. Recall that S_{n} acts on T_{n−1} by permuting the n copies of 𝔾_{m} and that the normalizer N(T_{n−1}) of T_{n−1} in PGL_{n} is isomorphic to T_{n−1} {> \atop ◁} S_{n}. We shall be interested in subgroups of N(T_{n−1}) of the form T_{n−1} {> \atop ◁} G, where G is a subgroup of S_{n}. These groups have two properties that will be important to us in the sequel: (i) T_{n−1} {> \atop ◁} G-varieties and their compressions can be constructed from G-lattices and (ii) certain (PGL_{n}, T_{n−1} {> \atop ◁} G)-sections will naturally come up in the proofs of Theorems 1.1 and 1.2. In this section we will focus on the relationship between T_{n−1} {> \atop ◁} G-varieties and G-lattices.

3.2. T_{n−1} {> \atop ◁} G-varieties and G-lattices. Suppose we are given a morphism

f: M ⟶ A_{n−1}

of G-lattices, where A_{n−1} is the root lattice defined in (1.5). Note that A_{n−1} ≃ X_{∗}(T_{n−1}) as an S_{n}-lattice (and hence as G-lattice), where X_{∗}(T_{n−1}) is the lattice of characters of T_{n−1}. We will always identify A_{n−1} with X_{∗}(T_{n−1}).

We will now associate to f a T_{n−1} {> \atop ◁} G-variety X_{f} as follows. Let G act on k[M] as usual and define a T_{n−1}-action on k[M] by putting

t(m) = f(m)(t) · m (t ∈ T_{n−1}, m ∈ M) .

One easily checks that, by k-linear extension of this rule, one obtains a well-defined action of T_{n−1} by automorphism on k[M]. Moreover, for t ∈ T_{n−1}, g ∈ G and m ∈ M, one calculates

t(gm) = f(gm)(t) · gm = [gf(m)](t) · gm = f(m)(t^{g}) · gm = [gt^{g}](m) ;

so the actions of G and T_{n−1} combine to yield a locally finite action of T_{n−1} {> \atop ◁} G on k[M] and thus an algebraic action on X_{f} = Spec k[M].

Lemma 3.1. The T_{n−1} {> \atop ◁} G-variety X_{f} is a generically free if and only if

(a) f is surjective and

(b) Ker(f) is a faithful G-lattice.

Proof. Condition (a) is equivalent to saying that the T_{n−1}-action on X_{f} is generically free; cf., e.g., [OV, Theorem 3.2.5]. To interpret condition (b) geometrically, note that k(X_{f}) = k(M) and k(X_{f}/T_{n−1}) = k(M)^{T_{n−1}} = k(Ker(f)). Thus condition (b) holds iff G acts faithfully on X_{f}/T_{n−1} or, equivalently, iff the G-action on X_{f}/T_{n−1} is and generically free (the two notions coincide for finite groups). The desired conclusion now follows from Lemma 2.1.

3.3. Compressions and G-lattices. In the sequel we will only be interested in generically free T_{n−1} {> \atop ◁} G-varieties. In particular, we will assume that f is surjective and expand it into an exact sequence of G-lattices:

0 ⟶ K = Ker(f) ⟶ M ⟶^{f} A_{n−1} ⟶ 0 ,  (3.1)

where K is a faithful G-lattice. For future reference, we extract the following equality from the proof of Lemma 3.1:

k(X_{f}/T_{n−1} {> \atop ◁} G) = k(X_{f})^{T_{n−1}{> \atop ◁}G} = [k(M)^{T_{n−1}}]^{G} = k(K)^{G} .  (3.2)

We can now obtain information about T_{n−1} {> \atop ◁} G-compressions of X_{f} by studying this sequence more closely.

Lemma 3.2. Suppose that the exact sequence (3.1) extends to a commutative diagram

[figure: 0; K; M; f; A_{n−1}; 0; 0; K_{0}; M_{0}; f_{0}; A_{n−1}; 0]

of G-lattices, where K_{0} is faithful, and the vertical map M_{0} ⟶ M is injective. Then there exists a T_{n−1} {> \atop ◁} G-compression X_{f} −→ X_{f_{0}}.

Proof. Since k(M_{0}) = k(X_{f_{0}}) is an T_{n−1} {> \atop ◁} G-invariant subfield of k(M) = k(X_{f}), it defines a dominant T_{n−1} {> \atop ◁} G-equivariant map X_{f} −→ X_{f_{0}}. Furthermore, by Lemma 3.1, the T_{n−1} {> \atop ◁} G-action on both X_{f} and X_{f_{0}} is generically free. Thus, the rational map X_{f} −→ X_{f_{0}} we have constructed is a T_{n−1} {> \atop ◁} G-compression.

3.4. Linearization and essential dimension. If the G-lattice M in Section 3.2 is a permutation lattice then the T_{n−1} {> \atop ◁} G-variety X_{f} is birationally linearizable. To see this, fix a ℤ-basis m_{1}, …, m_{r} of M that is permuted by G. Clearly, k(X_{f}) = k(M) = k(m_{1}, …, m_{r}). Thus, putting V_{f} = ∑_{i} km_{i} we obtain a T_{n−1} {> \atop ◁} G-invariant k-subspace of k(X_{f}) with k(X_{f}) = k(V_{f}).

A similar argument goes through if M to be permutation projective, i.e., M is a direct summand of a permutation G-lattice.

Lemma 3.3. If M is permutation projective in (3.1) then there is a T_{n−1} {> \atop ◁} Gcompression V −→ X_{f} with V a generically free linear T_{n−1} {> \atop ◁} G-variety. In particular, ed(T_{n−1} {> \atop ◁} G) ≤ rank K.

Proof. Suppose M ⊕ N = P, where P is a permutation G-lattice. Then the sequence (3.1) embeds in the obvious fashion in an exact sequence 0 ⟶ K ⊕ N ⟶ P = M ⊕ N ⟶ A_{n−1} ⟶ 0. In view of the foregoing and Lemma 3.2, this proves the first assertion.

