ON THE GORENSTEIN PROPERTY FOR MODULAR INVARIANTS

AMIRAM BRAUN

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

Abstract. Let G ⊂ GL(V) be a finite group, where V is a finite dimensional vector space over a field F of arbitrary characteristic. Let S(V) be the symmetric algebra of V and S(V)^{G} the ring of G-invariants. We prove here the following results: Theorem. Suppose that G contains no pseudo-reflection (of any kind).

(1) If S(V)^{G} is Gorenstein, then G ⊂ SL(V).

(2) If G ⊂ SL(V) then the Cohen-Macaulay locus of S(V)^{G} coincides with its Gorenstein locus. In particular if S(V)^{G} is Cohen-Macaulay then it is also Gorenstein.

This extends well known results of K. Watanabe in case (charF, |G|) = 1. A similar extension is given to a theorem of D. Benson. Our proof uses non-commutative algebra methods in an essential way.

1. Introduction

Let F be a field, V a finite dimensional F -vector space of dimension n and S(V) the symmetric algebra of V. Let G be a finite subgroup of GL(V). The following is a well known result of K. Watanabe [23] and [24]:

Theorem. [6, Theorems 6.4.9 and 6.4.10] Suppose that (|G|, charF) = 1. Then,

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

(2) Conversely, suppose that G contains no pseudo-reflection and S(V)^{G} is Gorenstein. Then G ⊂ SL(V).

It is natural to ask if a modular version of this theorem exists, namely when charF divides |G|. An obvious obstacle is that S(V)^{G} need not be Cohen-Macaulay in this case [3]. Still, suppose that S(V)^{G} is Cohen-Macaulay, then a version of the above in the modular case, will be of interest.

Date: March 18, 2009.

2 AMIRAM BRAUN

Our main purpose here is to establish such a version. The following is a key result.

Theorem A. Let V be a finite dimensional F -vector space over a field F of arbitrary characteristic. Let G ⊂ GL(V) be a finite group. Suppose that G contains no pseudoreflection (neither diagonalizable nor trasvection). Then the following are equivalent:

(1) G ⊂ SL(V),

(2) S(V)^{G} ≅ Hom_{C}(S(V^{G}, C), for every polynomial ring C such that S(V)^{G}/C is a finite extension and the homogenous generators of C have degrees divisible by |G|.

As a consequence we obtain the following two results that can be considered as modular versions of Watanabe’s theorem.

Theorem B. Let G ⊂ SL(V) be a finite group which contains no pseudo-reflection (of any type). Then, the Cohen-Macaulay locus of S(V)^{G} coincides with its Gorenstein locus. In particular, if S(V)^{G} is Cohen-Macaulay then it is also Gorenstein.

Theorem C. Suppose that G ⊂ GL(V) is a finite group which contains no pseudo-reflection (of any type) and S(V)^{G} is Gorenstein. Then G ⊂ SL(V).

Our methods for proving the previous results differ from the ones used in the non-modular case. Non-commutative constructs and methods related to Calabi-Yau algebras are used in an essential way (see e.g. [4], [5], [13]). A key point is the computation of ”the” Nakayama automorphism of the skew group ring S(V) ∗ G. For this, a lifting from a suitable finite dimensional Frobenius homomorphic image is employed. A somewhat similar method, in the presence of a PBW basis, is used in [10] and [5]. As a consequence a criterion for the C-symmetry property of S(V) ∗ G is established where C is the above polynomial subring of S(V)^{G}. The C-symmetry of S(V) ∗ G is then translated , using a result of [12], into the relevant properties of S(V)^{G} and G.

Let Λ(V) denote the exterior algebra on V (assuming charF ≠ 2). Similar arguments provide the following generalization of a Theorem of D. Benson [2, Theorem 5.3.2].

Theorem D. Suppose that G ⊂ GL(V) contains no transvections. If (S(V) ⊗_{F} Λ(V))^{G} is Cohen-Macaulay then it is also Gorenstein.

Attention is given to various possible definitions of Gorenstein, for this non-commutative ring. We also demonstrate, by an example, the necessity of the assumption on G.

ON THE GORENSTEIN PROPERTY FOR MODULAR INVARIANTS 3

The next result being a consequence of Theorem A and a theorem of G. Evans and P. Griffith [9], is an application in a different direction. Recall that the embedding dimension of S(V)^{G}, is given by e.dimS(V)^{G} ≡ dim_{F} m/m^{2}, where m is the unique graded maximal ideal of S(V)^{G}. It can be easily seen that it is also the minimal number of algebra generators of S(V)^{G}.

