THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE

AMIRAM BRAUN

Dept. of Mathematics, University of Haifa, Haifa, Israel 31905; abraun@math.haifa.ac.il

Abstract. Let G ⊂ SL(V) be a finite group, where V is a finite dimensional vector space over a field F with charF = p ≥ 0. Let S(V) be the symmetric algebra of V and S(V)^{G} the ring of G-invariants. We provide for each p > 0 an example of a group G, where S(V)^{G} is Cohen-Macaulay, but is not Gorenstein. This refutes a natural conjecture due to G. Kemper, E. Körding, G. Malle, B.H. Matzat, D. Vogel and G. Wiese. We also prove a ”local-global” criterion for the Gorenstein property to hold in S(V)^{G}. This is then used to characterize the Gorenstein property of S(F^{2})^{G} for any finite subgroups G of SL(2, F).

1. Introduction

Let G ⊂ GL(V) be a finite group, where V is a finite dimensional vector space over a field F with charF = p ≥ 0. The following is a classical result of K. Watanabe.

Theorem. [7, Theorems 6.4.10] Suppose that (|G|, p) = 1 and G contains no pseudoreflection. Then the following are equivalent:

(1) G ⊂ SL(V),

(2) S(V)^{G} is a Gorenstein ring.

In the modular case, that is if p divides |G|, S(V)^{G} may fail to be Cohen-Macaulay, let alone Gorenstein, thus a suitable generalization of the above to the modular case should require the Cohen-Macaulay property. In [18, Conjecture 5], Kemper et al. conjectured that one of the above implications is valid (assuming the Cohen-Macaulay property), in the modular case as well. Explicitly:

Conjecture. [18, Conjecture 5] Suppose G ⊂ SL(V). Then S(V)^{G} is Gorenstien if (and only if) it is Cohen-Macaulay.

Date: April 28, 2012.

2 AMIRAM BRAUN

This conjecture was verified in [18] on all 1916 examples of the database (which satisfy its hypothesis). A special case of the conjecture, assuming that G contains no transvections, was confirmed in [3, Theorem B] using non-commutative methods. A commutative proof for this special case was recently given by Fleischmann-Woodcock [12]. I was informed by A. Broer that [6] already contains a proof of this special case.

Our first objective here is to settle this conjecture in the negative. We have:

Example A. Let charF = p be an arbitrary prime number and n an odd integer satisfying (p, n) = 1, n ∤ (p − 1). Suppose also that F contains a primitive n-th root of unity. Let V = F^{n} be the natural SL(n, F)-module. Then there exists a cyclic subgroup G of SL(n, F), |G| = np, where S(V)^{G} is Cohen-Macaulay but is not Gorenstein.

Further (non-commutative) solvable and non-solvable counterexamples are given in Chapter 2.

On the positive side we have the following results.

Denote by G_{p} the G-subgroup generated by all p-sylov subgroup of G and by W (G) the G-subgroup generated by all pseudo-reflections (of all types).

Theorem B. Let G ⊂ GL(V) be a finite subgroup. Then there exists a subgroup H ⊆ G having the following properties:

(1) H ⊇ (G, G)G_{p}W (G),

(2) G/H is cyclic, and (|G/H|, p) = 1,

(3) S(V)^{H} is Gorenstein if it is Cohen-Macaulay,

(4) S(V)^{K} is Gorenstein if it is Cohen-Maculay, for each subgroup K satisfying H ⊇ K ⊇ W (G),

(5) If S(V)^{K} is Gorenstein, for a subgroup K satisfying K ⊇ W (G), then K ⊆ H.

The key to this result is the observation that the ”cyclic Gorenstein cover” of S(V)^{G} appears as a direct summand of the ring of G-semi-invariants Sz(S(V)). Recall that this subring of S(V) (also known as the ring of relative invariants) is given by ⊕_{λ}S(V)_{λ}, where λ ∈ Hom(G, F^{×}) and S(V)_{λ} = {s ∈ S(V)|g(s) = λ(g)s}. The notion of ”cyclic cover” is frequently used in commutative algebra and algebraic geometry (e.g. [20, Definition 2.49]).

The next consequence is motivated by Lie theory. There, the semi-center Sz(U(L)) is usually better behaved than the center Z(U(L)) of the enveloping algebra U(L). This

THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE 3

can be seen, for example, in Moeglin’s characteristic zero factoriality result [23], and in Braun-Vernik [5, Theorem C], in prime characteristic.

Theorem C. Suppose G ⊂ GL(V) has no diagonalizable pseudo-reflection. Then Sz(S(V)) is Gorenstein if it is Cohen-Macaulay.

Corollary D. Let G ⊂ GL(V) be a finite group. Then S(V)^{G^{(2)}G_{p}} is Gorenstein if it is Cohen-Macaulay.

Let S(V)^{G}/C be a finite extension with C normal. We say that S(V)^{G} is C-symmetric if S(V)^{G} ≅ Hom_{C}(S(V)^{G}, C) as S(V)^{G}-modules. In case S(V)^{G} is Cohen-Macaulay and C is an homogenous polynomial ring this property is equivalent to the Gorenstein property.

Our next result is probably the most important application of an extension of Theorem B. It characterizes the C- symmetry of S(V)^{G} in terms of ”local data”. Here < S > denotes the subgroup generated by S.

Theorem E. Let C be a normal domain with S(V)^{G}/C being a finite extension. Then the following are equivalent:

(1) S(V)^{G} is C-symmetric,

(2) S(V)^{<g,W (G)>} is C-symmetric for each p′-element g ∈ G ∖ (G, G)G_{p}W (G).

This result provides an easy proof to our extension of Watanabe’s theorem [3, Theorem A], as well as to his original theorem. The novelty here is that this is achieved without calculating the canonical module or the Nakayama automorphism. Instead we are reduced by Theorem E (using W (G) = 1) to the cyclic non-modular case. For that case one can either use Watanabe’s theorem or alternatively the semi-group technique for diagonalizable commutative finite groups (e.g. [7, Corollary 6.4.3, Theorem 6.3.5]).

Our counterexamples to the Kemper et al. conjecture are in the form of < W (G), g >= G, for some p′-element g in G ∖ (G, G)G_{p}W (G), in fact (G, G) = G_{p} = W (G) in most of these cases. However, they were firstly conceived by a non-commutative analysis of the conjecture. This will be explained in Chapter 3, in which the following non-commutative characterization of the C-symmetry of S(V)^{G} is obtained.

Proposition F. Let C ⊆ S(V)^{G} be as in Theorem E. Set N = (G, G)G_{p}W (G). Then the following are equivalent:

(1) S(V)^{G} is C-symmetric, (2) The skew group ring S(V)^{N} ∗ G/N is C-symmetric.

The methods for proving this are directly related to those appearing in [3] and thus they are not new. The prominent role of the subgroup (G, G)G_{p}W (G) is clearly seen from the last two results. This was previously (implicitly) observed by H. Nakajima [25] in his characterization of the UFD property of S(V)^{G}.

Our last result concerns the validity of the Kemper et al. conjecture for finite subgroups of SL(2, F). Let T (G) be the G-subgroup generated by all transvections. We have the following:

Theorem G. Let F be an arbitrary field with charF = p > 0. Let G ⊆ SL(2, F) be a finite subgroup and V = F^{2}, the natural SL(2, F)-module. Then

(i) S(V)^{G} is Gorenstein whenever V is an irreducible T (G)-module,

(ii) If V is a reducible T (G)-module and T G ≠ 1_{G} then S(V)^{G} is Gorenstein if and only if |G| divides |T (G)|(|T (G)| − 1),

(iii) There exists a unique maximal subgroup D of G, D ⊇ T (G) with |G|/|D| ≤ 2 and S(V)^{D} is Gorenstien,

(iv) In particular S(V)^{G} is Gorenstein in the following (not mutually disjoint) cases:

(1) T (G) = {1_{G}},

(2) G is an odd order group,

(3) F = F_{q}, where q = p^{s} and s is odd,

(4) p = 2.

(v) If p divides |G| then S(V)^{G} is an hypersurface in F^{3} whenever it is Gorenstein.

Our proof uses Theorem B (or equivalently Theorem E) together with a careful analysis of the possible actions of T (G) on V. The cases where V is an irreducible T (G)-module require separate treatment. All of this is carried out in Chapter 4.

2. Counterexamples

Let F be a field with charF = p > 0. We shall describe here a collection of finite groups G, some cyclic, some non-commutative, having the following properties:

(1) G ⊂ SL(V),

(2) S(V)^{G} is Cohen-Macaulay,

THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE 5

(3) S(V)^{G} is not Gorenstein.

This settles [18, Conjecture 5] in the negative.

Example 2.1.

Assume the p ≡ 3(mod 4) and that F contains a primitive 4-th root of unity ξ. Thus ξ^{2} = −1 ≠ 1 (since p is necessarily odd). Let V = F^{2} be the natural SL(2, F)-module with basis {x_{1}, x_{2}}.

Let τ = [\begin{matrix}ξ, & 0 \\ 0, & ξ^{−1} \end{matrix}], σ = [\begin{matrix}1, & 1 \\ 0, & 1 \end{matrix}] be matrices in SL(2, F) = SL(V).

Thus τ(x_{1}) = ξx_{1}, τ(x_{2}) = ξ^{−1}x_{2}, σ(x_{1}) = x_{1} + x_{2}, σ(x_{2}) = x_{2}.

We have:

τστ^{−1} = [\begin{matrix}ξ, & 0 \\ 0, & ξ^{−1} \end{matrix}] [\begin{matrix}1, & 1 \\ 0, & 1 \end{matrix}] [\begin{matrix}ξ^{−1}, & 0 \\ 0, & ξ \end{matrix}] = [\begin{matrix}1, & ξ^{2} \\ 0, & 1 \end{matrix}] = [\begin{matrix}1, & −1 \\ 0, & 1 \end{matrix}] = σ^{p−1}

Let G =< τ, σ >, be the subgroup of SL(2, F) generated by {σ, τ}. Then < σ > is a normal subgroup of G, | < σ > | = p and |G| = 4p. We extend the action of G on V into an action by automorphisms on the polynomial ring S(V) = F [x_{1}, x_{2}].

The next result is standard and its proof is included for completeness sake.

Lemma 2.2. T ≡ F [x_{1}, x_{2}]^{<σ>} = F [a, x_{2}], where a = x_{1}^{p} − x_{1}x_{2}^{p−1}.

Proof: σ(x_{1}) = x_{1} + x_{2} implies that:

Norm_{σ}(x_{1}) = x_{1}(x_{1} + x_{2})(x_{1} + 2x_{2})...(x_{1} + (p − 1)x_{2}) = x_{1}^{p} − x_{1}x_{2}^{p−1} = a.

Now by Galois theory:

[Q(F [x_{1}, x_{2}]) : Q(T)] = | < σ > | = p.

Also [Q(F [x_{1}, x_{2}]) : Q(F [a, x_{2}])] = dega = p. Consequently Q(T) = Q(F [a, x])). Since T is integral over F [a, x_{2}] and the latter is normal (being a polynomial ring), it now follows that T = F [a, x_{2}] □

Now clearly S(V)^{G} is a normal domain of Krull dimension 2. In particular S(V)^{G} is Cohen-Macaulay.

We shall next compute the action of τ on T:

τ(a) = (τ(x_{1}))^{p} − τ(x_{1})(τ(x_{2}))^{p−1} = (ξx_{1})^{p} − (ξx_{1})(ξ^{−1}x_{2})^{p−1} = ξ^{p}x_{1}^{p} − ξ^{2−p}x_{1}x_{2}^{p−1}.

6 AMIRAM BRAUN

Now p ≡ 3(mod 4) implies that p − (2 − p) = 2p − 2 ≡ 0(mod 4) and therefore ξ^{p} = ξ^{2−p}. Consequently τ(a) = ξ^{p}a and τ(x_{2}) = ξ^{−1}x_{2}.

Consider U ≡ F a + F x_{2}, a two dimensional F -subspace of T. Clearly S(U) = T = F [a, x_{2}], is a polynomial ring in the two variables {a, x_{2}}. The action of τ on T can therefore be regarded as the natural extension of the linear action of τ_{|U} on U, to S(U). Therefore

T^{<τ>} = S(U)^{<τ_{|U} >}. Now τ_{|U} = [\begin{matrix}ξ^{p}, & 0 \\ 0, & ξ^{−1} \end{matrix}] is the matrix representing τ_{|U} with respect

to the basis {a, x_{2}}. Clearly < τ_{|U} > contains no pseudo-reflections, | < τ_{|U} > | = 4 and (p, 4) = 1. Also det.τ_{|U} = ξ^{p−1} = −1 (using p ≡ 3(mod 4)). Consequently by Watanabe’s theorem S(U)^{<τ_{|U} >} = T^{<τ>} is not Gorenstein. Since S(V)^{G} = (S(V)^{<σ>})^{<τ>} = T^{<τ>}, the result follows.

Remark 2.3.

(1) Observe that if p is an odd prime then 𝔽_{p^{2}}, the Galois field of p^{2} elements, contains a primitive 4-th root of unity since 𝔽_{p^{2}} ∖ {0} is a cyclic group of order p^{2} − 1.