To complete the proof, recall that the essential dimension of an algebraic group G is defined as the smallest possible value dim(X/G), where X is a generically free G-variety so that there is a G-compression V −→ X with V a generically free linear G-variety; see [Re_{2}, Definition 3.5]. For G = T_{n−1} {> \atop ◁} G and X = X_{f}, we have dim(X/G) = rank K by (3.2).

4. Proof of Theorem 1.1

4.1. Reduction to a lattice-theoretic problem. The universal division algebra UD(n) is represented by a class c ∈ H^{1}(Z(n), PGL_{n}). We can write UD(n) = D ⊗_{Z(D)} Z(n) if and only if c lies in the image of the natural map

H^{1}(Z(D), PGL_{n}) ⟶ H^{1}(Z(n), PGL_{n})  (4.1)

Recall that for any finitely generated field extension L/k, an element of α ∈ H^{1}(L, PGL_{n}) may also be interpreted as a PGL_{n}-torsor, i.e., a generically free PGL_{n}- variety X_{α} such that k(X_{α})^{PGL^{n}} = L. Moreover, X_{α} is uniquely determined (up to birational isomorphism of PGL_{n}-varieties), and the central simple algebra corresponding to α can be recovered as the algebra of PGL_{n}-equivariant rational maps X_{α} −→ M_{n}. In particular, X_{c} = M_{n} × M_{n}, where PGL_{n} acts on M_{n} × M_{n} by simultaneous conjugation; to say that c lies in the image of the map (4.1) is equivalent to saying that there exists a PGL_{n}-compression M_{n} × M_{n} −→ X′ such that k(X′)^{PGL^{n}} = Z(D). For a more detailed discussion of these facts and further references, see [RY, Section 3].

Denote the linear subspace of M_{n} consisting of diagonal matrices by D_{n}. It is easy to see that D_{n} × M_{n} is a (PGL_{n}, N(T_{n−1}))-section of M_{n} × M_{n}, where N(T_{n−1}) = T_{n−1} {> \atop ◁} S_{n} is the normalizer of the maximal torus T_{n−1} = (𝔾_{m})^{n}/Δ in PGL_{n}, as in the previous section. (See Lemma 5.2 below for a more general fact.) Thus, in view of Lemma 2.2, we have the following Reduction 4.1. In order to prove Theorem 1.1 it is enough to show that there exists an N(T_{n−1})-compression

D_{n} × M_{n} −→ X  (4.2)

such that

(i) dim(X) = \frac{1}{2}(n − 1)(n − 2) + n − 1 or equivalently, dim(X/N(T_{n−1})) = trdeg_{k} k(X)^{N(T_{n−1})} = \frac{1}{2}(n − 1)(n − 2).

(ii) Z(n) = k(M_{n} × M_{n})^{PGL^{n}} = k(D_{n} × M_{n})^{N(T_{n−1})} is purely transcendental over k(X)^{N(T_{n−1})}.

Our construction of the compression (4.2) will be based on Lemma 3.2. In order to apply this lemma, we need to write the linear N(T_{n−1})-variety D_{n} × M_{n} birationally in the form X_{f}, where f is as in (3.1). Let x_{i} and y_{rs} be the standard coordinates on D_{n} and M_{n} respectively. In this coordinate system, the S_{n}-action on D_{n} × M_{n} is given by

σ(x_{i}) = x_{σ(i)} and σ(y_{rs}) = y_{σ(r)σ(s)} .

Thus, monomials in these coordinates and their inverses form an S_{n}-lattice isomorphic to M = U_{n} ⊕ U_{n}^{⊗2}, where U_{n} = ℤ^{n} be the standard permutation S_{n}-lattice. Moreover, an element t = (t_{1}, …, t_{n}) of T_{n−1} = (𝔾_{m})^{n}/Δ acts on monomials in x_{i}, y_{rs} by characters determined (multiplicatively) by

t(x_{i}) = x_{i} and t(y_{rs}) = t_{r}t_{s}^{−1}y_{rs} .

Denoting the standard basis of U_{n} by b_{1}, …, b_{n} and defining f: M = U_{n} ⊕ U_{n}^{⊗2} ⟶ A_{n−1} by f(b_{i}, b_{r}⊗b_{s}) = b_{r}−b_{s}, the above formulas give exactly the action of N(T_{n−1}) = T_{n−1} {> \atop ◁} S_{n} on X_{f} as described in (3.2). The exact sequence (3.1) for this f is the Formanek – Procesi exact sequence

0 ⟶ K = Ker(f) ⟶ U_{n} ⊕ U_{n}^{⊗2} ⟶^{f} A_{n−1} ⟶ 0 ;  (4.3)

see [F]. Note that

K ≃ U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2} .

Here, the first copy of U_{n} is mapped identically onto the first summand of U_{n} ⊕ U_{n}^{⊗2}, the second U_{n} corresponds to the sublattice of U_{n}^{⊗2} consisting of the monomials in y_{ii} ∈ K, and A_{n−1}^{⊗2} describes the kernel of f, restricted to the sublattice ⟨y_{rs} | r ≠ s⟩ ≃ U_{n} ⊗ A_{n−1} ⊂ U_{n}^{⊗2} ; cf. [B, p. 3573].

We now want to apply Lemma 3.2 to the above sequence, with K_{0} = ⋀^{2} A_{n−1}. Recall that ⋀^{2} A_{n−1} may be viewed as a sublattice A_{n−1}^{⊗2}; see (2.1). Let

φ: ⋀^{2} A_{n−1} ↪ U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2}  (4.4)

be the natural embedding of ⋀^{2} A_{n−1} into the third component of U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2}. We remark that for n > 3, S_{n} acts faithfully on ⋀^{2} A_{n−1}; in fact, ⋀^{2} A_{n−1} ⊗_{ℤ} ℚ is the irreducible S_{n}-representation corresponding to the partition (n − 2, 1^{2}) of n; see, e.g., [FH, Exerc. 4.6].

Combining Reduction 4.1 with Lemma 3.2, we obtain:

Reduction 4.2. Theorem 1.1 follows from Propositions 4.3 and 4.4 stated below.