Proposition E. Let G ⊂ SL(V) be a finite group with no pseudo reflection. Suppose that e.dimS(V)^{G} ≤ dim_{F} V + 2. Then S(V)^{G} is a complete intersection.

2. Polynomial Invariants

We shall assume throughout this paper that G ⊂ GL(V) is a finite group, where V is a n-dimensional F -vector space. Let S(V) = F ⊕_{i≥1} V^{i}, be the symmetric algebra of V. This is a polynomial ring in n variables (any fixed basis of V over F will do). We can choose, by Noether normalization theorem, a polynomial subring C = F [f_{1}, ..., f_{n}] of S(V)^{G} generated by homogenous elements, and having the following properties:

(1) S(V)^{G}/C is a finite extension,

(2) |G| divides k_{i} ≡ deg(f_{i}), for i = 1, ..., n.

Let m ≡ (f_{1}, ..., f_{n}) be the unique homogenous maximal ideal of S(V)^{G} and \overline{S(V)} ≡ S(V)/mS(V) ≡ F ⊕_{i≥1} \overline{S(V})_{i}.

Lemma 2.1. Retaining all the above notations we have:

(1) dim_{F} \overline{S(V)}_{d} = 1, where d = ∑_{i=1}^{n}(k_{i} − 1),

(2) \overline{S(V)}_{i} = 0, for i > d,

(3) For each 0 ≠ a ∈ \overline{S(V})_{i}, there exists b ∈ \overline{S(V)}_{d−i} such that ab = uα, where 0 ≠ u ∈ F and α is a fixed basis element of \overline{S(V)}_{d}.

Proof: Item (1) is fairly standard, and can be found e.g. in [20, Proposition 3.3.2]. Namely, the Poincaré polynomial of \overline{S(V)} is given by:

P (\overline{S(V)}, t) = ∏_{i=1}^{n}(1 − t^{k_{i}})/(1 − t) = ∏_{i=1}^{n}(1 + t + ... + t^{k_{i}−1}).

This also shows that there is no non-zero term of degree bigger than d in \overline{S(V)}, thus establishing item (2).

Item (3) is proved in [20, Theorem 5.4.1], and can also be deduced from the fact that \overline{S(V)} is a graded Artinian commutative Gorenstein ring. □

The next result appears in [2, p.59]. The novelty here is the observation that the restriction (charF, |G|) = 1 is not needed.

Lemma 2.2. Let G, S(V) be as above. Fix α so that \overline{S(V)}_{d} = F α. Then g(α) = det^{−1}(g)α, for each g ∈ G.

Proof: Since \overline{S(V)}_{d} = F α is one dimensional then g(α) = λ(g)α, where λ ∈ Hom(G, F ∖0) is a group homomorphism (a rational character in another terminology). We have to show that λ = det^{−1}.

We firstly deal with the case where g is diagonalizable. Clearly we may extend F to be algebraically closed. Also since \overline{S(V)}_{d} does not change if we alter the basis of V, we may therefore assume that V has a basis on which g acts diagonally by eigen values {λ_{1}, ..., λ_{n}} (not necessarily distinct). Denote by T r(g, S(V)_{j}) the trace of g on S(V)_{j} and similarly let T r(g, \overline{S(V)}_{j}) be the trace of g on (S(V)/mS(V))_{j} ≡ \overline{S(V)}_{j}. Then as a consequence of the fact that g acts diagonally on each S(V)_{j} we have:

∑_{j=0}^{∞} t^{j}T r(g, \overline{S(V)}_{j}) = ∏_{i=1}^{n}(1 − t^{k_{i}}) ∑_{j=0}^{∞} t^{j}T r(g, S(V)_{j}).

Now by either [6, Proof of Theorem 6.4.8, p. 284] or [2, Proposition 2.5.1] we have that

∑_{j=0}^{∞} T r(g, S(V)_{j}) = ∏_{i=1}^{n} 1/(1 − λ_{i}t).

Consequently:

Σ_{j=0}^{∞}t^{j}T r(g, \overline{S(V)}_{j}) = ∏_{i=1}^{n}((1 − t^{k_{i}})/(1 − λ_{i}t)) = ∏_{i=1}^{n}(1 + λ_{i}t + ... + λ_{i}^{−1}t^{k_{i}−1}),

where in the last equality we have used that λ_{i}^{k_{i}} = 1, for each i. This shows, since \overline{S(V)}_{d} = F α, that g(α) = det^{−1}(g)α.