(2) The smallest example one obtains by the previous method is for p = 3, where |G| = 4 · 3 = 12 and F = 𝔽_{3^{2}}.

Example 2.4.

Let F be a field with charF = p > 0. Let n be an integer satisfying (p, n) = 1 and n ∤ (p − 1). We also assume that F contains a primitive n-th root of unity. Let x_{1}, ..., x_{n} be a basis of the vector space V = F^{n}. Consider the following matrices in SL(n, F):

τ = [\begin{matrix}ξ, & 0, & ⋯ & 0 \\ 0, & ξ, & 0, & ⋯ \\ ⋮ & 0, & ... & 0 \\ 0, & ⋯ & 0, & ξ \end{matrix}] and σ = [\begin{matrix}1, & 1, & 0, & ⋯ \\ 0, & 1, & 0, & ⋯ \\ ⋮ & 0, & ... & 0 \\ 0, & ⋯ & 0, & 1 \end{matrix}]

Thus τ(x_{i}) = ξx_{i}, for 1 ≤ i ≤ n, and σ(x_{1}) = x_{1} + x_{2}, σ(x_{i}) = x_{i}, for 2 ≤ i ≤ n. Clearly τ^{n} = 1, σ^{p} = 1. Therefore since (p, n) = 1, G ≡< σ, τ > is a cyclic group with generator τσ and |G| = np. As before we extend the action of G to a group of automorphisms of the symmetric algebra S(V) = F [x_{1}, ..., x_{n}].

We shall next show that S(V)^{G} is Cohen-Macaulay but not Gorenstein.

Set T ≡ S(V)^{<σ>}. Then the same calculations as in Lemma 2.2 show that T = F [a, x_{2}, ..., x_{n}], where a = x_{1}^{p} − x_{1}x_{2}^{p−1}. Clearly T is a polynomial ring, and T^{<τ>} =

THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE 7

(S(V)^{<σ>})^{<τ>} = S(V)^{G}. Now since | < τ > | = n is prime to p = charF, then T^{<τ>} is a T^{<τ>} - direct summand of T. Therefore T^{<τ>} = S(V)^{G} is Cohen-Macaulay. Set U ≡ F a + F x_{2} + ... + F x_{n}, an n-dimensional F -subspace of T. Clearly S(U) = T. Now τ(a) = (τ(x_{1}))^{p} − (τ(x_{1}))(τ(x_{2}))^{p−1} = ξ^{p}x_{1}^{p} − (ξx_{1})(ξx_{2})^{p−1} = ξ^{p}a, and τ(x_{i}) = ξx_{i}, for 2 ≤ i ≤ n. Consequently the action of τ_{|U} on U with respect to the basis {a, x_{2}, ..., x_{n}}

is represented by the matrix τ_{|U} = [\begin{matrix}ξ^{p}, & 0, & ⋯ & 0 \\ 0, & ξ, & 0, & ⋯ \\ ⋮ & 0, & ... & 0 \\ 0, & ⋯ & 0, & ξ \end{matrix}] ∈ GL(U). Consequently detτ_{|U} =

ξ^{p+n−1} ≠ 1 (since n ∤ (p − 1)). It is also clear that < τ_{|U} > contains no pseudo-reflection of GL(U) and < τ_{|U} >= n is prime to p. Consequently by Watanabe’s theorem S(U)^{<τ_{|U} >} is not Gorenstein.

Finally the action of τ on T = S(U) can be regarded as the natural extension of the linear action of τ_{|U} on U to S(U). Therefore T^{<τ>} = S(U)^{<τ_{|U}} > is not Gorenstein. Hence since S(V)^{G} = (S(V)^{<σ>})^{<τ>} = T^{<τ>} = S(U)^{<τ_{|U} >} then S(V)^{G} is not Gorenstein.□

Remark 2.5.

(1) If p is an odd prime then n = p + 1 satisfies n ∤ (p − 1) as well as (p, n) = 1, and the Galois field 𝔽_{p^{2}} contains a primitive n-th root of unity.

(2) If p ≡ 2(mod 3), then n = 3 will do and |G| = 3p is a cyclic subgroup of SL(3, F). Also p^{2} − 1 ≡ 4 − 1 ≡ 0(mod 3) and 𝔽_{p^{2}} contains a primitive 3-rd root of unity. In particular p = 2, |G| = 6 and F = 𝔽_{4} provide a rather small counterexample.

Non-solvable examples follow from the next theorem. We omit the proof due to its length.

Theorem 2.6. Let G ⊂ SL(n, F) be a finite group with T (G) = SL(n, 𝔽_{q}) being properly contained in G and |𝔽_{q}| ≥ 4. Let V = F^{n} be the natural SL(n, F)-module. Then:

(1) S(V)^{G} is Cohen-Macaulay but not Gorenstein, provided n is prime, n ≠ p ≡ charF and (n, q − 1) = 1,

(2) S(V)^{G} is Gorenstein if q ≡ 1(mod n).

Examples of a similar nature to Theorem 2.6 in case p = 2 are given next.

Example 2.7. Suppose charF = 2. Let n be an odd prime number and ξ ∈ F be a primitive n-th root of unity. Consider G =< ξI_{V} , SL(n, 𝔽_{2}) > a finite subgroup of SL(n, F) and V = F^{n} the natural SL(n, F)-module. Then S(V)^{G} is Cohen-Macaulay but not Gorenstein.

8 AMIRAM BRAUN

Proof: Let E = 𝔽_{2}^{n} = 𝔽_{2}x_{1} + ... + 𝔽_{2}x_{n} be the natural GL(n, 𝔽_{2}) = SL(n, 𝔽_{2}) module. Hence V = F^{n} = E ⊗_{𝔽_{2}} F. We have (e.g. [2, Theorem 8.1.1]) that S(V)^{SL(n,𝔽_{2})} = S(V)^{GL(n,𝔽_{2})} = F [c_{n,0}, c_{n,1}, .., c_{n,n−1}], where c_{n,i} is the i-th Dickson invariant. Consequently S(V)^{G} = F [c_{n,0}, ..., c_{n,n−1}]^{<g>} where g ≡ ξI_{V}. Let U ≡ F c_{n,0} + ⋯ + F c_{n,n−1}. Then U is a n-dimensional F -subspace of the polynomial ring S(V)^{GL(n,𝔽_{2})}, and S(U) its symmetric algebra satisfies S(U) = F [c_{n,0}, ..., c_{n,n−1}]. Now degc_{n,i} = 2^{n} − 2^{i} implying that g(c_{n,i}) = ξ^{2^{n}−2^{i}}c_{n,i}, for i = 0, ..., n − 1. Therefore g_{|U} is represented with respect to the basis c_{n,0}, ..., c_{n,n−1} by the following matrix:

[\begin{matrix}ξ^{2^{n}−1}, & 0, & ⋯ & ⋯ \\ 0, & ... & 0, & ⋯ \\ ⋮ & 0, & ξ^{2^{n}−2^{i}}, & 0 \\ ⋮ & ⋮ & ⋯ & ... \end{matrix}] .

Since (2^{n} − 1) + ⋯ + (2^{n} − 2^{i}) + ⋯ + (2^{n} − 2^{n−1}) = n2^{n} − (2^{n} − 1) and ξ^{n} = 1, we get that detg_{|U} = ξ^{−(2^{n}−1)} = ξ^{−1} ≠ 1, where the last equality follows from 2^{n} ≡ 2(mod n).

Now ξ^{2^{n}−1} = ξ grants that |g_{|U}| = |ξ| = n and is prime to p = charF. Therefore if we can show that < g_{|U} > contains no pseudo-reflection, the result will follow from Watanabe’s theorem since S(U)^{<g_{|U} >} = S(V)^{G}. Now if 1 ≤ k < n, then (k(2^{n} − 2^{i}), n) = 1, if i ≠ 1. Therefore (g_{|U})^{k} will not contain 1 on its diagonal except at the (2, 2) spot, thus given that n ≥ 3, (g_{|U})^{k} cannot be a pseudo-reflection. □

Remark 2.8.

(1) Since n|(2^{n−1} − 1) for an odd prime integer n, 𝔽_{2^{n−1}} already contains a primitive n-th root of unity ξ and any F ⊇ 𝔽_{2^{n−1}} will do in the previous example.

(2) The smallest example obtained in this way is for n = 3, F ⊇ 𝔽_{2^{3−1}}, G =< ξI_{V} , SL(3, 𝔽_{2}) >. Here |G| = 504.

3. Positive Results

The main purpose of the present chapter is to prove a generalization of Theorem B and its consequences Theorem C, Corollary D and Theorem E. The latter result shows that the C-symmetry of S(V)^{G} is equivalent to the C-symmetry of rings of invariants of special subgroups S(V)^{<g,W (G)>}, where g runs on all p′-elements of G ∖ (G, G)G_{p}W (G). This explains the nature of the counterexamples given in Chapter 2, almost all are of this

THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE 9

subgroup type. This result also provides a new non-computational proof of [3, Theorem A], by reducing it to Watanabe’s non-modular Theorem. It can also be used to give a proof of Watanabe’s theorem by a reduction to a non-modular cyclic group case, a case for which a combinatorial proof is also available [7, Theorem 6.3.5 and 6.4.2].

Let Sz(S(V)) ≡ ⊕_{λ}S(V)_{λ}, where λ runs on all elements of Hom(G, F^{×}), be the ring of G-semi-invariants. Let Ĝ ≡ {λ ∈ Hom(G, F^{×})|S(V)_{λ} ≠ 0}. Clearly Ĝ is a subgroup of Hom(G, F^{×}).

The next result is fairly easy in case the field F contains a primitive |G|-th root of unity. The general case requires a little more care.

Lemma 3.1. Let H be a subgroup of Ĝ and D(H) ≡ {g ∈ G|λ(g) = 1, for all λ ∈ H}. Then ⊕_{λ∈H}S(V)_{λ} = S(V)^{D(H)} and G/D(H) ≅ H.

Proof: ⊕_{λ∈H}S(V)_{λ} ⊆ S(V)^{D(H)} follows from λ(g) = 1, for each g ∈ D(H) and λ ∈ H. Clearly (G, G) ⊆ kerλ, for each λ ∈ Hom(G, F^{×}). Similarly for each g ∈ G with |g| = p^{e} we have λ(g) = 1. Thus (G, G)G_{p} ⊆ D(H) and therefore G/D(H) is a finite group with (|G/D(H)|, p) = 1.

We shall next show G/D(H) ≅ H. Now H, being commutative and finite, has the following direct sum decomposition into cyclic subgroups H =< λ_{1} > × ⋯ × < λ_{r} >, with |λ_{i}| = n_{i}, for i = 1, ..., r. Define

φ : G/D(H) → (λ_{1}(G), ..., λ_{r}(G)) ⊂ (F^{×}, ..., F^{×}),

by

φ(g(D(H)) = (λ_{1}(g), ..., λ_{r}(g)), for each g ∈ G.

Clearly φ is a homomorphism and λ_{i}(g) = 1, for i = 1, ..., r implies that λ(g) = 1, for each λ ∈ H and so g ∈ D(H). Thus φ is also an injection. Consequently |G/D(H)| ≤ |λ_{1}(G)| ⋯ |λ_{r}(G)|. Now (λ_{i}(g))^{n_{i}} = λ_{i}^{n_{i}}(g) = 1, implying that λ_{i}(g) is a n_{i}-th root of unity, for 1 ≤ i ≤ r and g ∈ G. Consequently |λ_{i}(G)| ≤ n_{i}, for 1 ≤ i ≤ r. Therefore |G/D(H)| ≤ n_{1} ⋯ n_{r} = |H|. Since Q(S(V)_{λ}) is one dimensional over Q(S(V)^{G}) we have [Q(⊕_{λ∈H}S(V)_{λ}) : Q(S(V)^{G})] = |H|. Also (S(V)^{D(H)})^{G/D(H)} = S(V)^{G}, showing by Galois theory that [Q(S(V)^{D(H)}) : Q(S(V)^{G})] = |G/D(H)| ≤ |H|. Consequently since Q(⊕_{λ∈H}S(V)_{λ}) ⊆ Q(S(V)^{D(H)}), we get that |G/D(H)| = |H|, Q(⊕_{λ∈H}S(V)_{λ}) = Q(S(V)^{D(H)}) and |λ_{i}(G)| = n_{i}, for i = 1, ..., r. Thus λ_{i}(G) =< ξ_{i} >, where ξ_{i} is a primitive n_{i}-th root of unity for i = 1, ..., r. Therefore since G/D(H) ≅< ξ_{1} > × ⋯ × < ξ_{r} >,

10 AMIRAM BRAUN

G/D(H) has a direct sum decomposition where each generator has its minimal polynomial split in F. Therefore since G/D(H) is commutative, we can decompose S(V)^{D(H)} into weight spaces with respect to the G/D(H)-action. Equivalently S(V)^{D(H)} ⊆ Sz(S(V)) ∩ Q(⊕_{λ∈H}(S(V)_{λ}) = ⊕_{λ∈H}S(V)_{λ}. □

Corollary 3.2. Let H be a subgroup of Ĝ. Then ⊕_{λ∈H}S(V)_{λ} is normal. In particular Sz(S(V)) is normal.