Proposition 4.3. For odd n ≥ 5, k(U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2}) ≃ k(⋀^{2} A_{n−1})(y_{1}, …, y_{r}) as S_{n}-fields, where the elements y_{i} are S_{n}-invariant and transcendental over k(⋀^{2} A_{n−1}). In particular, k(U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2})^{S^{n}} is rational over k(⋀^{2} A_{n−1})^{S^{n}}.

Proposition 4.4. For odd n, there exists a commutative diagram of S_{n}-lattices:

[figure: 0; U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2}; U_{n} ⊕ U_{n}^{⊗2}; f; A_{n−1}; 0; 0; _{⋀_{2} A_{n−1}}; φ; L; f_{0}; A_{n−1}; 0]

Here the first row is the Formanek – Procesi sequence (4.3).

Indeed, Proposition 4.4 in conjunction with Lemma 3.2 yields an N(T_{n−1})- compression

D_{n} × M_{n} −→^{≃} X_{f} −→ X = X_{f_{0}} ,

and formula (3.2) further implies that k(D_{n} × M_{n})^{N(T_{n−1})} = k(X_{f})^{N(T_{n−1})} = k(U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2})^{S^{n}} and k(X)^{N(T_{n−1})} = k(⋀^{2} A_{n−1})^{S^{n}}. Thus, condition (i) in Reduction 4.1 is clearly satisfied and Proposition 4.3 ensures that (ii) holds as well.

4.2. Solution of the lattice-theoretic problem. Our proofs of Propositions 4.3 and 4.4 will be based on the following result from [LL, Section 3.5]. If G is a finite group, H is a subgroup of G and M a ℤ[H]-module then M ↑_{H}^{G}= ℤ[G] ⊗_{ℤ[H]} M will denote the induced ℤ[G]-module.

Proposition 4.5. For odd n, there is an isomorphism of S_{n}-lattices

Sym^{2} A_{n−1} ⊕ U_{n} ⊕ ℤ ≃ ℤ↑_{S_{n−2}×S_{2}}^{S_{n}} ⊕U_{n} ⊕ ℤ ,

where ℤ has the trivial S_{n}-action. In particular, Sym^{2} A_{n−1} ⊕ U_{n} ⊕ ℤ is a permutation lattice.

Proof of Proposition 4.3. First, applying Proposition 2.3 to the obvious sequence 0 ⟶ U_{n} ⊕ A_{n−1}^{⊗2} ⟶ U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2} ⟶ U_{n} ⟶ 0, we see that

k(U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2}) ≃ k(U_{n} ⊕ A_{n−1}^{⊗2})(t_{1}, …, t_{n})

as S_{n}-fields.

Next, sequence (2.1) for M = A_{n−1} combined with Proposition 4.5 gives rise to an exact sequence of S_{n}-lattices

0 ⟶ ⋀^{2} A_{n−1} ⟶ A_{n−1}^{⊗2} ⊕ U_{n} ⊕ ℤ ⟶ P ⟶ 0 ,

where P = Sym^{2} A_{n−1} ⊕ U_{n} ⊕ ℤ is permutation. Applying Proposition 2.3 to this sequence, we deduce that

k(A_{n−1}^{⊗2} ⊕ U_{n} ⊕ ℤ) ≃ k(⋀^{2} A_{n−1})(x_{1}, …, x_{m})

as S_{n}-fields.

Finally,

k(U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2}) ≃ k(U_{n} ⊕ A_{n−1}^{⊗2})(t_{1}, …, t_{n})
= k(A_{n−1}^{⊗2} ⊕ U_{n} ⊕ ℤ)(t_{1}, …, t_{n−1})
≃ k(⋀^{2} A_{n−1})(x_{1}, …, x_{m}, t_{1}, …, t_{n−1})

as S_{n}-fields, which proves the first assertion of Proposition 4.3. The second assertion is an immediate consequence of the first.

Proof of Proposition 4.4. Recall that the embedding φ of (4.4) is defined as the composition

φ: ⋀^{2} A_{n−1} ↪^{ψ} A_{n−1}^{⊗2} ↪ U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2} ,

where ψ is the injection from (2.1) (with M = A_{n−1}) and the second map identifies A_{n−1}^{⊗2} with the third component of U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2}. We aim to show that φ together with sequence (4.3) will give rise to a commutative diagram as in the statement of Proposition 4.4. In other words, our goal is to show that the class in Ext_{ℤ[S_{n}]}(A_{n−1}, U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2}) corresponding to the extension (4.3) belongs to the image of the map

φ_{∗}: Ext_{ℤ[S_{n}]}(A_{n−1}, ⋀^{2} A_{n−1}) ⟶ Ext_{ℤ[S_{n}]}(A_{n−1}, U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2}) .

In fact, we will prove:

Lemma 4.6. For odd n, the map φ_{∗} is surjective.

Proof. We will tacitly use the following standard facts from homological algebra, valid for any finite group G:

• If V is a G-module and M an H-module for some subgroup H ≤ G then Ext_{ℤ[G]}^{∗}(V, M ↑_{H}^{G}) ≃ Ext_{ℤ[H]}^{∗}(V |_{H}, M); see [HS, Prop. IV.12.3] and [Br, Prop. III.5.9]. For V = ℤ, the trivial G-module, this isomorphism is the “Shapiro isomorphism” H^{∗}(G, M ↑_{H}^{G}) ≃ H^{∗}(H, M). In case, M is actually a G-module, the restriction map Res_{H}^{G}: H^{∗}(G, M) ⟶ H^{∗}(H, M) factors through the Shapiro isomorphism:

Res_{H}^{G} : H^{∗}(G, M) ⟶^{μ_{∗}} H^{∗}(G, M↑_{H}^{G}) ⟶^{∼} H^{∗}(H, M) ,

where μ: M ⟶ M↑_{H}^{G} sends m ↦ ∑_{g∈G/H} g ⊗ g^{−1}m; see [Br, p. 81].

• For any G-lattices V and W, Ext_{ℤ[G]}^{∗}(V, W) ≃ H^{∗}(G, V^{∗} ⊗ W), where ⊗ = ⊗_{ℤ} and V^{∗} = Hom_{ℤ}(V, ℤ) is the dual G-lattice; see [Br, Prop. III.2.2].