Now, if charF = 0 then each g ∈ G is diagonalizable and the previous argument provides the required result. Suppose next that charF = p > 0. If g^{p^{k}} = 1 then λ(g)^{p^{k}} = 1 and also (det^{−1}(g))^{p^{k}} = 1. Consequently λ(g) = 1 = det^{−1}(g).

Suppose finally that g^{p^{r}k} = 1 with (k, p) = 1. Then g^{p^{r}} ≡ h is diagonalizable and by the previous argument λ(h) = det^{−1}(h). Consequently λ(g)^{p^{r}} = (det^{−1}(g))^{p^{r}} and therefore λ(g) = det^{−1}(g). □

Remark 2.3. Let α̃ ∈ S(V)_{d} be an homogenous pre-image of α. Then we cannot prove the equality g(α̃) = det^{−1}(g)α̃. This fact is an easy consequence, in the non-modular case, of the diagonal action of g in S(V)_{d}.

ON THE GORENSTEIN PROPERTY FOR MODULAR INVARIANTS 5

Notation 2.4. Let a_{1} = 1, ..., a_{r} = α̃ be a collection of homogenous elements in S(V), so that their images \overline{a}_{1} = 1, ..., \overline{a}_{r} = α, form a basis of \overline{S(V)} over F. We clearly may choose a_{i} to satisfy deg(a_{i}) = deg(\overline{a}_{i}) for i = 1, ..., r and order them via deg(a_{i}) ≤ deg(a_{i+1}) for i = 1, ..., r − 1. In particular deg(a_{i}) < d = deg(α̃) for i < r.

Using the graded Nakayama Lemma, we conclude that {a_{1}, ...a_{r}} is a free basis of S(V) over C. Consequently {a_{i}h|i = 1, ..., r, h ∈ G} form a free basis of the skew group ring R ≡ S(V) ∗ G over C.

Remark 2.5. Let R and C be as above. Then Hom_{C}(R, C) is endowed with the following R-bimodule structure: r.f.t(x) = f(txr), for each r, t, x ∈ R and f ∈ Hom_{C}(R, C).

Definition 2.6. We retain all the notations of 2.4. Let ϕ ∈ Hom_{C}(R, C) be defined by:

ϕ(a_{i}h) = \{ \begin{matrix}0, if i < r or h ≠ 1_{G} \\ 1, if i=r and h = 1_{G} \end{matrix} , and ϕ is extended to R via C-linearity.

Let \overline{ϕ} ∈ Hom_{F} (R/mR, F) be the induced map, namely \overline{ϕ}(\overline{r}) = \overline{ϕ(r)} ∈ C/m = F, where \overline{r} = r + mR. Then clearly,

\overline{ϕ}(\overline{a}_{i}h) = \{ \begin{matrix}0, if i < r or h ≠ 1_{G} \\ 1, if i=r and h = 1_{G} \end{matrix}

Remark 2.7. It is easily checked that ϕ is a homogenous element in the ℤ-graded C-module Hom_{C}(R, C) and consequently R.ϕ is a graded sub-module.

Lemma 2.8. Set \overline{R} ≡ R/mR = \overline{S(V)} ∗ G. Then Hom_{F} (\overline{R}, F) = \overline{R}.\overline{ϕ}.

Proof: Since dim_{F} \overline{R} = dim_{F} Hom_{F} (\overline{R}, F) we only have to show that the left \overline{R}-module map: \overline{r} ⟶ \overline{r}.\overline{ϕ}, for each \overline{r} ∈ \overline{R}, is injective.

Let 0 ≠ \overline{r} = ∑_{h∈G} ∑_{i=0}^{d} s_{i,h}h, where s_{i,h} ∈ \overline{S(V)}_{i}. Suppose that s_{j,g} ≠ 0 for some j ≤ d and g ∈ G. Then by Lemma 2.1(3), there exists t ∈ \overline{S(V)}_{d−j} with ts_{j,g} = uα, where 0 ≠ u ∈ F. Recall that \overline{ϕ}(g^{−1}\overline{S(V)}h) = 0 if h ≠ g and that ts_{i,h} is not in \overline{S(V)}_{d} if i ≠ j. Consequently:

\overline{r}.\overline{ϕ}(g^{−1}t) = \overline{ϕ}(g^{−1}t\overline{r}) = \overline{ϕ}(g^{−1}ts_{j,g}g) = u\overline{ϕ}(g^{−1}(α)g^{−1}g) = udet^{−1}(g^{−1})\overline{ϕ}(α) = udet(g) ≠ 0. □

Corollary 2.9. Hom_{C}(R, C) = R.ϕ and R ≅ Hom_{C}(R, C), as left R-modules.