Proof: This follows from ⊕_{λ∈H}S(V)_{λ} = S(V)^{D(H)} and Sz(S(V)) = ⊕_{λ∈Ĝ}S(V)_{λ} = S(V)^{D(Ĝ)}. □

The next result is well known and is included for lack of a convenient reference.

Lemma 3.3. Let B be a normal domain and G ⊆ Aut(B), a finite group of automorphisms acting on B. Let A = B^{G}. Let I be a reflexive ideal in A. Then (IB)^{∗∗} ∩ A = I, where (−)^{∗} = Hom_{B}(−, B).

Proof: Since A is a Krull domain we have I = ∩_{v}I_{v}, where v runs on all height one prime ideals in A (e.g. [2, Lemma 3.4.1]. Let {p_{1}, ..., p_{n}} be the set of all height one prime ideals in A containing I. Since B_{p_{i}} is a finitely generated free A_{p_{i}}-module, it follows that A_{p_{i}} is a A_{p_{i}}-direct summand of B_{p_{i}}, implying that I_{p_{i}}B ∩ A_{p_{i}} = I_{p_{i}}. Moreover for each height one v in A with v ≠ p_{i}, i = 1, ..., n, we have I_{v}B_{v} ∩ A_{v} = B_{v} ∩ A_{v} = A_{v} = I_{v}. Now (IB)^{∗∗} = ∩_{q}(IB)_{q}, where q runs on all height one primes in B. Since the extension B/A satisfies the going-down property, we have (IB)^{∗∗} = ∩_{v}(IB)_{v}, where v runs on all height one primes in A. Consequently:

(IB)^{∗∗} ∩ A ⊆ (∩_{i=1}^{n}I_{p_{i}}B_{p_{i}} ∩ A_{p_{i}}) ∩ (∩_{v≠p_{i}}I_{v}B_{v} ∩ A_{v}) = ∩_{i=1}^{n}I_{p_{i}} ∩ (∩_{v≠p_{i}}I_{v}) = I.

Since the reverse inclusion is clear, the result follows. □

The next standard Lemma is useful.

Lemma 3.4. Let A ⊂ B ⊂ D be finite extensions of normal domains. Let I be an ideal of A. Then ((IB)^{∗∗}D)^{∗∗} = (ID)^{∗∗}, where (IB)^{∗∗} = Hom_{B}(Hom_{B}(IB), B) and (ID)^{∗∗} = Hom_{D}(Hom_{D}(ID), D).

Proof: This follows form (IB)^{∗∗} = ∩_{q}(IB), where q runs in all height one primes of B, (ID)^{∗∗} = ∩_{v}(ID)_{v}, where v runs on all height one primes in D, and the going down property between D and B. □

THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE 11

Our next result is an application of Samuel’s Galois descent(e.g. [13, Chapter 4]) combined with the result of Dress [11].

Lemma 3.5. Let G ⊂ GL(V) be a finite group acting as an automorphisms group on S(V). Let I be a reflexive ideal in S(V)^{G}. Then there exists a weight α ∈ Ĝ having the following properties:

(1) I ≅ S(V)_{α},

(2) α(g) = 1 for each g ∈ W (G).

Proof: By [11], S(V)^{W (G)} is a factorial domain, implying that (IS(V)^{W (G)})^{∗∗} = aS(V)^{W (G)} with a ∈ S(V)^{W (G)}. Consequently by Lemma 3.4

aS(V) = (aS(V)^{W (G)}S(V))^{∗∗} = ((IS(V)^{W (G)})^{∗∗}S(V))^{∗∗} = (IS(V))^{∗∗}.

Now the G-stability of I implies the same for (IS(V))^{∗∗} and therefore a is a weight vector with respect to the G-action that is a ∈ S(V)_{λ}, for some λ ∈ Ĝ. Therefore by Lemma 3.3 we get I = (IS(V))^{∗∗} ∩ S(V)^{G} = aS(V) ∩ S(V)^{G} = aS(V)_{λ^{−1}}. Consequently I ≅ S(V)_{α}, where α ≡ λ^{−1}. Also a ∈ S(V)^{W (G)} implies that λ(g) = 1 for each g ∈ W (G), and hence α(g) = 1 as well. □

Lemma 3.6. Suppose T ⊃ S ⊃ C are module-finite extensions of normal domains. Then: (Hom_{S}(T, S)⊗_{S}Hom_{C}(S, C))^{∗∗} ≅ Hom_{C}(T, C), as T -modules, where (−)^{∗} = Hom_{T} (−, T).

Proof: Denote w = Hom_{C}(S, C). Then by the adjunction isomorphism (e.g. [21, Corollary 3.2]) we have:

Hom_{S}(T, w) = Hom_{S}(T, Hom_{C}(S, C)) ≅ Hom_{C}(T ⊗_{S} S, C) ≅ Hom_{C}(T, C) as T -modules.

Therefore we need to show the isomorphism:

(Hom_{S}(T, S) ⊗_{S} w)^{∗∗} ≅ Hom_{S}(T, w), as T -modules.

Let q be a height one prime ideal in C. Then S_{q} is a Dedekind domain and in particular Gorenstein. Consequently by [4, Proposition 2.6] S_{q} ≅ Hom_{C_{q}}(S_{q}, C_{q}) = w_{q}. Alternatively S_{q} is a principle ideal ring, and in particular it is a factorial domain. Since Hom_{C_{q}}(S_{q}, C_{q}) is isomorphic to a reflexive ideal in S_{q}, it must be principal. Consider the mapping:

ψ : Hom_{S}(T, S) ⊗_{S} w → Hom_{S}(T, w),

Defined by:

ψ(f ⊗ z)(t) = f(t)z, for f ∈ Hom_{S}(T, S), t ∈ T, z ∈ w.

12 AMIRAM BRAUN

It is easy to check that ψ is a T -module homomorphism. We have the localized map:

ψ_{q} : (Hom_{S}(T, S) ⊗_{S} w)_{q} → (Hom_{S}(T, w))_{q}

Since w_{q} ≅ S_{q} we have:

α : (Hom(T, S) ⊗_{S} w)_{q} = Hom_{S_{q}}(T_{q}, S_{q}) ⊗_{S_{q}} w_{q} ≅ Hom_{S_{q}}(T_{q}, S_{q}).

Also

β : (Hom_{S}(T, w))_{q} = Hom_{S_{q}}(T_{q}, w_{q}) ≅ Hom(T_{q}, S_{q}).

Now it is easily checked that β · ψ_{q} · α^{−1} is the identity map, implying that ψ_{q} is an isomorphism for each q. Now it is standard (e.g. [2, Lemma 3.4.1]) for a finitely generated T -module N that N^{∗∗} = ∩_{p}N_{p}, where p runs on all height one prime of T. Consequently by the ”going down” property between T on C we also have N^{∗∗} = ∩_{q}N_{q}, where q runs on all height one prime ideals of C. Therefore by taking N = Hom_{S}(T, S) ⊗_{S} w), ψ induces an isomorphism between (Hom_{S}(T, S) ⊗_{S} w)^{∗∗} and Hom_{S}(T, w). □

Versions of the next corollary appear in [14, Lemma 2.7] and [32, Corollary 1.7(i)]. However we do not assume any Cohen-Macaulay, Gorenstein or local properties.

Corollary 3.7. Suppose T ⊃ S ⊃ C are module-finite extensions of normal domains, and T/S is unramified in codimension one. Then:

(T ⊗_{S} Hom_{C}(S, C))^{∗∗} ≅ Hom_{C}(T, C), as T − modules.

Proof: Clearly the result follows from Lemma 3.6, once we show that T ≅ Hom_{S}(T, S). Now it is standard (and goes back to Auslander-Buchsbaum) that this follows from the unramified assumption. Indeed, this assumption is equivalent to the fact that T_{q}/S_{q} is a separable extension, for each height one prime ideal q in S. Consequently Q(T)/Q(S) is a finite separable field extensions. Therefore the inverse different D_{T/S}^{−1} is well defined and D_{T/S}^{−1} ≅ Hom_{S}(T, S) as T -modules (e.g. [2, 3.10]). Now by [2, Theorem 3.10.2] the unramified assumption is equivalent to D_{T/S} = T. Consequently D_{T/S}^{−1} = T, and the result follows. □

The next result is crucial. It identifies the ”cyclic Gorenstein cover” of S(V)^{G} as a direct summand of Sz(S(V)).

Let C ⊆ S(V)^{G} be a normal domain with S(V)^{G}/C, being a finite extension. Then Hom_{C}(S(V)^{G}, C) ≅ I, a reflexive ideal in S(V)^{G}. Therefore by Lemma 3.5 I ≅ S(V)_{α}, for some α ∈ Ĝ, and α(W (G)) = 1. Let k ≡ |α| and set T ≡ ⊕_{i=0}^{k−1}S(V)_{α^{i}}.

Proposition 3.8. Retaining the above notation, we have Hom_{C}(T, C) ≅ T as T -modules.

Proof: Let D ≡ {g ∈ G|α(g) = 1}. Then, since α^{i}(g) = 1, for each i we have D = D(< α >) in the notations of Lemma 3.1. Consequently by Lemma 3.1 T = S(V)^{D}. In particular T is a normal domain and T^{G/D} = S(V)^{G}. Also α(W (G)) = 1 implies that T ⊆ S(V)^{W (G)}. Consequently T/S(V)^{G} is an unramified extension in codimension one [29, I. No.7, Proposition 21] and we have by Corollary 3.7:

(T ⊗_{S(V)^{G}} Hom_{C}(S(V)^{G}, C))^{∗∗} ≅ Hom_{C}(T, C), as T -modules.

Therefore

(T ⊗_{S(V)^{G}} S(V)_{α})^{∗∗} ≅ Hom_{C}(T, C).

Clearly we have

T_{α} ≡ {t ∈ T |g.t = α(g)t, for every g ∈ G} = S(V)_{α}.

Consider the multiplication map:

ψ : T ⊗_{S(V)^{G}} T_{α} ↠ T T_{α} ⊆ T.

Then for every height one prime ideal v in S(V)^{G} we have the localized onto map:

ψ_{v} : (T ⊗_{S(V)^{G}} T_{α})_{v} ↠ (T T_{α})_{v} = ((T T_{α})_{v})^{∗∗} = ((T T_{α})^{∗∗})_{v} = T_{v},

where the last equality follows from Nakajima’s [25, Lemma 2.2] since T/S(V)^{G} is unramified in codimension one. Now since (T ⊗_{S(V)^{G}} T_{α})_{v} = T_{v}⊗_{S(V)_{v}^{G}} (T_{α})_{v} is a free S(V)_{v}^{G}-module of rank [T_{v} : S(V)_{v}^{G}] we conclude by [27, Proposition 1.14] that ψ_{v} is an isomorphism. Consequently ψ : T ⊗_{S(V)^{G}} T_{α} → T, is an isomorphism at each height one prime ideal v in S(V)^{G}. Therefore (T ⊗_{S(V)^{G}} T_{α})^{∗∗} ≅ T, as T -modules. The result follows. □

Lemma 3.9. Keeping the notation of Proposition 3.8, assume that K ⊇ W (G) is a Gsubgroup with S(V)^{K} being C-symmetric. Then K ⊆ D.

Proof: Recall I ≅ Hom_{C}(S(V)^{G}, C) is a reflexive ideal of S(V)^{G}. Also as in Lemma 3.5 I = aS(V)_{α}, where a ∈ S(V)_{λ}, α = λ^{−1} and (IS(V))^{∗∗} = aS(V). Now S(V)^{G} ⊂ S(V)^{K} ⊆ S(V)^{W (G)}, implying that S(V)^{K}/S(V)^{G} is unramified in codimentsion one. Hence by Corollary 3.7:

(S(V)^{K} ⊗_{S(V)^{G}} Hom_{C}(S(V)^{G}, C))^{∗∗} ≅ Hom_{C}(S(V)^{K}, C) ≅ S(V)^{K}.

Consequently (S(V)^{K} ⊗_{S(V)^{G}} I)^{∗∗} ≅ S(V)^{K}, as S(V)^{K}-modules. Now the multiplication map induces the isomorphism (S(V)^{K}⊗_{S(V)^{G}}I)^{∗∗} ≅ (S(V)^{K}I)^{∗∗}. Consequently (S(V)^{K}I)^{∗∗}

14 AMIRAM BRAUN

is a principal ideal in S(V)^{K}, (S(V)^{K}I)^{∗∗} = bS(V)^{K} with b ∈ S(V)^{K}. Therefore by Lemma 3.4:

aS(V) = (IS(V))^{∗∗} = ((IS(V)^{K})^{∗∗}S(V))^{∗∗} = (bS(V)^{K}S(V))^{∗∗} = bS(V).

Therefore a = bε with ε ∈ F^{×}. Now g(b) = b for all g ∈ K implies the same for a and therefore that α(g) = 1, for each g ∈ K. Thus K ⊆ D ≡ {g ∈ G|α(g) = 1}. □

The next result implies Theorem B if C is taken to be an homogenous polynomial subring of S(V)^{G} with S(V)^{G}/C being finite.