• If V and W are both permutation G-lattices then Ext_{ℤ[G]}(V, W) = 0; cf. [L, Propositions 1.1, 1.2].

Armed with these facts, we proceed as follows. First, Ext_{ℤ[S_{n}]}(A_{n−1}, U_{n}) = 0, because U_{n} ≃ ℤ↑\begin{matrix}S & n \\ S & n−1 \end{matrix} and A_{n−1}|_{S_{n−1}} ≃ U_{n−1}. Therefore, it suffices to show that the map

ψ_{∗}: Ext_{ℤ[S_{n}]}(A_{n−1}, ⋀^{2} A_{n−1}) ⟶ Ext_{ℤ[S_{n}]}(A_{n−1}, A_{n−1}^{⊗2})

is surjective. But the extension (2.1) (for M = A_{n−1}) gives rise to an exact sequence

Ext_{ℤ[S_{n}]}(A_{n−1}, ⋀^{2} A_{n−1}) ⟶^{ψ_{∗}} Ext_{ℤ[S_{n}]}(A_{n−1}, A_{n−1}^{⊗2}) ⟶ Ext_{ℤ[S_{n}]}(A_{n−1}, Sym^{2} A_{n−1}) .

Therefore, it suffices to prove:

Ext_{ℤ[S_{n}]}(A_{n−1}, Sym^{2} A_{n−1}) = 0 for odd n.

For this, we use the isomorphism Sym^{2} A_{n−1} ⊕ U_{n} ⊕ ℤ ≃ ℤ↑_{G}^{S_{n}} ⊕U_{n} ⊕ ℤ of Proposition 4.5, where we have put G = S_{n−2} × S_{2} for simplicity. This isomorphism entails

Ext_{ℤ[S_{n}]}(A_{n−1}, Sym^{2} A_{n−1}) ≃ Ext_{ℤ[S_{n}]}(A_{n−1}, ℤ↑_{G}^{S_{n}})
≃ Ext_{ℤ[G]}(A_{n−1}|_{G}, ℤ)
≃ H^{1}(G, A_{n−1}^{∗}) .

Dualizing the augmentation sequence 0 ⟶ A_{n−1} ⟶ U_{n} ⟶^{ϵ} ℤ ⟶ 0 we obtain an exact sequence 0 ⟶ ℤ ⟶^{ϵ^{∗}} U_{n} ⟶ A_{n−1}^{∗} ⟶ 0, where ϵ^{∗}(1) = ∑_{i} e_{i} , the sum of the natural basis elements of U_{n}. This sequence, viewed as exact sequence of G-lattices, in turn yields an exact sequence

H^{1}(G, U_{n}) = 0 ⟶ H^{1}(G, A_{n−1}^{∗}) ⟶ H^{2}(G, ℤ) ⟶ H^{2}(G, U_{n}) .

Thus, it suffices to show that H^{2}(G, ℤ) ⟶ H^{2}(G, U_{n}) is injective. As a G-module, U_{n} = V ⊕ W, where V = ⨁_{i=1}^{n−2} ℤe_{i} ≃ ℤ↑_{S_{n−3}×S_{2}}^{G} and W = ℤe_{n−1} ⊕ ℤe_{n} ≃ ℤ↑_{S_{n−2}}^{G}. Therefore, the Shapiro isomorphism gives

H^{2}(G, U_{n}) ≃ H^{2}(S_{n−3} × S_{2}, ℤ) ⊕ H^{2}(S_{n−2}, ℤ)

and the map H^{2}(G, ℤ) ⟶ H^{2}(G, U_{n}) becomes the restriction map

Res_{S_{n−3}×S_{2}}^{G} × Res_{S_{n−2}}^{G}: H^{2}(G, ℤ) ⟶ H^{2}(S_{n−3} × S_{2}, ℤ) ⊕ H^{2}(S_{n−2}, ℤ) .

This map is indeed injective, as is easily seen by identifying H^{2}(G, ℤ) in the usual fashion with Hom(G, ℚ/ℤ), and similarly for the subgroups S_{n−3} × S_{2} and S_{n−2}. This finishes the proof of the Lemma, and hence, of Proposition 4.4 and of Theorem 1.1.

5. Proof of Theorem 1.2

5.1. General observations. The following notations will be used throughout this section:

F will be a field containing k;

A will be a finite-dimensional central simple algebra with center F; L will be a strictly maximal commutative subfield of A; n denotes the degree of A, so dim_{F} A = n^{2} and [L : F] = n; G is the Galois group of the normal closure L^{norm} of L over F; T_{n−1} denotes the diagonal maximal torus of PGL_{n}.

Reduction 5.1. In the course of proving Theorem 1.2 we may assume without loss of generality that F is a finitely generated field extension of k.

Proof. Indeed, choose a primitive element x for the extension L/F and complete 1, x, …, x^{n−1} to an F -basis e_{1}, …, e_{n^{2}} of A, where e_{i} = x^{i−1} for i = 1, …, n. Let c_{rs}^{t} be the structure constants for A in this basis, i.e.,

e_{r}e_{s} = ∑_{t=1}^{n^{2}} c_{rs}^{t}e_{t}  (5.1)

for every r, s = 1, …, n^{2}. Then A = A_{0} ⊗_{F_{0}} F, where F_{0} = k(c_{rs}^{t}) and A_{0} is the n^{2}-dimensional F_{0}-algebra spanned by e_{1}, …, e_{n^{2}} with multiplication given by (5.1). Moreover, A_{0} is a central simple algebra of degree n with center F_{0} and A_{0} contains the strictly maximal subfield L_{0} = F_{0}(x) whose normal closure L_{0}^{norm} has Galois group G over F_{0}. Clearly, τ(A) ≤ τ(A_{0}). Thus, after replacing A by A_{0} we may assume that F is finitely generated extension of k.

Next we pass from central simple algebras to generically free PGL_{n}-varieties, as we did at the beginning of Section 4. Recall that a central simple algebra A of degree n with center F defines a class H^{1}(F, PGL_{n}). This class, in turn, gives rise to a PGL_{n}-torsor X_{A} over F. Since we are assuming F is a finitely generated extension of k, X_{A} is a generically free PGL_{n}-variety; moreover, F = k(X_{A}/PGL_{n}) and A can be recovered from X_{A} as the algebra of PGL_{n}-equivariant rational maps X_{A} −→ M_{n}; cf. [RY, Section 3].