Proof: R is a finitely generated free C-module and consequently:

(2.1)  Hom_{C}(R, C)/mHom_{C}(R, C) ≅ Hom_{C/m}(R/mR, C/m) = \overline{R}.\overline{ϕ},

where the last equality is due to Lemma 2.8. Therefore:

Hom_{C}(R, C) = R.ϕ + mHom_{C}(R, C).

Now R = S(V) ∗ G = F G ⊕_{i≥1} V^{i}F G, is an ℕ-graded ring with grading which extends and is compatible with the natural one on S(V) = F ⊕_{i≥1} V^{i} and on C. Consequently R is a finitely generated ℕ-graded C-module and Hom_{C}(R, C) is a finitely generated ℤ-graded C-module. Also by its very definition ϕ is a graded C-homomorphism with deg(ϕ) = −d, implying that both Hom_{C}(R, C) and R.ϕ are finitely generated graded C-module with finitely many non-zero negative components. Thus the ℕ-graded Nakayama Lemma (e.g. [21, Lemma 3.3.5] and Hom_{C}(R, C) = R.ϕ + mHom_{C}(R, C) imply the required equality Hom_{C}(R, C) = R.ϕ. Finally, the map r ⟶ r.ϕ for r∈ R, is an onto map between the two finitely generated free C-modules of equal rank, R and R.ϕ = Hom_{C}(R, C). Therefore by a standard result (e.g. [22, Exercise 1.16, p.7]) this map is an isomorphism. □

Proposition 2.10. We retain all the notations from Definition 2.6 and on. Then, g.ϕ = det(g)ϕ.g for each g ∈ G and s.ϕ = ϕ.s for each s ∈ S(V).

Proof: Firstly we shall establish that g.\overline{ϕ} = det(g)\overline{ϕ}.g for each g ∈ G. This will be verified by its evaluation on each basis element \overline{a}_{i}h of \overline{R}, for i = 1, ..., r, h ∈ G. Now if h ≠ g^{−1} and i = 1, ..., r we have:

(g.\overline{ϕ})(\overline{a}_{i}h) = ϕ(\overline{a}_{i}hg) = 0.

Similarly, since gh ≠ 1_{G}:

det(g)\overline{ϕ}.g(\overline{a}_{i}h) = det(g)\overline{ϕ}(g\overline{a}_{i}h) = det(g)\overline{ϕ}(g(\overline{a}_{i})gh) = 0.

If h = g^{−1} and i < r then:

(g.\overline{ϕ})(\overline{a}_{i}g^{−1}) = ϕ(\overline{a}_{i}g^{−1}g) = ϕ(\overline{a}_{i}) = 0.

Similarly, since g(\overline{a}_{i}) ∈ \overline{S(V)}_{deg(\overline{a}_{i})} and deg(\overline{a}_{i}) < d = deg(α), then g(\overline{a}_{i}) = ∑_{i} f_{l}\overline{a}_{l}, where f_{l} ∈ F and deg(\overline{a}_{l}) = deg(\overline{a}_{i}) < d, for each l. Therefore \overline{ϕ}(g(\overline{a}_{i})) = 0 and we have:

det(g)\overline{ϕ}.g(\overline{a}_{i}g^{−1}) = det(g)\overline{ϕ}(g\overline{a}_{i}g^{−1}) = det(g)\overline{ϕ}(g(\overline{a}_{i})gg^{−1}) = det(g)\overline{ϕ}(g(\overline{a}_{i})) = 0.

So if i = 1, ..., r and h ≠ g^{−1}, or if h = g^{−1} and i < r we have:

(g.\overline{ϕ} − det(g)\overline{ϕ}.g)(\overline{a}_{i}h) = 0.