Theorem 3.10. Let G ⊂ GL(V) be a finite group and S(V)^{G}/C be a finite extension, where C is a normal domain. Then there exists a finite subgroup D ≡ D(C) of G having the following properties:

(1) D ⊇ (G, G)G_{p}W (G),

(2) G/D is cyclic and (|G/D|, p) = 1,

(3) S(V)^{D} is C-symmetric,

(4) S(V)^{K} is C-symmetric, for each subgroup K with D ⊇ K ⊇ W (G),

(5) Suppose S(V)^{K} is C-symmetric, where K is a subgroup with K ⊇ W (G). Then K ⊆ D.

Proof: Let I ≅ Hom_{C}(S(V)^{G}, C) ≅ S(V)_{α} be as in Proposition 3.8. Set T = ⊕_{i=0}^{k−1}S(V)_{α^{i}}, where k = |α|, the order of α in Hom(G, F^{×}). Let D ≡ {g ∈ G|α(g) = 1}. G_{p} is generated by elements of p-power order. Such generators are mapped by each element of Hom(G, F^{×}) to 1, implying that α(G_{p}) = 1 and G_{p} ⊆ D. Similarly (G, G) ⊆ D. So combined with α(W (G)) = 1 (by Lemma 3.5), we get that (1) is established

Set H =< α >, then D = D(H), in the notation of Lemma 3.5, and by this Lemma G/D ≅< α > is a cyclic group. Since (|G/G_{p}|, p) = 1 this establishes (2).

Now T ≅ S(V)^{D} by Lemma 3.1 and (3) follows from Proposition 3.8.

Given a subgroup K satisfying W (G) ⊆ K ⊆ D, we have that S(V)^{G} ⊆ T = S(V)^{D} ⊆ S(V)^{K} ⊆ S(V)^{W (G)}, and in particular S(V)^{K} is unramified in codimension one over T. Therefore by Corollary 3.7 (applied to the triple C ⊆ T ⊂ S(V)^{K}), we have that:

(S(V)^{K} ⊗_{T} Hom_{C}(T, C))^{∗∗} ≅ Hom_{C}(S(V)^{K}, C).

Now by Proposition 3.8 Hom_{C}(T, C) ≅ T, and therefore the previous isomorphism reduces to S(V)^{K} ≅ Hom_{C}(S(V)^{K}, C), and (4) is satisfied.

Finally (5) follows from Lemma 3.9. □

Remark 3.11.

Different C′s can produce different D′s. This is easily seen by taking C = S(V)^{G} and then S(V)^{G} is trivially C-symmetric implying that G = D. However if C is in addition an homogenous polynomial subring of S(V)^{G}, the C-symmetry of S(V)^{G} is equivalent in case it is Cohen-Macaulay to its Gorenstein property. This is because Hom_{C}(S(V)^{G}, C) is then isomorphic to the canonical module of S(V)^{G}.

The next result establishes Theorem C of the Introduction.

Theorem 3.12. Suppose G has no diagonalizable pseudo-reflections. Then Sz(S(V)), the ring of G-relative invariants, is Gorenstein if it is Cohen-Macaulay.

Proof: By assumptions W (G) is generated by transvections (if there are any) and consequently W (G) ⊆ G_{p}. Since G_{p} ⊆ {g ∈ G|λ(g) = 1}, for each λ ∈ Ĝ, we get that G_{p} ⊆ D(Ĝ) and therefore W (G) ⊆ G_{p} ⊆ D(Ĝ). Clearly D(Ĝ) ⊆ D(< α >) ≡ D, where α is defined as in Proposition 3.8 and Theorem 3.10. Consequently W (G) ⊆ D(Ĝ) ⊆ D ⊆ G, implying that S(V)^{W (G)} ⊇ S(V)^{D(Ĝ)} ⊇ S(V)^{D} ⊇ S(V)^{G} and therefore S(V)^{D(Ĝ)}/S(V)^{D} = T is an unramified extension in codimentsion one.

By applying now Corollary 3.7 to the triple C ⊆ T ⊆ S(V)^{D(Ĝ)} we get that

(S(V)^{D(Ĝ)} ⊗_{T} Hom_{C}(T, C))^{∗∗} ≅ Hom_{C}(S(V)^{D(Ĝ)}, C).

Now since T ≅ Hom_{C}(T, C), by Proposition 3.8, the previous isomorphism is reduced to S(V)^{D(Ĝ)} ≅ Hom_{C}(S(V)^{D(Ĝ)}, C). Since S(V)^{D(Ĝ)} = Sz(S(V)) by Lemma 3.1, the C-symmetry of Sz(S(V)) is established. Now if C is in addition a polynomial ring and Sz(S(V)) is Cohen-Macaulay, then Hom_{C}(Sz(S(V)), C) is isomorphic to the canonical module of Sz(S(V)), and therefore Sz(S(V)) is Gorenstein.□

The next result is Corollary D of the Introduction.

Corollary 3.13. Let G ⊂ GL(V) be a finite group. Then S(V)^{G^{(2)}G_{p}} is Gorenstein if it is Cohen-Macaulay.

Proof: Let G_{1} ≡ (G, G)G_{p}. Clearly G_{1} ⊆ SL(V) and therefore G_{1} does not contain diagonalizable pseudo-reflections, that is W (G_{1}) ⊆ (G_{1})_{p}. Clearly (G_{1})_{p} = G_{p}.

Now by applying Theorem 3.10(1) to the group G_{1} we have:

D(G_{1}, C) ⊇ (G_{1}, G_{1})(G_{1})_{p}W (G_{1}) = (G_{1}, G_{1})(G_{1})_{p} ⊇ W (G_{1}).

16 AMIRAM BRAUN

Therefore, by 3.10(4), S(V)^{(G_{1},G_{1})(G_{1})_{p}} is C-symmetric. Since (G_{1}, G_{1})(G_{1})_{p} = G^{(2)}G_{p}, the result follows if C is an homogenous polynomial subring of S(V)^{G} and S(V)^{G}/C is a finite extension. □

The next result is Theorem E of the Introduction.

Theorem 3.14. Let G ⊂ GL(V) be a finite group and S(V)^{G}/C a finite extension with C being normal. Then the following are equivalent:

(1) S(V)^{G} is C-symmetric,

(2) S(V)^{<g,W (G)>} is C-symmetric, for each p′-element g ∈ (G, G)G_{p}W (G).

Proof: Assume that (1) holds then D = G (in the notation of Theorem 3.10). Item (2) follows from Theorem 3.10(4) by taking K =< g, W (G) >.

Conversely assume by negation that the implication (2) ⇒ (1) fails. Therefore D = D(C) ≠ G. Let ḡ be a generator of G/D. Since |ḡ| = k = |G/D| is prime to p (Theorem 3.10(2)), there exists e so that g^{p^{e}} ≡ g_{1} ∉ D (ḡ_{1} is still a generator of G/D), and (|g_{1}|, p) = 1. Since (G, G)G_{p}W (G) ⊆ D, we also have that g_{1} ∉ (G, G)G_{p}W (G). Consequently by (2), S(V)^{<g_{1},W (G)>} is C-symmetric and by Theorem 3.10 (5) this implies that < g_{1}, W (G) >⊆ D, a contradiction. □

Our next result gives a proof of [3, Theorem A], by reducing it to a very special case of Watanabe’s non-modular theorem. This case, where the group is non-modular and cyclic, can also be handled by normal semi-group methods (e.g. [7, Theorem 6.3.5, Theorem 6.4.2]).

Theorem 3.15. Let G ⊂ GL(V) be a finite group with W (G) = {1}. Let C ⊆ S(V)^{G} be any Gorenstein normal domain with S(V)^{G}/C a finite extension. Then the following is equivalent:

(1) S(V)^{G} is C-symmetric,

(2) G ⊂ SL(V).

Proof: Assuming (1) and let g ∈ G. Let e be an integer such that g^{p^{e}} ≡ g_{1} is a p′ element. Then (| < g_{1} > |, p) = 1 and consequently S(V)^{<g_{1}>} is Cohen-Macaulay. Then theorem 3.14(1) implies that S(V)^{<g_{1}>} = S(V)^{<g_{1},W (G)>} is C-symmetric. Now since C is Gorenstein, Hom_{C}(S(V)^{<g_{1}>}, C) is isomorphic to the canonical module S(V)^{<g_{1}>}, so by its C-symmetry S(V)^{<g_{1}>} is Gorenstein. This implies either by Watanabe’s theorem or by [7,

THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE 17

Theorem 6.4.2] that g_{1} ∈ SL(V), so det(g_{1}) = 1. Since g_{1} = g^{p^{e}}, it follows that det(g) = 1 and (2) is established.

In order to show the reversed implication, let g ∈ G be a p′-element in G∖(G, G)G_{p}W (G). Then by Watanabe’s theorem (or the above special case), S(V)^{<g>} is Gorenstein and hence S(V)^{<g>} is C-symmetric. Since W (G) = {1}, the condition of Theorem 3.14(2) is satisfied and therefore by Theorem 3.14(1) S(V)^{G} is C-symmetric. □

Remark 3.16.

Clearly the above method does not compute the canonical module. In particular it improves [3, Theorem A], by relaxing the polynomial assumption on C.

4. Non-commutative Heuristics

We explain here the non-commutative reasons why the Kemper et al. conjecture fails in general. We also explore possible general conditions under which the conjecture holds. We also illuminate the important role of the subgroup (G, G)G_{p}W (G) in the present paper.

Let C ⊆ S(V)^{G} be any homogenous polynomial subring with S(V)^{G}/C a finite extension. Consider the following properties:

(1) G ⊂ SL(V),

(2) S(V) ∗ G is C-symmetric,

(3) S(V)^{G} is C-symmetric,

(4) End_{S(V)^{G}}S(V) is C-symmetric.

The arguments in [3] show that (1) and (2) are equivalent. The implication (1) ⇒ (2) is a consequence of [3, Proposition 2.11]. The reverse implication (2) ⇒ (1) appears in the proof of [3, Theorem 2.25] (in the 2-nd and 3-rd paragraphs).

Another observation we had in [3] is that (3) and (4) are also equivalnet. Explicitly, the implication (3) ⇒ (4) holds since S(V)_{q} is a finitely generated projective S(V)_{q}^{G}-module for each height one prime ideal q of C, and then use [3, Lemma 2.19]. The reverse implication (4) ⇒ (3) follows since End_{S(V)_{q}^{G}}S(V)_{q} = (End_{S(V)^{G}}S(V))_{q} is a trivial Azumaya algebra for each height one prime ideal q of C. So, by the lifting property of Azumaya algebras, S(V)_{q} is a finitely generated projective (End_{S(V)^{G}}S(V))_{q}-module. One applies now [3, 2.19].

Therefore the Kemper et al. conjecture amounts in the presence of the Cohen-Macaulay property of S(V)^{G} to the implication (2) ⇒ (4).

18 AMIRAM BRAUN

There exists a natural injection ψ : S(V)∗G → End_{S(V)^{G}}, defined by ψ(sg)(t) = sg(t) for each s, t ∈ S(V), g ∈ G. Moreover ψ(S(V) ∗ G) and End_{S(V)^{G}}S(V) have the same quotient rings and End_{S(V)^{G}} is ”the” maximal order which contains ψ(S(V) ∗ G).

However, there is no general result (to the best of our knowledge) which grants the Csymmetry of End_{S(V)^{G}}S(V) out of the one of S(V) ∗ G (or of ψ(S(V) ∗ G)). This is the main reason why counterexamples to the Kemper et al. conjecture do exist. The noncommutative reason why the conjecture holds in the absence of transvections is that in this case ψ(S(V) ∗ G) = End_{S(V)^{G}}S(V).

One may wonder whether the equality ψ(S(V) ∗ G) = End_{S(V)^{G}}S(V) holds in a more general situation. This will imply the validity of the Kemper et al. conjecture in more general circumstances. It follows from the next result that this is not the case. Similarly the ”height one projective” property of S(V) yields the same conclusion.

Proposition 4.1. The following conditions are equivalent:

(1) G has no transvections,

(2) tr(S(V)) ⫅̸ v, for each height one prime ideal v in S(V)^{G} where tr(S(V)) ≡ {∑_{g∈G} g(t)|t ∈ S(V)},

(3) S(V)_{v} is a projective S(V)_{v} ∗G-module, for each height one prime ideal v in S(V)^{G},

(4) S(V)_{v} ∗ G is hereditary, for each height one prime ideal v in S(V)^{G}.

Proof: The equivalence of (2) and (4) is due to Rosen (e.g. [28, Theorem 40.13], where it is also attributed to Auslander-Rim).

The implication (1) ⇒ (2) appears in [3, Proposition 3.7], while its converse (2) ⇒ (1) follows from [26, Corollary 6.4.3] since (2) then implies that the inertial subgroup of any height one prime ν of S(V)^{G} contains no p-order elements. This implies that G has no transvections.

Also the equivalence of (2) and (3) is a consequence of [1]. Explicitly, recall that S(V) is an ”height one projective” S(V)∗G-module if and only if S(V)_{v} is a projective S(V)_{v}∗G module for each height one v in S(V)^{G}. Now by [1] this holds if and only if ∑ g(xg^{−1}) = 1_{S(V)_{v}}, for some x ∈ S(V)_{v}. Equivalently ∑ g(xg^{−1}(t)) = t, for each t ∈ S(V)_{v} and therefore if and only if ∑ g(x) = 1 holds, as needed. □

Corollary 4.2. Suppose G ⊂ SL(V) and ψ(S(V) ∗ G) = End_{S(V)^{G}}S(V). Then G has no transvections and consequently G has no pseudo-reflection of any kind.

THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE 19

Proof: Let v be any height one prime in S(V)^{G}. Then End_{S(V)_{v}^{G}}S(V)_{v} ≅ S(V)_{v} ∗G is an Azumaya algebra and therefore a maximal order and in particular hereditary. Consequently by Proposition 4.1 G has no transvection. □

We end this chapter with a non-commutative characterization of the Gorenstein property of S(V)^{G}. Recall that G_{p} is the subgroup of G generated by all p-sylov subgroups of G (where p = charF) and W (G) is the subgroup of G generated by all pseudo-reflections.

Proposition 4.3. Let C ⊂ S(V)^{G} be an homogenous polynomial subring with S(V)^{G}/C a finite extension. Suppose S(V)^{G} is Cohen-Macaulay. Let N = (G, G)G_{p}W (G). Then the following are equivalent:

(1) S(V)^{G} is Gorenstein,

(2) S(V)^{N} ∗ G/N is C-symmetric.

Proof: Since S(V)^{W (G)}/S(V)^{G} is unramified in codimension one (e.g.[29, I. No. 7 Proposition 21]), the same holds for S(V)^{N}/S(V)^{G}. Therefore by [3, Proposition 2.20] we have S(V)^{N} ∗ G/N = End_{S(V)^{G}}S(V)^{N}. Now suppose S(V)^{G} is Gorenstein, equivalently Hom_{C}(S(V)^{G}, C) ≅ S(V)^{G}. Let q be an arbitrary height one prime ideal of C. Then S(V)_{q}^{N} is a projective S(V)_{q}^{G}-module implying that S(V)^{N} is a ”height one projective” S(V)^{G}-module. Therefore by [3, Lemma 2.19] the above isomorphism implies (2).

The implication (2)⇒ (1) follows similarly since M ≡ S(V)^{N} is an ”height one projective” A ≡ End_{S(V)^{G}}S(V)^{N}-module. Consequently by [3, Lemma 2.19] End_{A}(M) = S(V)^{G} is Csymmetric. □

5. Finite Subgroups of SL(2, F)

Our main objective here is to find out the extent of validity of the Kemper et al. conjecture, for finite subgroups of SL(2, F), where F is an arbitrary field of positive charctacteristic p. Qualitatively speaking, most of the groups do satisfy the conjecture. Our argument also shows why and how counterexamples appear. For SL(n, F) with n > 2 such an analysis does not exist (as yet), due partially to the difficulty in analyzing irreducible primitive actions of T (G), the G-subgroup generated by all transvections. The following theorem is our main result.

Theorem 5.1. Let F be an arbitrary field with charF = p > 0. Let G ⊆ SL(2, F) be a finite subgroup and V = F^{2}, the natural SL(2, F)-module. Then

20 AMIRAM BRAUN

(i) S(V)^{G} is Gorenstein whenever V is an irreducible T (G)-module,

(ii) If V is a reducible T (G)-module and T (G) ≠ 1_{G} then S(V)^{G} is Gorenstein if and only if |G| divides |T (G)|(|T (G)| − 1),

(iii) There exists a unique maximal subgroup D of G, D ⊇ T (G) with |G|/|D| ≤ 2 and S(V)^{D} is Gorenstien,

(iv) In particular S(V))^{G} is Gorenstein in the following (not mutually disjoint) cases:

(1) T (G) = {1_{G}},

(2) G is an odd order group,

(3) F = F_{q}, where q = p^{s} and s is odd,

(4) p = 2.

(v) If p divides |G| then S(V)^{G} is an hypersurface in F^{3} whenever it is Gorenstein.

Our proof will proceed in steps. We shall first establish some reductions which will be solely used in the irreducible primitive case. The cases where T (G) acts irreducibly on V will be handled separately. The reducible case will be handled in the main course of the proof.

Our first reduction result can be obtained in several ways.

Lemma 5.2. Let V_{0} be a k-finite dimensional G-module and F ⊃ k, a field extension. Set V ≡ V_{0} ⊗_{k} F, and G acts trivially on F. Then the Gorenstein property of S(V_{0})^{G} ascends to S(V)^{G}.

Proof: Let w be the canonical S(V_{0})^{G}-module. By [7, Exercise 3.3.31] w⊗_{k} F is a canonical S(V_{0})^{G} ⊗_{k} F -module. Now the Gorenstein property of S(V_{0})^{G} amounts to S(V_{0})^{G} ≅ w as S(V_{0})^{G}-modules and consequently S(V_{0})^{G} ⊗_{k} F is a canonical S(V_{0})^{G} ⊗_{k} F -module. Now clearly S(V) = S(V_{0})⊗_{k} F and consequently S(V)^{G} = S(V_{0})^{G}⊗_{k} F. Hence S(V)^{G} is its own canonical module. Since S(V)^{G} is Cohen-Macaulay by [7, Theorem 2.1.10], the Gorenstien property of S(V)^{G} now follows. □

Remark 5.3.

Such an ascend of the unique factorization property from S(V_{0})^{G} to S(V)^{G} is not true in general.

Our second reduction result is well known. Failing to find a reference for it, we provide a proof.

THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE 21

Lemma 5.4. Let G ⊂ GL(V) be a finite group, where V is an irreducible F -finite dimensional G-module. Then there exists a subfield k ⊂ F, with [k : 𝔽_{p}] < ∞ and a faithfully irreducible G-module V_{0} satisfying V = V_{0} ⊗_{k} F.

Proof: Denote by F G the group algebra of G over F and set

Ann_{F G}V = {x ∈ F G|x.V = 0} = M.

Then M is a two-sided maximal ideal in F G and F G/M ≅ End_{F} (V) = M_{n}(F). So we can identify V with a column in M_{n}(F). Clearly 𝔽_{p}G ⊗_{𝔽_{p}} F = F G. Let N ≡ M ∩ 𝔽_{p}G = Ann_{𝔽_{p}G}V. Then the exactness of (−)⊗_{𝔽_{p}}F shows that NF G = N ⊗_{𝔽_{p}}F ⊆ M and therefore F G/M = (𝔽_{p}G/N) · F. Therefore F G/M is a central extension of 𝔽_{p}G/N, implying that 𝔽_{p}G/N is a simple ring and its center C ≡ Z(𝔽_{p}G/N) ⊆ Z(F G/M). This shows (by [10, Cor. 3.6, p.54]) that F G/M = 𝔽_{p}G/N ⊗_{C} S, where S is the centralizer of 𝔽_{p}G/N in F G/M. It follows from the previous equality that F = S, that is 𝔽_{p}G/N ⊗_{𝔽_{p}} F = F G/M = M_{n}(F). Let k be a maximal subfield of the division ring D satisfying 𝔽_{p}G/N = M_{r}(D). Then [𝔽_{p}G/N : 𝔽_{p}] < ∞ grants that [k : 𝔽_{p}] < ∞, moreover [𝔽_{p}G/N ⊗_{𝔽_{p}} k = M_{n}(k). Then M_{n}(k) ⊗_{k} F = M_{n}(F) shows that V_{0} ⊗_{k} F = V, for some V_{0} an irreducible M_{n}(k)-module. Clearly V_{0} is also an irreducible G-module with G ⊂ GL(V_{0}). □

We now present a special case one needs to consider.

Proposition 5.5. Suppose G ⊂ SL(2, F) is a finite group with T (G) = SL(2, 𝔽_{q}), where q = p^{s} and 𝔽_{q} ⊆ F. Let V = F^{2} be the natural SL(2, F)-module. Then S(V)^{G} is Gorenstein.

Proof: Fix a basis {x, y} of V. Then by [26, Ex.1, p.104] the two Dickson invariants of S(V)^{GL(2,𝔽_{q})} are c_{2,0} = (xy^{q} − yx^{q})^{q−1} and c_{2,1} = \frac{xy^{q^{2}}−yx^{q^{2}}}{xy^{q}−yx^{q}}. Moreover by [2, Theorem

8.2.1] \{u = xy^{q} − yx^{q}, c_{21}\} are the two generators of S(V)^{SL(2,𝔽_{q})}. Let g = [\begin{matrix}a, & b \\ c, & d \end{matrix}] ∈ G be

arbitrary. Then gT (G)g^{−1} = T (G) implies that:

g [\begin{matrix}1, & 1 \\ 0, & 1 \end{matrix}] g^{−1} = [\begin{matrix}1 − ac, & a^{2} \\ −c^{2}, & 1 + ac \end{matrix}] ∈ SL(2, 𝔽_{q}),

as well as

g [\begin{matrix}1, & 0 \\ 1, & 1 \end{matrix}] g^{−1} = [\begin{matrix}1 + bd, & −b^{2} \\ d^{2}, & 1 − bd \end{matrix}] ∈ SL(2, 𝔽_{q}).

Consequently ac, a^{2}, c^{2}, b^{2}, d^{2}, bd ∈ 𝔽_{q}. Also 1 = detg = ad − bc. So assuming without loss of generality that a ≠ 0, we get that b = aβ, c = aγ, d = aδ, with β, γ, δ ∈ 𝔽_{q} and

g = [\begin{matrix}a, & 0 \\ 0, & a \end{matrix}] [\begin{matrix}1, & β \\ γ, & δ \end{matrix}], with h ≡ [\begin{matrix}1, & β \\ γ, & δ \end{matrix}] ∈ GL(2, 𝔽_{q}).

Consequently g(c_{21}) = (aI_{V})(h(c_{21})) = (aI_{V})(c_{21}) = a^{q^{2}−q}c_{21}. Also if g_{1} = [\begin{matrix}a_{1}, & b_{1} \\ c_{1}, & d_{1} \end{matrix}] ∈

GL(2, 𝔽_{q}), then a_{1}^{q} = a_{1}, b_{1}^{q} = b_{1}, c_{1}^{q} = c_{1}, d_{1}^{q} = d_{1} imply that: g_{1}(u) = g_{1}(x)g_{1}(y)^{q} − g_{1}(y)g_{1}(x)^{q} = (a_{1}x + b_{1}y)(c_{1}x + d_{1}y)^{q} − (c_{1}x + d_{1}y)(a_{1}x + b_{1}y)^{q} = (a_{1}d_{1} − b_{1}c_{1})(xy^{q} − yx^{q}) = det(g_{1})u. Consequently, g(u) = (aI_{V})(h(u)) = det(h)a^{degu}u = a^{q+1}a^{−2}u = a^{q−1}u, where the second to last equality follows from 1 = det(g) = a^{2}det(h).

Suppose by negation that D ≠ G where D is as in Theorem 3.10. Let g ∈ G where (|g|, p) = 1 and ḡ is a generator of G/D(G). Now

S(V)^{<g,T (G)>} = (S(V)^{T (G)})^{<g>} = (S(V)^{SL(2,𝔽_{q})})^{<g>} = (F [u, c_{21}])^{<g>}.

Observe that F [u, c_{21}] = S(U), where U = F u + F c_{21}, is a 2-dimensional F -vectorspace. Moreover g acts linearly on U and extending this action to S(U) induces the original action of g on F [u, c_{21}]. Also the matrix of g_{|U} with respect to the basis {u, c_{21}} is given by

the matrix [\begin{matrix}a^{q−1}, & 0 \\ 0, & a^{q^{2}−q} \end{matrix}], implying that detg_{|U} = a^{q^{2}−1}. Now a^{2} = det(h)^{−1} ∈ 𝔽_{q} and

therefore [𝔽_{q}(a) : 𝔽_{q}] = 1 or 2. So |𝔽_{q}(a)| = q or q^{2}. In both cases we have a^{q^{2}−1} = 1. Consequently since (|g_{|U}|, p)) = 1, we get by Watanabe’s theorem that S(V)^{<g,T (G)>} = F [u, c_{21}]^{<g>} is Gorenstein implying that g ∈ D, an obvious contradiction. □

Remark 5.6.

(i) A similar result with G ⊆ SL(n, F)), T (G) = SL(n, 𝔽_{q}), and n > 2 does not hold in general. This is another source for counterexamples as seen in Theorem 2.6.

(ii) In fact S(V)^{G} is an hypersurface in Proposition 5.5. Indeed the argument showed

that each g ∈ G is represented by the matrix g_{|U} = [\begin{matrix}a^{q−1}, & 0 \\ 0, & a^{q^{2}−q} \end{matrix}] on U and a^{q^{2}−1} =

a^{q−1}a^{q^{2}−q} = 1. Since a^{q−1} = b^{q−1} = c^{q−1} = d^{q−1}, this presentation is independent of the particular choice of a. This shows that < g_{|U}|g ∈ G > acts as a subgroup of the unique cyclic subgroup of order \frac{q^{2}−1}{q−1} = q + 1 of F^{×}. Let g′ ∈ G where g′_{|U} is a generator of this subgroup.