The algebras A we are concerned with in the context Theorem 1.2 are of a special form: they have a maximal subfield L/F such that Gal(L^{norm}/F) is the given group G. We would like to know how this extra structure is reflected in the geometry of the PGL_{n}-variety X_{A}. The following lemma gives a partial answer. We continue to let T_{n−1} denote the diagonal maximal torus of PGL_{n}, as in Section 3.1.

Lemma 5.2. X_{A} has a (PGL_{n}, T_{n−1} {> \atop ◁} G)-section.

Note that the group G comes with a natural permutation representation of G on the n embeddings L ↪ L^{norm} over F. This permutation representation gives an embedding α: G ↪ S_{n} that we use to define the semidirect product T_{n−1} {> \atop ◁} G. Note that α is only defined up to an inner automorphism of S_{n}, since the n embedding L ↪ L^{norm} are not naturally in a 1-1 correspondence with {1, …, n}. Relabeling these embeddings (or, equivalently, reordering the roots of a defining polynomial for L/F) will cause T_{n−1} {> \atop ◁} G to be replaced by a conjugate subgroup of PGL_{n}; the corresponding section will be translates of each other by elements of S_{n} ⊂ PGL_{n}.

Thus it is sufficient to verify that such a section exists for a particular numbering of the embeddings L ↪ L^{norm}.

Proof. Suppose L = F (r) for some r ∈ L and r = r_{1}, r_{2}, …, r_{n} are the conjugates of r in L^{norm}. As we remarked above, the order of the roots is not intrinsic; however, we choose it at this point, and it will not be changed in the sequel. The group G permutes r_{1}, …, r_{n} transitively via a permutation representation α: G ↪ S_{n}.

Now let x_{1}, …, x_{n} be commuting independent variables over k. The symmetric group acts on the polynomial ring k[x_{1}, …, x_{n}] by permuting these variables; composing this action with α, we obtain a (permutation) action of G on k[x_{1}, …, x_{n}]. Note that for every f(x_{1}, …, x_{n}) ∈ k[x_{1}, …, x_{n}]^{G}, we have f(r_{1}, …, r_{n}) ∈ F; we shall write this element of F as f_{r}.

Recall that A is the algebra of PGL_{n}-equivariant rational maps X_{A} −→ M_{n}. We claim that

S = \{x ∈ X_{A} : \begin{matrix}r(x) = diag(λ_{1}, …, λ_{n}) is a diagonal matrix \\ and \\ f(λ_{1}, …, λ_{n}) = f_{r}(x) for every f ∈ k[x_{1}, …, x_{n}]^{G} \end{matrix} \}

is an (PGL_{n}, T_{n−1}{> \atop ◁}G)-section S of X_{A}. Note that S is a T_{n−1}{> \atop ◁}G-invariant subvariety of X_{A}: indeed, T_{n−1} acts trivially on the set of diagonal matrices, and f_{r} ∈ F is a PGL_{n}-invariant rational function on X_{A}. Thus we need to show that

(i) PGL_{n}x intersects S for x in general position in X_{A} and

(ii) gs ∈ S ⟹ g ∈ T_{n−1} {> \atop ◁} G for s in general position in S;

cf. Section 2.2.

Let π: X_{A} −→ X_{A}/PGL_{n} be the rational quotient map. Recall that k(X_{A}/PGL_{n}) = k(X_{A})^{PGL^{n}} = F. Suppose p(t) = t^{n} + a_{1}t^{n−1} + ⋯ + a_{n} is the minimal polynomial of r ∈ L over F. Note a_{1}, …, a_{n} ∈ F are PGL_{n}-invariant rational functions on X_{A}; in particular, for x ∈ X_{A} in general position, the matrix r(x) ∈ M_{n} satisfies the polynomial p_{x}(t) = t^{n} + a_{1}(x)t^{n−1} + ⋯ + a_{n}(x) ∈ k[t]. Since p(t) is an irreducible polynomial over F, its discriminant δ is a non-zero element of F, i.e., a non-zero PGL_{n}-invariant rational function X_{A}. This means that for x ∈ X_{A} in general position (i.e., away from the zero locus of δ and the indeterminacy locus of r), the n × n-matrix r(x) has distinct eigenvalues. We conclude that in this case p_{x}(t) is the characteristic polynomial for r(x); in particular, the eigenvalues of r(x) are precisely the roots of p_{x}(t).

To see what these eigenvalues are more explicitly, let Y −→ X_{A}/PGL_{n} be a rational map of varieties induced by the field extension L^{norm}/F. Then r_{1}, …, r_{n} ∈ L^{norm} are rational functions on Y. Thus we have the following diagram of rational maps

\xymatrix{X_{A} \ar@{-->}[dd]_{π} \ar@{-->}[r]^{r} & M_{n} \\  & Y \ar@{-->}[r]^{r_{i}} \ar@{-->}[dl] & k \\ X_{A}/PGL_{n}}

Suppose x be a point of X_{A} in general position and y is a point of Y lying above π(x). Since r_{1}, …, r_{n} are distinct elements of L^{norm} = k(Y), λ_{1} = r_{1}(y), …, λ_{n} = r_{n}(y) are the n distinct roots of p_{x}(t), i.e., the n distinct eigenvalues of the n × n-matrix r(x).

Proof of (i): In view of the above discussion, we may assume without loss of generality that r(x) is diagonalizable. In other words, the PGL_{n}-orbit of r(x) in M_{n} contains the diagonal matrix diag(λ_{1}, …, λ_{n}) or equivalently, r(x′) = diag(λ_{1}, …, λ_{n}) for some x′ ∈ PGL_{n}x. It remains to show that x′ ∈ S. Indeed, for any f ∈ k[x_{1}, …, x_{n}]^{G}, we have

f_{r}(x′) = f(r_{1}(y), …, r_{n}(y)) = f(λ_{1}, …, λ_{n}) ,

as desired. This completes the proof of (i).