ON THE GORENSTEIN PROPERTY FOR MODULAR INVARIANTS 7

Next we consider:

(g.\overline{ϕ} − det(g)\overline{ϕ}.g)(αg^{−1}) = \overline{ϕ}(αg^{−1}g) − det(g)\overline{ϕ}(gαg^{−1}) = \overline{ϕ}(α) − det(g)\overline{ϕ}(g(α)gg^{−1}) =

= 1 − det(g)\overline{ϕ}(det^{−1}(g)α) = 1 − \overline{ϕ}(α) = 1 − 1 = 0.

All in all g.\overline{ϕ} = det(g)\overline{ϕ}.g, for each g ∈ G. Consequently, by isomorphism (2.1),

(2.2)  T (g) ≡ g.ϕ − det(g)ϕ.g ∈ mHom_{C}(R, C), for each g ∈ G.

Now by exactly the same computations T (g)(a_{i}h) = 0 for i = 1, ..., r and h ≠ g^{−1}. Next we shall check T (g)(α̃g^{−1}). Indeed,

T (g)(α̃g^{−1}) = (g.ϕ)(α̃g^{−1}) − det(g)(ϕ.g)(α̃g^{−1}) = ϕ(α̃g^{−1}g) − det(g)ϕ(gα̃g^{−1}) =
= 1 − det(g)ϕ(g(α̃)gg^{−1}) = 1 − det(g)ϕ(g(α̃)).

Now g(α̃) ∈ S(V)_{d} and hence g(α̃) = ∑_{i=1}^{r} c_{i}a_{i} with c_{i} ∈ C. So by degree arguments deg(c_{i}) + deg(a_{i}) = deg(g(α̃)) = deg(α̃) = d. Consequently c_{r} ∈ F, and since deg(a_{i}) < deg(α̃) if i < r, then ϕ(g(α̃)) = c_{r}ϕ(α̃) = c_{r} and T (g)(α̃g^{−1}) = 1 − det(g)c_{r} ∈ F. Now by equation (2.2), T (g)(α̃g^{−1}) ∈ m, and consequently T (g)(α̃g^{−1}) = 0. Finally if i < r and h = g^{−1} we have:

T (g)(a_{i}g^{−1}) = (g.ϕ−det(g)ϕ.g)(a_{i}g^{−1}) = ϕ(a_{i}g^{−1}g)−det(g)ϕ(ga_{i}g^{−1}) = 0−det(g)ϕ(g(a_{i})gg^{−1}).

Now deg(g(a_{i}) = det(a_{i}) < d, consequently g(a_{i}) = ∑_{l} c_{l}a_{l} with c_{l} ∈ C, where deg(c_{l}) + deg(a_{l}) = deg(a_{i}). Thus deg(a_{l}) ≤ deg(a_{i}) < d, implying that ϕ(a_{l}) = 0 for each l and therefore ϕ(g(a_{i}) = 0. That is T (g)(a_{i}g^{−1}) = 0, for each i < r. All in all g.ϕ − det(g)ϕ.g ≡ T (g) = 0 for each g ∈ G, and the first part of Proposition 2.10 is verified.

We next observe that for s, b ∈ S(V):

(s.ϕ)(bh) = ϕ(bhs) = ϕ(bh(s)h) = 0, if h ≠ 1_{G}.

Similarly:

(ϕ.s)(bh) = ϕ(sbh) = 0, if h ≠ 1_{G}.

Also, if h = 1_{G} we have:

(s.ϕ)(b) = ϕ(bs) = ϕ(sb) = (ϕ.s)(b).

All in all s.ϕ = ϕ.s for each s ∈ S(V). □

Remark 2.11. We are indebted to K. Brown for pointing out that a similar result to Proposition 2.10 was obtained by his student M. Macleod.

8 AMIRAM BRAUN

The next remark provides the necessary details regarding the notions of Nakayama automorphism and free Frobenius extensions. For a fuller account see [14] or [5, Section 2.2].

Remark 2.12. (1) Suppose that Hom_{C}(R, C) = R.f. Then R and Hom_{C}(R, C) are both finitely generated free C-modules of the same C-rank, and by [22, p. 7] l-ann_{R}f = {0}. Consequently, since Hom_{C}(R, C) is also an R-bimodule, f induces a Nakayama automorphism ν_{f}, of R, via f.r = ν_{f}(r).f, for each r ∈ R. Suppose also that Hom_{C}(R, C) = R.h, then clearly u.f = h, for some unit element u ∈ R. Then ν_{h}(r) = uν_{f}(r)u^{−1}, for each r ∈ R. This follows from l-ann_{R}f = {0} and the following chain of equalities:

uν_{f}(r).f = u.f.r = h.r = ν_{h}(r).h = ν_{h}(r)u.f.