So g′_{|U} = [\begin{matrix}ξ, & 0 \\ 0, & ξ^{−1} \end{matrix}] with ξ^{k} = 1. Then S(V)^{G} = F [u, c_{2,1}]^{<g>} = F [u^{k}, c_{2,1,}^{k}, uc_{2,1}], which is

clearly an hypersurface in F^{3}.

Our next result is also connected to the irreducible primitive case analysis. It actually shows that T (G) is its own normalizer in SL(2, F) in this case. This may already be known. Our proof establishes this by using invariant theory.

Proposition 5.7. Suppose G ⊂ SL(2, F) is a finite group with T (G) ≅ SL(2, 5), T (G) ⊆ SL(2, 𝔽_{9}), and 𝔽_{9} ⊆ F (here p = 3). Let V = F^{2} be the natural SL(2, F)-module. Then G = T (G) and consequently S(V)^{G} is a polynomial ring.

Proof: The group T (G) can be generated by the elements [\begin{matrix}1, & 0 \\ ξ^{2}, & 1 \end{matrix}], [\begin{matrix}1, & 1 \\ 0, & 1 \end{matrix}] in SL(2, 𝔽_{9})

where ξ is the generator of the multiplicative group 𝔽_{9} ∖ {0}. By [19, p. 79]

S(V)^{T (G)} = F [f_{10}, f_{12}], where f_{10} = x_{1}^{9}x_{2} − x_{1}x_{2}^{9}, f_{12} = x_{1}^{12} + x_{1}^{10}x_{2}^{2} − x_{1}^{6}x_{2}^{6} + x_{1}^{2}x_{2}^{10} − x_{1}^{12}

(this was obtained by using INVAR).

As in Proposition 5.5, take g = [\begin{matrix}a, & b \\ c, & d \end{matrix}] ∈ G. Then g [\begin{matrix}1, & 1 \\ 0, & 1 \end{matrix}] g^{−1} ∈ T (G) ⊆ SL(2, 𝔽_{9})

and g [\begin{matrix}1, & 1 \\ ξ^{2}, & 1 \end{matrix}] g^{−1} ∈ T (G) will give a^{2}, c^{2}, ac, b^{2}, d^{2}, bd ∈ 𝔽_{9}. This together with ad−bc = 1

leads to g = [\begin{matrix}a, & 0 \\ 0, & a \end{matrix}] [\begin{matrix}1, & β \\ γ, & δ \end{matrix}], where h = [\begin{matrix}1, & β \\ γ, & δ \end{matrix}] ∈ GL(2, 𝔽_{9}). Also by 1 = det(g) =

a^{2}det(h), we have a^{−2} = det(h) ∈ 𝔽_{9}.

We get that g(f_{10}) = a^{10}h(t_{10}) = a^{10}a^{−2}f_{10} = a^{8}f_{10}. Now F [f_{10}, f_{12}] = S(V)^{G} is g-stable, implying by degree reasoning that g(f_{12}) = εf_{12} for some ε ∈ F.

Since g(f_{12}) = a^{12}h(f_{12}), we shall next consider h(f_{12}). We have

h(f_{12}) = (x_{1} + βx_{2})^{12} + (x_{1} + βx_{2})^{10}(γx_{1} + δx_{2})^{2} − (x_{1} + βx_{2})^{6}(γx_{1} + δx_{2})^{6}+
+(x_{1} + βx_{2})^{2}(γx_{1} + δx_{2})^{10} − (γx_{1} + δx_{2})^{12} =
= (x_{1}^{9} + β^{9}x_{2}^{9})(x_{1}^{3} + β^{3}x_{2}^{3}) + (x_{1}^{9} + β^{9}x_{2}^{9}))(x_{1} + βx_{2})(γx_{1} + δx_{2})^{2}−
−(x_{1} + βx_{2})^{6}(γx_{1} + δx_{2})^{6} + (x_{1} + βx_{2})^{2}(γx_{1} + δx_{2})(γ^{9}x_{1}^{9} + δ^{9}x_{2}^{9})−
−(γ^{9}x_{1}^{9} + δ^{9}x_{2}^{9})(γ^{3}x_{1}^{3} + δ^{3}x_{2}^{3}).

24 AMIRAM BRAUN

Therefore the only contribution to the monomial x_{1}^{6}x_{2}^{6} will come from the term

−(x_{1} + βx_{2})^{6}(γx_{1} + δx_{2})^{6} = −(x_{1}^{3} + β^{3}x_{2}^{3})^{2}(γ^{3}x_{1}^{3} + δ^{3}x_{2}^{3})^{2}.

So the contribution is exactly

−x_{1}^{6}x_{2}^{6}(δ^{3} − β^{3}γ^{3})^{2} = x_{1}^{6}x_{2}^{6}(δ − βγ)^{6} = −x_{1}^{6}x_{2}^{6}(det(h))^{6} = −(a^{−2})^{6}x_{1}^{6}x_{2}^{6} = −a^{−12}x_{1}^{6}x_{2}^{6}.

Consequently −ε = −a^{12}a^{−12} = −1. This shows that ε = 1 and g(f_{12}) = f_{12}.

Also the coefficient of x_{1}^{12} in the right hand side of h(f_{12}) (using γ^{8} = 1) is

(1 + γ^{2} − γ^{6} + γ^{10} − γ^{12}) = 1 + γ^{2} − γ^{6} + γ^{2} − γ^{4} = 1 + 2γ^{2} − γ^{6} − γ^{4} =

= 1 − γ^{2}(1 + γ^{2} + γ^{4}) = 1 − \frac{γ^{2}((γ^{2})^{3} − 1)}{γ^{2} − 1} = 1 − \frac{γ^{8} − γ^{2}}{γ^{2} − 1} = 2 = −1

and therefore the coefficient of x_{1}^{12} in g(f_{12}) = f_{12} is a^{12}(−1). Hence a^{12} = −1. Since a^{2} ∈ 𝔽_{9} we have a^{16} = 1 implying that a^{4} = −1. Consequently g(f_{10}) = a^{8}f_{10} = f_{10}. To conclude g acts as an identity on F [f_{10}, f_{12}] = S(V)^{T (G)}. Therefore by Galois theory < g, T (G) >= T (G) and g ∈ T (G). □

Recall (e.g. [9, p.261]) that V is an imprimitive G-module if V = Ind_{H}^{G}(W), where W is a H-module for some proper subgroup H of G. V is a primitive G-module if V is not imprimitive.

The following is our first positive result.

Theorem 5.8. Let G ⊆ SL(2, F) be a finite group and V = F^{2} the natural SL(2, F)- module. Suppose that V is an irreducible primitive T (G)-module. Then S(V)^{G} is a hypersurface in F^{3} (or a polynomial ring).

Proof: Clearly V is also an irreducible G-module. Therefore by Lemma 5.4, there exists a G-module V_{0} and a subfield k ⊂ F with [k : 𝔽_{p}] < ∞ such that G ⊂ SL(2, k) and V = V_{0} ⊗_{k} F.

We next show that V_{0} is also an irreducible T (G)-module. Indeed by [8, 49.2], V_{0} is a completely reducible T (G)-module. If V_{0} = W_{1} ⊕ W_{2}, where W_{i} is a proper T (G)- module for i = 1, 2, then V = V_{0} ⊗_{k} F = (W_{1} ⊗_{k} F) ⊕ (W_{2} ⊗_{k} F), contradicting the irreducibility of V as a T (G)-module. Similarly V_{0} is a primitive T (G)-module. Indeed if V_{0} = Ind_{H}^{G}(W) ≡ W ⊗_{kH} kG, for some proper subgroup H of T (G), then

V = V_{0} ⊗_{k} F = (W ⊗_{kH} kG) ⊗_{k} F = (W ⊗_{k} F) ⊗_{F H} F G = Ind_{H}^{G}(W ⊗_{k} F),

THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE 25

in contradiction to the primitivity of V as T (G)-module. Now by applying [33] or [16], there exists an element a ∈ GL(V_{0}, k) = GL(2, k) such that aT (G)a^{−1} ∈ {SL(2, 𝔽_{q}), S_{p}(2, 𝔽_{q}), SU(2, 𝔽_{q})} or aT (G)a^{−1} is isomorphic to SL(2, 5) regarded as a subgroup of SL(2, 𝔽_{9}) ⊆ SL(2, k).

Since SL(2, 𝔽_{q}) = S_{p}(2, 𝔽_{q}) = SU(2, 𝔽_{q}) (e.g. [15, Proposition 3.1, Corollary 11.29]) we need to consider first aT (G)a^{−1} = SL(2, q). Let G_{1} ≡ aGa^{−1}. Then T (G_{1}) = aT (G)a^{−1} = SL(2, q) and G_{1} ⊆ SL(2, k). Therefore by Proposition 5.5 S(V_{0})^{G_{1}} is Gorenstein. Hence S(V_{0})^{G} is Gorenstein and by Lemma 5.4 S(V)^{G} is Gorenstein, and therefore it is a hypersurface in F^{3} by 5.6(ii). If aT (G)a^{−1} ≅ SL(2, 5) regarded as a subgroup of SL(2, 𝔽_{9}) ⊆ SL(2, k). Then in G_{1} ≡ aGa^{−1} we have T (G_{1}) = aT (G)a^{−1} and by Proposition 5.7 G_{1} = T (G_{1}). Hence by [19, Theorem 8.1] S(V_{0})^{G_{1}} is a polynomial ring implying that S(V_{0})^{G} is a polynomial ring. This shows by [19, p.62] that S(V)^{G} is a polynomial ring and in particular Gorenstein. □

Our next objective is to handle finite groups G ⊂ SL(2, F) with T (G) acting irreducibly and imprimitively on the natural SL(2, F)-module F^{2} = V. Since it is well known (e.g. [33]) that T (G) is monomial in this case, we restrict our attention to this subfamily. Recall that H ⊂ GL(V) is called monomial if V has a basis with respect to which the matrix of each element of H has exactly one non-zero entry in each row and column.

The next easy result disposes off the charF = p ≠ 2 case.

Lemma 5.9. Let H ⊂ SL(2, F) be a monomial group generated by transvections and p ≠ 2. Then H = {1}.

Proof: Let {y_{1}, y_{2}} be a basis of V = F^{2} with respect to which H is monomial. Let

g ∈ H be a transvection. Then g = [\begin{matrix}λ, & 0 \\ 0, & δ \end{matrix}] or [\begin{matrix}0, & λ \\ δ, & 0 \end{matrix}]. The first case can’t hold since

g − I_{V} = [\begin{matrix}λ − 1, & 0 \\ 0, & δ − 1 \end{matrix}] and so either λ = 1 or δ = 1 but since g ∈ SL(2, F), δ = λ^{−1} and

we get, in both cases, that g = 1. If g = [\begin{matrix}0, & λ \\ δ, & 0 \end{matrix}] then δ = −λ^{−1} and g−I_{V} = [\begin{matrix}−1, & λ \\ −λ^{−1}, & −1 \end{matrix}],

a matrix whose determinant is 2 and is non-zero since p ≠ 2. So rank(g − I_{V}) > 1, which is a contradiction. □

This limits the possibilities of T (G) to the p = 2 case.

Proposition 5.10. Let G ⊂ SL(2, F) be a finite group where charF = 2. Suppose T (G) is an irreducible monomial subgroup. Then G = T (G).

26 AMIRAM BRAUN

Proof: Let {x_{1}, x_{2}} be the basis of V = F^{2} with respect to which T (G) is irreducible and monomial. Let {g_{1}, ..., g_{n}} be the set of transvections generating T (G). The irreducibility

forces n > 1. So either g_{i} = [\begin{matrix}∗, & 0 \\ 0, & ∗ \end{matrix}], or [\begin{matrix}0, & ∗ \\ ∗, & 0 \end{matrix}]. The first possibility leads to g_{i} = 1. So

g_{i} = [\begin{matrix}0, & α_{i} \\ α_{i}^{−1}, & 0 \end{matrix}], for i = 1, ..., n. Let g = [\begin{matrix}a, & b \\ c, & d \end{matrix}] ∈ G. Since g normalizes T (G), we have

for each i = 1, ..., n:

[\begin{matrix}a, & b \\ c, & d \end{matrix}] [\begin{matrix}0, & α_{i} \\ α_{i}^{−1}, & 0 \end{matrix}] [\begin{matrix}d, & −b \\ −c, & a \end{matrix}] = [\begin{matrix}bdα_{i}^{−1} − acα_{i}, & −b^{2}α_{i}^{−1} + a^{2}α_{i} \\ d^{2}α_{i}^{−1} − c^{2}α_{i}, & −dbα_{i}^{−1} + caα_{i} \end{matrix}] ∈ T (G).

So this matrix must have the form [\begin{matrix}∗, & 0 \\ 0, & ∗ \end{matrix}] or [\begin{matrix}0, & ∗ \\ ∗ & 0 \end{matrix}].

In the first case b^{2}α_{i}^{−1} = a^{2}α_{i}, d^{2}α_{i}^{−1} = c^{2}α_{i}. So using p = 2 we get b = aα_{i}, d = cα_{i}, in contradiction to ad − bc = 1.