Proof of (ii): Let x be a point of S in general position. We may assume without loss of generality that the eigenvalues λ_{i} of the diagonal matrix r(x) = diag(λ_{1}, …, λ_{n}) are distinct. Suppose gx ∈ S for some g ∈ PGL_{n}. Then gr(x)g^{−1} is again diagonal; hence, g ∈ T_{n−1} {> \atop ◁} S_{n}. In other words, g = tσ, where t is a diagonal matrix and σ is a permutation matrix; our goal is to show that σ ∈ G. Indeed, by the definition of S, σ has the property that f(λ_{σ(1)}, …, λ_{σ(n)}) = f(λ_{1}, …, λ_{n}) for every f ∈ k[x_{1}, …, x_{n}]^{G}. Since G-invariant regular functions separate the orbits of the permutation G-action on affine n-space (this is true for any finite group action on an affine variety; see, e.g., [PV, Section 0.4]), the points (λ_{σ(1)}, …, λ_{σ(n)}) and (λ_{1}, …, λ_{n}) are in the same G-orbit for this action. On the other hand, since λ_{1}, …, λ_{n} are distinct, this is only possible if σ ∈ G. This completes the proof of (ii) and thus of Lemma 5.2.

Remark 5.3. One can show that the converse of Lemma 5.2 is also true: if X_{A} has a (PGL_{n}, T_{n−1} {> \atop ◁} G)-section then A contains a strictly maximal subfield L such that Gal(L^{norm}/F) = G. Since this result is not needed in the sequel, we omit the proof.

5.2. Conclusion of the proof. We are now ready to finish the proof of Theorem 1.2. Let A be a central simple algebra of degree n and let X_{A} denote the PGL_{n}-variety associated to A. Recall that τ(A) = ed(X_{A}, PGL_{n}); see [Re_{2}, Theorem 8.8 and Lemma 9.1]. Moreover, if X has a (PGL_{n}, H)-section S then ed(X, PGL_{n}) ≤ ed(S, H) ≤ ed(H); see [Re_{2}, Lemma 4.1 and Definition 3.5]. Applying these inequalities to the situation described by Lemma 5.2, with H = T_{n−1} {> \atop ◁} G, we see that

τ(A) ≤ ed(T_{n−1} {> \atop ◁} G) .  (5.2)

Thus Theorem 1.2 is a consequence of the following:

Lemma 5.4. Suppose G is a transitive subgroup of S_{n} generated by the subgroup H = G ∩ S_{n−1} together with elements g_{1}, …, g_{r}. Assume that either r ≥ 2 or H ≠ {1}.

Then ed(T_{n−1} {> \atop ◁} G) ≤ r|G| − n + 1.

Proof. By Lemma 3.3, it suffices to construct an exact sequence (3.1) with M permutation projective and K faithful having rank K = r|G| − n + 1. To this end, note that U_{n} ≃ ℤ[G/H] as G-lattices. Let : ℤ[G] ↠ ℤ[G/H] = U_{n} denote the canonical epimorphism; the kernel of is ℤ[G]ωH, where ωH denotes the augmentation ideal of ℤ[G]; cf. [Pa]. Then ∑_{i} ℤ[G]\overline{(g_{i} − 1)} = A_{n−1}; see [Pa, Lemma 3.1.1]. Therefore, we obtain an epimorphism of G-lattices

f: M = ℤ[G]^{r} ↠ A_{n−1} , (α_{1}, …, α_{r}) ↦ ∑_{i}^{r} α_{i}\overline{(g_{i} − 1)} .

Put K = Ker f; so K certainly has the required rank. For faithfulness, we may consider K ⊗ ℚ instead of K and work over the semisimple algebra ℚ[G]. Since f ⊗ ℚ and ⊗ ℚ are split, we have ℚ[G]-isomorphisms (A_{n−1} ⊗ ℚ) ⊕ (K ⊗ ℚ) ≃ ℚ[G]^{r} and (A_{n−1} ⊗ ℚ) ⊕ ℚ ⊕ ℚ[G]ωH ≃ ℚ[G]. Therefore,

K ⊗ ℚ ≃ ℚ[G]^{r−1} ⊕ ℚ ⊕ ℚ[G]ωH .

If r ≥ 2 then ℚ[G]^{r−1} is G-faithful, and if H ≠ {1} then ωH ⊗ ℚ is H-faithful and so ℚ[G]ωH ≃ (ωH ⊗ ℚ)↑_{H}^{G} is G-faithful. In either case, K ⊗ ℚ is faithful, and hence so is K, as desired.

6. Algebras of degree four

Recall that Corollary 1.4 asserts that d(4) equals 4 or 5. Whether the true value of d(4) is four or five is an open question. The purpose of this section is to show that this question cannot be resolved by the methods of this paper.

For the rest of this section we will identify the Klein 4-group V = (ℤ/2ℤ) × (ℤ/2ℤ) with the subgroup of S_{4} generated by (12)(34) and (13)(24). Let A_{3} be the augmentation (or root) lattice of S_{4}, restricted to V; see (1.5). In other words,

A_{3} ≃ ωV ,  (6.1)

where ωV is the augmentation ideal of the group ring ℤ[V].

We now briefly recall how we arrived at the bound d(4) ≤ 5. First of all, since UD(4) is a V-crossed product, d(4) = τ(UD(4)) ≤ ed(T_{3} {> \atop ◁} V); see (5.2). Secondly, Lemma 3.2 tells us that ed(T_{3} {> \atop ◁} V) ≤ rank(K_{0}), for any commutative diagram

[figure: 0; K; M; f; A_{3}; 0; 0; K_{0}; φ; M_{0}; f_{0}; A_{3}; 0]  (6.2)

of V-lattices with M permutation projective, K_{0} faithful, and φ is injective. Finally, in the course of the proof of Lemma 5.4 (with G = V and r = 2) we constructed a particular diagram (6.2) with M = M_{0} = ℤ[V]^{2} and rank(K_{0}) = 5. This gave us the bound d(4) ≤ 5. The question we will now address is whether or not one can sharpen this bound by choosing a different diagram (6.2). The following proposition shows that the answer is “no”.

Proposition 6.1. Let (6.2) be a commutative diagram of V = (ℤ/2ℤ) × (ℤ/2ℤ)- lattices with M permutation projective. Then K_{0} is faithful and rank K_{0} ≥ 5.

Note that in the setting of Lemma 3.2 we assumed that (i) K_{0} is faithful and (ii) φ is injective. Here we see that (i) is automatic and (ii) is irrelevant for the rank estimate.

Proof. Since M is permutation projective, we have H^{1}(H, M) = 0 = H^{−1}(H, M) for every subgroup H of V. (This condition is actually equivalent to M being permutation projective; see [C-TS, Proposition 4].) Consequently, (6.2) yields a commutative diagram

[figure: 0 = H^{1}(H, M); H^{1}(H, A_{3}); δ; H^{2}(H, K); H^{1}(H, A_{3}); δ_{0}; H^{2}(H, K_{0}); φ_{∗}]

Thus, δ_{0} is mono. Since H^{1}(V, A_{3}) ≃ ℤ/4ℤ and H^{1}(H, A_{3}) ≃ ℤ/2ℤ for any nonidentity cyclic subgroup H of V (see [LL, Lemma 4.3]), we obtain

ℤ/4ℤ ↪ H^{2}(V, K_{0})  and  H^{2}(H, K_{0}) ≠ 0 for all 1 ≠ H ≤ V.  (6.3)

Similarly, 0 = H^{−1}(H, M) implies H^{−1}(H, A_{3}) ↪ Ĥ^{0}(H, K_{0}). Using the identification (6.1), we have H^{−1}(V, A_{3}) ≃ ωV/(ωV)^{2} ≃ ℤ/2ℤ ⊕ ℤ/2ℤ. Thus:

ℤ/2ℤ ⊕ ℤ/2ℤ ↪ Ĥ^{0}(H, K_{0}) .  (6.4)

We will show that (6.3) forces K_{0} to be faithful, and (6.3) and (6.4) together imply that rank K_{0} ≥ 5. The discussion below could be shortened somewhat by a reference to [N]; however, for the sake of completeness, we will give a self-contained argument.

Lemma 6.2. Let L be a V-lattice, 1 ≠ x ∈ V and L_{±} = {l ∈ L | xl = ±l}. If L|_{⟨x⟩} = L_{+} ⊕ L_{−} then 2 · H^{2}(V, L) = 0.

Proof. Since L_{+} and L_{−} are V-sublattices of L, we may assume L = L_{+} or L = L_{−}. Write V = ⟨x, y⟩. Then ⟨y⟩-sublattices of L are stable under V. Therefore, we may assume that L is indecomposable as a ⟨y⟩-lattice. This leaves the following possibilities for L: ℤ_{±}↑_{⟨x⟩}^{V} or ℤ_{λ} for some λ ∈ Hom(V, ℤ). In each case, 2 · H^{2}(V, L) = 0 is easy to verify.

Lemma 6.2, in combination with the first condition in (6.3), implies that K_{0} is faithful. Consequently, \overline{K_{0}} = K_{0}/K_{0}^{V} is faithful as well, and so rank \overline{K_{0}} ≥ 2. In addition, we know by (6.4) that Ĥ^{0}(V, K_{0}) = K_{0}^{V}/(∑_{V} v)K_{0} is not cyclic. Hence, neither is K_{0}^{V}, which forces rank K_{0}^{V} ≥ 2 and thus rank K_{0} ≥ 4. Suppose, by way of contradiction, that equality holds here, i.e., \overline{K_{0}} = K_{0}/K_{0}^{V} and K_{0}^{V} both have rank 2. By the well-known classification of finite subgroups of GL_{2}(ℤ), the action of V on \overline{K_{0}} is given by either diag: the matrices (\begin{matrix}1 \\ & −1 \end{matrix}) and (\begin{matrix}−1 \\ & 1 \end{matrix}); so \overline{K_{0}} ≃ ℤ_{+,−} ⊕ ℤ_{−,+}, or

non-diag: the matrices (\begin{matrix}& 1 \\ 1 \end{matrix}) and (\begin{matrix}& −1 \\ −1 \end{matrix}); so \overline{K_{0}} ≃ ℤ_{−}↑_{H}^{V} for some cyclic

H ≤ V. In case non-diag, H^{2}(V, \overline{K_{0}}) ≃ H^{2}(H, ℤ_{−}) ≃ Ĥ^{0}(H, ℤ_{−}) = 0, and hence H^{2}(V, K_{0}^{V}) maps onto H^{2}(V, K_{0}). But H^{2}(V, K_{0}^{V}) ≃ H^{2}(V, ℤ)^{2} ≃ Hom(V, ℚ/ℤ)^{2} ≃ (ℤ/2ℤ)^{4}. Thus H^{2}(V, K_{0}^{V}) is annihilated by 2, and hence so is H^{2}(V, K_{0}), contradicting (6.3). Therefore, diag must hold:

\overline{K_{0}} = K_{0}/K_{0}^{V} ≃ ℤ_{+,−} ⊕ ℤ_{−,+} .

The action of V on K_{0} is given by matrices

c = ( 1_{2×2} 0 γ ; \begin{aligned}1 \\ & −1 \end{aligned} )  and  d = ( 1_{2×2} δ 0 ; \begin{aligned}−1 \\ & 1 \end{aligned} )

with γ, δ ∈ M_{2×1}(ℤ) and 0 = ( \begin{matrix}0 \\ 0 \end{matrix} ). By Lemma 6.2, γ ≠ 0 and δ ≠ 0. Conjugating by a suitable matrix of the form ( \begin{matrix}1_{2×2} & 0 ρ \\ 1 & 2×2 \end{matrix} ) we can ensure that the entries of γ are 0 or

1, and similarly for δ. If γ = ( \begin{matrix}1 \\ 1 \end{matrix} ) or ( \begin{matrix}1 \\ 0 \end{matrix} ) then conjugating, respectively, by ( 1 1 1 ; 1_{2×2} )

or ( {1 \atop 1} ; 1_{2×2} ), we can replace γ by γ = ( \begin{matrix}0 \\ 1 \end{matrix} ). Thus we may assume that γ = ( \begin{matrix}0 \\ 1 \end{matrix} ). If

δ = ( \begin{matrix}1 \\ 1 \end{matrix} ), then conjugating by ( 1 1 1 ; 1_{2×2} ), we replace δ by ( \begin{matrix}1 \\ 0 \end{matrix} ) without changing c. This leaves us with two cases to consider:

\underline{δ = ( \begin{matrix}0 \\ 1 \end{matrix} )}: Then c = ( \begin{matrix}1 \\ & c′ \end{matrix} ) and d = ( \begin{matrix}1 \\ & d′ \end{matrix} ) with c′ = ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) and d′ = ( \begin{matrix}1 & 1 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ).