In the second case bd = acα_{i}^{2}, for i = 1, ..., n. Since α_{1} ≠ α_{2} (and therefore α_{1}^{2} ≠ α_{2}^{2})

we get that bd = ac = 0. If a = 0, then ad − bc = 1 implies d = 0 and g = [\begin{matrix}0, & b \\ c, & 0 \end{matrix}]. Also

g ∈ SL(2, F) so c = b^{−1}. Consequently g(x_{1} + bx_{2}) = bx_{2} + bcx_{1} = x_{1} + bx_{2}, as well as

g(x_{1}) = bx_{2} = x_{1} + (x_{1} + bx_{2}), so the matrix of g with respect to \{x_{1}, x_{1} + bx_{2}\} is [\begin{matrix}1, & 1 \\ 0, & 1 \end{matrix}].

That is g is a transvection and hence g ∈ T (G).

Suppose therefore that b = 0. Then ad − bc = 1 forces c = 0 and g = [\begin{matrix}a, & 0 \\ 0, & a^{−1} \end{matrix}]. Also

g [\begin{matrix}0, & α_{i} \\ α_{i}^{−1}, & 0 \end{matrix}] = [\begin{matrix}0, & aα_{i} \\ a^{−1}α_{i}^{−1}, & 0 \end{matrix}] ∈ G, for each i. However as in the previous case one

observes that [\begin{matrix}0, & aα_{i} \\ a^{−1}α_{i}^{−1}, & 0 \end{matrix}] is a transvection and therefore it is T (G). Consequently

g = [\begin{matrix}0, & aα_{i} \\ a^{−1}α_{i}^{−1}, & 0 \end{matrix}] [\begin{matrix}0, & α_{i} \\ α_{i}^{−1}, & 0 \end{matrix}] ∈ T (G), as needed. □

We can now state the second possibility for the irreducible T (G)-action.

Theorem 5.11. Let G ⊂ SL(2, F) be a finite group with T (G) acting imprimitively and irreducibly on the natural module V = F^{2}. Then S(V)^{G} is a polynomial ring.

Proof: We have by (e.g. [33]) that T (G) is monomial. Alternatively, by assumption and [8, Cor. (50.4)] T (G) acts transitively on V. That is V = M_{1} ⊕ ⋯ ⊕ M_{k}, T (G) acts

THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE 27

by permutations on {M_{1}, ..., M_{k}} and Orbit(M_{1}) = {M_{1}, ...M_{k}}. Clearly if gM_{i} = M_{i}, for i = 1, ..., k, for each transvection g in T (G), then T (G)M_{i} = M_{i}, for i = 1, ..., k, in violation of the transitivity. So let g be a transvection in T (G) with GM_{i} = M_{j} with i ≠ j. Say i = 1. Suppose dim_{F} M_{1} > 1 and {m_{1}, ..., m_{r}} is a basis of M_{1} with r > 1. Then g(m_{1}) − m_{1}, g(m_{2}) − m_{2} ∈ (g − I_{V})(V), they are non-zero and are linearly independent as can easily be seen. This violates the pseudo-reflection property of g. Consequently by Lemma 5.9 p = 2 and by Proposition 5.10 we get T (G) = G. Therefore by [24, Theorem 2.4] S(V)^{G} is a polynomial ring. □

Our next two results will be used in the treatment of the reducible T (G)-module case, in the proof of Theorem 5.1.

Lemma 5.12. Let charF = p and ξ ∈ F a primitive m-th root of unity, where (m, p) = 1. Let W ⊂ F be a n-dimensional 𝔽_{p}-subspace of F with ξW ⊆ W. Then m|p^{n} − 1.

Proof: Let C_{m} be the cyclic group of order m and g ∈ C_{m} a generator. The map ϕ : G → End_{𝔽_{p}}W, where ϕ(g^{j}) ≡ ξ^{j}I_{W}, for each j, makes W into a C_{m}-module. (m, p) = 1 grants that W is a completely reducible 𝔽_{p}C_{m}-module, where the latter denotes the group algebra of C_{m} over 𝔽_{p}. Thus W = W_{1} ⊕ ⋯ ⊕ W_{r}, where W_{i} is an irreducible 𝔽_{p}C_{m}-module, for i = 1, ..., r. Set d_{i} ≡ dim_{𝔽_{p}}W_{i}. The result will follow once we show that m|p^{d_{i}} − 1 for each i.

Let ϕ_{i}(g) ≡ ϕ(g)_{|W_{i}} that is ϕ_{i} : G → GL(W_{i}), for each i. If ϕ_{i} is not injective this will give an integer k ≠ m, k|m with ξ^{k}w = w, for each w ∈ W_{i}. This will contradict the m-th primitivity of ξ. Therefore ϕ_{i} : C_{m} → GL(W_{i}) makes W_{i} into a faithful irreducible C_{m} module, for i = 1, ..., r. Let M_{i} ≡ Ann_{𝔽_{p}C_{m}}W_{i} = {x ∈ 𝔽_{p}C_{m}|xW_{i} = 0}. Then M_{i} is a maximal ideal in 𝔽_{p}C_{m}, and 𝔽_{p}C_{m}/M_{i} is a finite field extension of 𝔽_{p}. Also by commutativity of 𝔽_{p}C_{m} and Schur’s Lemma we have 𝔽_{p}C_{m}/M_{i} ≅ W_{i} as 𝔽_{p}C_{m}-module and End_{𝔽_{p}C_{m}}(W_{i}) = 𝔽_{p}C_{m}/M_{i} for each i. Since ϕ_{i} is an injection, it follows that C_{m} is isomorphic to a subgroup of the multiplicative group of 𝔽_{p}C_{m}/M_{i}. Consequently since |𝔽_{p}C_{m}/M_{i}| = |W_{i}| = p^{d_{i}}, we get m|p^{d_{i}} − 1, for each i = 1, ..., r. □

Lemma 5.13. Let V = F^{2} = F x_{1} + F x_{2} be the natural SL(2, F) module and σ_{i} =

[\begin{matrix}1, & α_{i} \\ 0, & 1 \end{matrix}] ∈ SL(2, F), a transvection, for i = 1, ..., n. Assume that \{α_{1}, ..., α_{n}\} are lin-

early independent over 𝔽_{p}. Then S(V)^{<σ_{1},...,σ_{n}>} = F [a_{n}, x_{2}], where a_{n} is an homogenous polynomial (in x_{1}, x_{2}) of degree p^{n}, having x_{1}^{p^{n}} as one of its monomials.

28 AMIRAM BRAUN

Proof: The proof is carried out by induction. The case n = 1 appears already in Lemma 3.1. Indeed σ_{1}(x_{1}) = x_{1} +α_{1}x_{2}, σ_{1}(x_{2}) = x_{2} imply that a_{1} = x_{1}^{p} −x_{1}(α_{1}x_{2})^{p−1} ∈ S(V)^{<σ_{1}>}. Now | < σ_{1} > | = p together with a Galois field reasoning imply F [a_{1}, x_{2}] = S(V)^{<σ_{1}>}.

Suppose by induction that S(V)^{<σ_{1},...,σ_{n−1}>} = F [a_{n−1}, x_{2}], where deg(a_{n−1}) = p^{n−1}, and with x_{1}^{p^{n−1}} as one of its monomials. σ_{n}σ_{i} = σ_{i}σ_{n} for i = 2, ..., n imply that F [a_{n−1}, x_{2}] is σ_{n}-stable. Therefore, since σ_{n} preserve degrees, σ_{n}(a_{n−1}) = αa_{n−1} +βx_{2}^{p^{n−1}}, α, β ∈ F. Now σ_{n}(x_{1}) = x_{1} + α_{n}x_{2}, σ_{n}x_{2} = x_{2} imply by plugging into σ_{n}(a_{n−1}) that x_{1}^{p^{n−1}} is one of the monomials appearing in σ_{n}(a_{n−1}) and therefore α = 1.

We next observe that β ≠ 0. To see this observe the fact that α_{n} ∉ Span_{𝔽_{p}}{α_{1}, ..., α_{n−1}} implies σ_{n} ∉< σ_{1}, ..., σ_{n−1} >. Therefore by Galois considerations F [a_{n−1}, x_{2}] = S(V)^{<σ_{1},...,σ_{n−1}>} ⫌ S(V)^{<σ_{1},...,σ_{n}>}. Thus σ_{n}(a_{n−1}) ≠ a_{n−1}, as claimed.

Finally set a_{n} ≡ a_{n−1}^{p} − a_{n−1}(βx_{2}^{p^{n−1}})^{p−1}. As in the first paragraph σ_{n}(a_{n}) = a_{n}, so a_{n} ∈ S(V)^{<σ_{1},...,σ_{n}>}, F [a_{n}, x_{2}] is a polynomial ring and since | < σ_{n} > | = p, [F (a_{n−1}, x_{2}) : F (a_{n}, x_{2})] = p. Thus F (a_{n}, x_{2}) = Q(S(V)^{<σ_{1},...,σ_{n}>}) and since S(V)^{<σ_{1},...,σ_{n}>} is integral over F [a_{n}, x_{2}] we get F [a_{n}, x_{2}] = S(V)^{<σ_{1},...,σ_{n}>}. Clearly x_{1}^{p^{n}} appears as one of the monomials of a_{n}. □

We finally arrive at:

The proof of Theorem 5.1 The case T (G) = {1_{G}} follows from [3, Theorem B]. In fact (|G|, p) = 1 in this case, so we are reduced to the non-modular case. Therefore we may assume that T (G) ≠ {1_{G}}. Let V = F^{2} be the natural SL(2, F)-module. If V is an irreducible T (G)-module then Theorems 5.8 and 5.11 imply that S(V)^{G} is Gorenstein and even a hypersurface in F^{3}. We may therefore assume that V is a reducible T (G)-module. Consequently there exists a basis

\{x_{1}, x_{2}\} of V with respect to which T (G) ⊆ [\begin{matrix}F, & F \\ 0, & F \end{matrix} ]. Therefore every transvection of T (G)

must be in [\begin{matrix}1, & F \\ 0, & 1 \end{matrix} ], implying that T (G) ⊆ [\begin{matrix}1, & F \\ 0, & 1 \end{matrix} ]. Consequently T (G) is elementary

abelian. So

T (G) =< σ_{1} > × ⋯ × < σ_{n} > , σ_{i} = [\begin{matrix}1, & α_{i} \\ 0, & 1 \end{matrix}], i = 1, ..., n

and {α_{1}, ...α_{n}} are linearly independent over 𝔽_{p}. Set W = 𝔽_{p}α_{1}+⋯+𝔽_{p}α_{n}, so dim_{𝔽_{p}}W = n. Consequently |T (G)| = p^{n}.

Let g = [\begin{matrix}a, & b \\ c, & d \end{matrix}] ∈ G. Then gσ_{i}g^{−1} ∈ G ⊆ [\begin{matrix}1, & F \\ 0, & 1 \end{matrix} ] forces

[\begin{matrix}a, & b \\ c, & d \end{matrix}] [\begin{matrix}1, & α_{i} \\ 0, & 1 \end{matrix}] [\begin{matrix}d, & −b \\ −c, & a \end{matrix}] = [\begin{matrix}1 − acα_{i}, & a^{2}α_{i} \\ −c^{2}α_{i}, & 1 + acα_{i} \end{matrix}] ∈ [\begin{matrix}1, & F \\ 0, & 1 \end{matrix} ]

and therefore c = 0 and g = [\begin{matrix}λ, & ϵ \\ 0, & λ^{−1} \end{matrix}] with λ, ϵ ∈ F. Now since gσ_{i}g^{−1} = [\begin{matrix}1, & λ^{2}α_{i} \\ 0, & 1 \end{matrix}] ∈

T (G), for i = 1, ..., n, we get that λ^{2}α_{i} ∈ W for i = 1, ..., n. Consequently by Lemma 5.12 |λ^{2}| divides p^{n} − 1.

We next observe that (|G|/|T (G)|, p) = 1 and that G/T (G) is isomorphic to a (cyclic) subgroup of the cyclic group H = {α ∈ F |α^{2(p^{n}−1)} = 1}. Indeed if g^{p} ∈ T (G) then

g^{p} = [\begin{matrix}λ^{p}, & ϵ′ \\ 0, & λ^{−p} \end{matrix}] ∈ [\begin{matrix}1, & F \\ 0, & 1 \end{matrix} ], so λ = 1 and g ∈ [\begin{matrix}1, & ϵ \\ 0, & 1 \end{matrix}] ∈ T (G). This shows that

(|G/T (G)|, p) = 1. Say g = g_{1}h with h ∈ T (G), where g = [\begin{matrix}λ, & ϵ \\ 0, & λ^{−1} \end{matrix}], g_{1} = [\begin{matrix}λ_{1}, & ϵ_{1} \\ 0, & λ_{1}^{−1} \end{matrix}].

Then g_{1}^{−1}g = [\begin{matrix}λ_{1}^{−1}λ, & λ_{1}^{−1}ϵ − ϵ_{1}λ^{−1} \\ 0, & λ_{1}λ^{−1} \end{matrix}] = h ∈ T (G) ⊆ [\begin{matrix}1, & F \\ 0, & 1 \end{matrix} ] and λ = λ_{1}. Therefore

the map ϕ : G/T (G) → H, which is defined by ϕ(g(T (G)) = λ, is a well defined injective homomorphism with its image in H since |λ^{2}| divides p^{n} − 1.