Therefore, K_{0} ≃ ℤ ⊕ (A_{3} ⊗_{ℤ} ℤ_{λ}), where λ: V ⟶ ℤ is the map sending the elements of V acting via c and d both to −1. Tensoring the augmentation sequence 0 ⟶ A_{3} = ωV ⟶ ℤ[V] ⟶ ℤ ⟶ 0 with ℤ_{λ} we obtain an exact sequence 0 ⟶ A_{3} ⊗ ℤ_{λ} ⟶ ℤ[V] ⊗ ℤ_{λ} = ℤ[V] ⟶ ℤ_{λ} ⟶ 0. This sequence in turn implies that H^{2}(V, A_{3} ⊗ ℤ_{λ}) ≃ H^{1}(V, ℤ_{λ}), and the inflation-restriction sequence easily gives H^{1}(V, ℤ_{λ}) = ℤ/2ℤ. Thus, H^{2}(V, K_{0}) = H^{2}(V, ℤ) ⊕ H^{2}(V, A_{3} ⊗ ℤ_{λ}) ≃ (ℤ/2ℤ)^{3}, contradicting (6.3).

\underline{δ = ( \begin{matrix}1 \\ 0 \end{matrix} )}: In this case, cd = \binom{1_{2×2} 1_{2×2}}{−1_{2×2}}. Letting H denote the cyclic subgroup of

V acting via cd, we have K_{0}|_{H} ≃ ℤ[H]^{2}. Thus, H^{2}(H, K_{0}) = 0, contradicting (6.3).

This completes the proof of the proposition.

Acknowledgment. The work on this article was started while the authors were attending the Noncommutative Algebra program at MSRI in the Fall of 1999. The authors would like to thank the organizers of this program and MSRI staff for their hospitality and support.

References

[B] E. Beneish, Induction theorems on the stable rationality of the center of the ring of generic matrices, Trans. Amer. Math. Soc. 350 (1998), 3571 – 3585.

[Bou] N. Bourbaki, Algèbre, chap. 1–3, Hermann, Paris, 1970.

[Br] K. S. Brown, Cohomology of Groups, Springer-Verlag, New York, 1982. [C-TS] J.-L. Colliot-Thélène and J.-J. Sansuc, La R-équivalence sur les tores, Ann. scient. Éc. Norm.

Sup. 10 (1977), 175–230.

[F] E. Formanek, The center of the ring of 3×3 generic matrices, Linear and Multilinear Algebra 7 (1979), 203–212.

[FH] W. Fulton and J. Harris, Representation Theory, A First Course, Springer-Verlag, New York, 1991.

[HS] P. J. Hilton and U. Stammbach, A Course in Homological Algebra, 2nd ed., Springer-Verlag, New York, 1997.

[LL] N. Lemire, M. Lorenz, On certain lattices associated with generic division algebras, preprint, 1999. Available at http://www.math.temple.edu/˜ lorenz/papers/lattices.html

[L] H. W. Lenstra, Rational functions invariant under a finite abelian group, Invent. math. 25 (1974), 299–325.

[M] K. Masuda, On a problem of Chevalley, Nagoya Math. J. 8 (1955), 59–63.

[N] L. A. Nazarova, Integral representations of Klein’s four-group, Soviet Math. Dokl. 2 (1961), 1304–1307.

[OV] A. L. Onishchik, E. B. Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag, 1990.

[Pa] D. S. Passman, The Algebraic Structure of Group Rings, John Wiley & Sons, New York, 1977.

[Po] V. V. Popov, Sections in Invariant Theory, Proceedings of the Sophus Lie Memorial Conference, Scandinavian University Press, 1994, 315–362.

[PV] V. L. Popov, E. B. Vinberg, Invariant Theory, in Encyclopedia of Math. Sciences 55, Algebraic Geometry IV, edited by A. N. Parshin and I. R. Shafarevich, Springer-Verlag, 1994.

[Pr] C. Procesi, Non-commutative affine rings, Atti Acc. Naz. Lincei, S. VIII, v. VIII, fo. 6 (1967), 239–255.

[Re_{1}] Z. Reichstein, On a theorem of Hermite and Joubert, Canadian Journal of Math., 51 (1)

(1999), 69–95.

[Re_{2}] Z. Reichstein, On the notion of essential dimension for algebraic groups, Transformation

groups, to appear. Available at http://ucs.orst.edu/˜ reichstz/pub.html.

[RY] Z. Reichstein, B. Youssin, Splitting fields of G-varieties, preprint.

Available at http://ucs.orst.edu/˜ reichstz/pub.html.

[Ros_{1}] M. Rosenlicht, Some basic theorems on algebraic groups, American Journal of Math., 78

(1956), 401–443.

[Ros_{2}] M. Rosenlicht, A remark on quotient spaces, Anais da Academia Brasileira de Ciências 35

(1963), 487–489.

[Row_{1}] L. H. Rowen, Polynomial Identities in Ring Theory, Academic Press, 1980. [Row_{2}] L. H. Rowen, Brauer factor sets and simple algebras, Trans. Amer. Math. Soc., 282, no. 2

(1984), 765–772.

Department of Mathematics, Temple University, Philadelphia, PA 19122-6094

   E-mail address: lorenz@math.temple.edu

Department of Mathematics, Oregon State University, Corvallis, OR 97331

   E-mail address: zinovy@math.orst.edu