Let ḡ ≡ gT (G) be a generator of G/T (G) with g = [\begin{matrix}λ, & ϵ \\ 0, & λ^{−1} \end{matrix}]. By Lemma 5.13 we have

S(V)^{T (G)} = S(V)^{<σ_{1},...,σ_{n}>} = F [a_{n}, x_{2}]. Now since gT (G)g^{−1} = T (G), F [a_{n}, x_{2}] is g-stable and since g preserve degree’s we get g(a_{n}) = αa_{n} + βx_{2}p^{n} with α, β ∈ F. Since a_{n} has x_{1}^{p^{n}} as one of it monomials and since g(x_{1}) = λx_{1} + ϵx_{2} we get that x_{1}^{p^{n}} appears with coefficient

λ^{p^{n}} in g(a_{n}), that is α = λ^{p^{n}}. If λ^{p^{n}} = 1 then λ = 1 and then g = [\begin{matrix}1, & ϵ \\ 0, & 1 \end{matrix}] ∈ T (G), which

was excluded.

Suppose firstly that λ ≠ −1, then λ^{p^{n}} ≠ (−1)^{p^{n}} = −1 (this also holds if p = 2 since λ ≠ 1). Consequently λ^{2p^{n}} ≠ 1. Let b = a_{n} + \frac{λ^{p^{n}}β}{λ^{2p^{n}}−1}x_{2}^{p^{n}}, then

g(b) = λ^{p^{n}}a_{n} + βx_{2}^{p^{n}} + \frac{λ^{p^{n}}β}{λ^{2p^{n}} − 1}λ^{−p^{n}}x_{2}^{p^{n}} = λ^{p^{n}}(a_{n} + \frac{λ^{p^{n}}β}{λ^{2p^{n}} − 1}x_{2}^{p^{n}}) = λ^{p^{n}}b.

Clearly L = F b + F x_{2} is a 2-dimensional vector space over F and S(L) = F [b, x_{2}] = F [a_{n}, x_{2}] = T. With respect to the basis {b, x_{2}} of L, g_{|L} is represented by the matrix [\begin{matrix}λ^{p^{n}}, & 0 \\ 0, & λ^{−1} \end{matrix}]. Let |λ| = k, then |g_{|L}| = |λ| = k and |λ^{2}| divides (p^{n} − 1) by Lemma 5.12,

implying that k|2(p^{n} − 1).

If p = 2 and k = 2m, then λ^{2m} = 1, implying that λ^{m} = 1, a contradiction. Hence (k, 2) = 1. So k|(2^{n} − 1), and therefore detg_{|L} = 1. Therefore by Watanabe’s theorem S(V)^{G} = S(V)^{<g_{|L}>} is always Gorenstein in case p = 2. Also \frac{|G|}{|T (G)|}|(|T (G)| − 1) holds in this case. This settles item (4).

If p ≠ 2 then (p, 2(p^{n} − 1)) = 1, hence λ^{ip^{n}} = 1 for i ≤ k forces i = k. This shows that < g_{|L} > has no diagonal pseudo-reflections, so since < g_{|L} >= k and (k, p) = 1 we get by Watanabe’s theorem that S(V)^{G} = S(L)^{<g_{|L}>} is Gorenstein if and only if k|(p^{n} − 1), that is \frac{|G|}{|T (G)|}|(|T (G)| − 1).

Let D = D(C) be the subgroup appearing in Theorem 3.10. Here C is an homogenous polynomial subring with S(V)^{G}/C being finite. Since |λ^{2}| divides (p^{n} − 1), the above reasoning shows that S(V)^{<g^{2},T (G)>} = S(L)^{<g_{|L}^{2}>} is Gorenstein and therefore g^{2} ∈ D(G). Consequently \frac{|G|}{|D(G)|} ≤ 2 (if λ ≠ −1) as claimed. Consequently if |G| is odd, the same holds for |G/T (G)|, and we get that D = G and S(V)^{G} is therefore Gorenstein (if λ ≠ −1) and (2) is verified.

Given the assumption of item (3) and suppose λ ∉ 𝔽_{p}(λ^{2}) then [𝔽_{p}(λ) : 𝔽_{p}(λ^{2})] = 2 in contradiction to [F : 𝔽_{p}] = s being odd. Thus λ ∈ 𝔽_{p}(λ^{2}) and since λ^{2}W ⊆ W we have λW ⊆ W, that is k = |λ| divides (p^{n} − 1), by Lemma 5.12. Now by the previous reasoning det(g_{|L}) = 1 and S(V)^{G} = S(L)^{<g_{|L}>} is Gorenstein and item (3) holds (if λ ≠ −1).

Suppose next that λ = −1. we shall show that S(V)^{G} is Gorenstein in this case, so the

restrictions imposed for proving the previous items will be removed. Now g = [\begin{matrix}−1, & ϵ \\ 0, & −1 \end{matrix}]

and p ≠ 2, imply that g^{2} ∈ T (G). Hence \frac{|G|}{|T (G)|} = 2. Observe that since |T (G)| = p^{n} is odd, then \frac{|G|}{|T (G)|}|(|T (G)| − 1) also holds in this case.

Let m = (a_{n}, x_{2}) be the unique graded maximal ideal in F [a_{n}, x_{2}] = T, then e ≡ m ∩ F [a_{n}, x_{2}]^{<g>} is the unique graded maximal ideal in F [a_{n}, x_{2}]^{<g>} = S(V)^{G}. Hence F [a_{n}, x_{2}]_{e} = F [a_{n}, x_{2}]_{m}. Now since λ = −1, then λ^{−1} = −1 and g(a_{n}) = −a_{n} + βx_{2}^{p^{n}},

g(x_{2}) = −x_{2}. Consequently g is represented by the matrix [\begin{matrix}−1, & 0 \\ 0, & −1 \end{matrix}] on the F -vector

space m/m^{2} = m_{e}/m_{e}^{2} with respect to the basis {a_{n} + m^{2}, x_{2} + m^{2}}. Consequently by [31, Theorem 2] F [a_{n}, x_{2}]_{e}^{<g>} is Gorenstein. Therefore by [7, Excercise 3.6.20(c)] F [a_{n}, x_{2}]^{<g>} =

THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE 31

S(V)^{<g,T (G)>} is Gorenstein. Therefore by Theorem 3.10, g ∈ D and hence G = D, that is S(V)^{G} is Gorenstein.

Our final task is to verify that whenever S(V)^{G} is Gorenstein then it is a hypersurface

in F^{3}. If λ ≠ −1 then S(V)^{G} = S(L)^{<g_{|L>}} and g_{|L} = [\begin{matrix}λ^{p^{n}}, & 0 \\ 0, & λ^{−1} \end{matrix}] with det_{(}g_{|L}) = 1.

Therefore λ^{p^{n}} = λ. Consequently since S(L) = F [b, x_{2}] we conclude that S(L)^{<g_{|L}>} = F [b^{k}, x_{2}^{k}, bx_{2}] which is clearly a hypersurface, and in fact a simple singularity (of type A_{k−1}).

The argument in case λ = −1 is somewhat more involved since the action of < g > is not linearizable (as opposed to case λ ≠ −1). For this purpose we consider the topology F ≡ {F_{n} = m^{n} ∩ S(V)^{G}, n = 1, 2, ...} on S(V)^{G} = S(U)^{<g_{|U} >} = T^{<g>}. It is easily checked that F and E ≡ {e^{n}} are equivalent topologies on S(V)^{G}, where e = m ∩ S(V)^{G} (e.g. [30, p.8]) and also

⊕_{n}F_{n}/F_{n+1} = grad_{F}S(V)^{G}) = grad_{E}(S(U))^{<g_{|U} >} = S(m/m^{2})

Consequently, since g_{|m/m^{2}} = [\begin{matrix}−1, & 0 \\ 0, & −1 \end{matrix}], grad_{F}(S(V)^{G}) is generated by the 3 homoge-

nous elements {ā_{n}^{2}, x̄_{2}^{2}, ā_{n}x̄_{2}}. Now if we choose t so that F_{t} ⊆ e^{2} then ⊕_{n≤t−1}F_{n}/F_{n+1} and consequently S(V)^{G}/e^{2} are generated by 3 elements (or less). Therefore the same holds for e/e^{2}, and dim_{F} e/e^{2} ≤ 3. Consequently edimS(V)^{G} ≤ 3 and it is standard (e.g.[17, Proposition 2.1]) that e can be then generated by 3 (or less) homogenous elements implying that S(V)^{G} is the coordinate ring of the affine hypersurface in F^{3}. □

Remark 5.14.

One can use the proof of the reducible case of Theorem 5.1 to give an example, for each p ≥ 5, with the following properties: |G| = 2(p − 1)p, G ⊂ SL(2, F), F ⊇ 𝔽_{p^{2}}, S(V)^{G} is Cohen-Macaulay but not Gorenstein, where V ≡ F^{2} and G ∩ Z(GL(V)) = {1}.

6. Acknowledgements

Thanks to Yuval Ginosar for stimulating conversations.

32 AMIRAM BRAUN

References

[1] E. Aljadeff, Y. Ginosar, Induction from elementary abelian subgroups, J. Algebra 179 (1996) 599–606.

[2] D. J. Benson, Polynomial invariants of finite groups, London Math. Society Lecture Ser., Vol. 190, Cambridge University Press, Cambridge 1993.

[3] A. Braun, On the Gorenstein property for modular invariants, J. Algebra 345 (2011) 81–99.

[4] A. Braun, On symmetric, smooth and Calabi-Yau algebras, J. Algebra 317 (2007) 519–533.

[5] A. Braun, G. Vernik, On the center and semi-center of enveloping algebras in prime characteristic, J. Algebra 322 (2009) 1830–1858.

[6] A. Broer, The direct summand property in modular invariant theory, Transformation Groups, Vol. 10, No. 1 (2005) 5–27.

[7] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge Study in Adv. Math., Cambridge University Press, 1998.

[8] C. W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras, Wiley Classics Library, 1988.

[9] C. W. Curtis, I. Reiner, Methods of representation theory, Vol. I, John Wiley & Sons, New-York,1981.

[10] F. DeMeyer, E. Ingraham, Separable algebras over commutative rings, Lecture Notes in Math., Vol. 181, Springer-Verlag, 1971.

[11] A. Dress, On finite groups generated by pseudoreflections, J. Algebra 11 (1969) 1–5.

[12] P. Fleischmann, C. Woodcock, Relative invariants, ideal classes and quasi-canonical modules of modular rings of invariants, J. Algebra 348 (2011) 110–134.

[13] R. M. Fossum, The divisor class group of a Krull domain, Springer-Verlag, Berlin-Heidelberg-New York, 1973.

[14] P. Griffith, Some results in local rings on ramification in low codimension, J. Algebra 137 (1991) 473– 490.

[15] L. C. Grove, Classical groups and geometric algeba, Graduate Studies in Mathematics, Vol. 39 A.M.S, 2002.

[16] W. Kantor, Subgroups of classical groups generated by long root elements, Trans. Amer. Math. Soc. 248 (1979) 347–379.

[17] G. Kemper, Loci in quotients by finite groups, pointwise stabilizers and the Buchsbaun property, J. reine angew. Math., 547 (2002) 69–96.

[18] G. Kemper, E. Körding, G. Malle, B. H. Matzat, D. Vogel, G. Wiese, A database of invariant rings, Experiment. Math. 10 (2001) 537–542.

[19] G. Kemper, G. Malle, The finite irreducible linear groups with polynomial ring of invariants, Transformation Groups, Vol. 2, No. 1 (1997) 57–89.

[20] J. Kollár, S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math., Cambridge University Press (1998).

[21] S. Maclane, Homology, Springer-Verlag, Berlin, 1975.

[22] H. Matsumura, Commutative ring theory, Cambridge University Press, 1986.

[23] G. Moeglin, Factorialite dans les algebras, envelopantes, C. R. Acad. Sci. Paris A 282 (1976) 1269–1272.

THE GORENSTEIN PROPERTY AND THE KEMPER ET AL. CONJECTURE 33

[24] H. Nakajima, Invariants of finite groups generated by pseudoreflections in positive characteristic, Tsukuba J. Math 3 (1979) 109–122.

[25] H. Nakajima, Relative invariants of finite groups, J. Algebra 79 (1982) 218–234.

[26] H. D. Neusel, L. Smith, Invariant theory of finite groups, Math. Surveys Monogr., Vol. 94, American Mathematical Society, 2002.

[27] M. Orzech, C. Small, The Brauer group of commutative rings, Marcel Dekker, 1975.

[28] I. Reiner, Maximal Orders, Academic Press, London - New York, 1975.

[29] J-P. Serre, Local Fields, Springer, 1995.

[30] K. Watanabe, Certain invariant subrings are Gorenstein I, Osaka J. Math. II (1974) 1–8.

[31] K. Watanabe, Certain invariant subrings are Gorenstein II, Osaka J. Math. II (1974) 379–388.

[32] D. Weston, Divisorial properties of the canonical module for invariant subrings, Comm. Algebra 19 (1991) 2641–2666.

[33] A. E. Zalesskii, V. N. Serežkin, Linear groups generated by transavections, Math. USSR Izv. 10 (1976) 25–46.