(2) Corollary 2.9 and Proposition 2.10 actually show that R is a free Frobenius extension of C, in the language of [14] or [5], with ν_{ϕ}(g) = det^{−1}(g)g for each g ∈ G and ν_{ϕ}(s) = s for s ∈ S(V).

(3) The method used in Proposition 2.10 amounts to performing a lifting from the F - finite dimensional Frobenius homomorphic image \overline{R}. This idea was employed in [10] and [5].

Definition 2.13. Let A be a ring which is a finitely generated module over a central subring C. We say that A is a symmetric C-algebra if Hom_{C}(A, C) ≅ A as A-bimodules. Equivalently Hom_{C}(A, C) = A.f, with r.f = f.r for each r ∈ A.

Remark 2.14. The notion of C-symmetry is extensively dealt with in [4], [5] and [13], being a key one in the context of Calabi-Yau algebras.

For our purpose we will need the following definition.

Definition 2.15. [13, p. 8] Let A be a ring which is a finitely generated C-module, with C being a central subring of A. Assume also that A is C-reflexive, that is Hom_{C}(Hom_{C}(A, C), C) ≅ A. Let M be a finitely generated left A-module which is Creflexive. We say that M is a height one projective if M_{p} is a projective A_{p}-module, for each height one prime ideal p in C.

The following result of O. Iyama is of importance to us.

Lemma 2.16. [12, 5.4.3(1)], [13, Proposition 2.4(4).] Let A be a C-algebra which is finitely generated reflexive C-module and M a finitely generated height one projective left A-module. Then End_{A}M is C-symmetric if A is C-symmetric.

ON THE GORENSTEIN PROPERTY FOR MODULAR INVARIANTS 9

The next result is a consequence of [2, Lemma 3.9.1] (see also [19, Lemma 2.6]).

Lemma 2.17. Let G ⊂ GL(V) be a finite group with dim_{F} V being finite. Suppose that G contains no pseudo-reflection (of any kind). Then the extension S(V)/S(V)^{G} is unramified in co-dimension one.

Proof: Let p be a height one prime ideal in S(V)^{G}, and P be a height one prime ideal in S(V) satisfying P ∩ S(V)^{G} = p. Suppose that the inertia subgroup G_{t} ≡ {g ∈ Stabilizer(P) | g(x) − x ∈ P, for each x ∈ S(V)} is not trivial. Then [2, lemma 3.9.1] and its argument show the existence of a non-trivial pseudo-reflection. Thus G_{t} = {1_{G}}. Equivalently for each 1 ≠ g ∈ Stabilizer(P), there exists s ∈ S(V) so that g(s) − s is not in P. Now if g is not in Stabilizer(P) then clearly g(s) − s is not in P for some s ∈ P. Therefore the extension S(V)_{p}/S(V)_{p}^{G} satisfies the conditions of [8, Proposition 1.2.(5), p.80] and S(V)_{p}/S(V)_{p}^{G} is a separable (thus unramified) extension. □

Corollary 2.18. Suppose that G ⊂ GL(V) contains no pseudo-reflection (of any kind). Let C ⊆ S(V)^{G} be a polynomial subring with S(V)^{G}/C being a finite extension. Then:

(1) R_{p} is a trivial Azumaya algebra, for each height one prime ideal p in C,

(2) S(V) ∗ G ≅ End_{S(V)^{G}}S(V),

(3) S(V) is a height one projective left R ≡ S(V) ∗ G-module.

Proof: By assumption and Lemma 2.17, the extension S(V)_{p}/S(V)_{p}^{G} is an unramified extension for each height one prime p in C (the shift from height one primes in S(V)^{G} to ones in C is granted by the Going Down property between S(V)^{G} and C). Consequently, by [8, Proposition 1.2(3), p. 80], S(V)_{p}/S(V)_{p}^{G} is a Galois extension and

R_{p} = S(V)_{p} ∗ G ≅ End_{S(V)_{p}^{G}}S(V)_{p}.

Since S(V)_{p}^{G} is a Dedekind domain, S(V)_{p} is a finitely generated projective S(V)_{p}^{G}-module and (1) is therefore established.

Now R = S(V) ∗ G is a finitely generated free C-module and in particular R is a reflexive C-module. Consequently R = ∩_{p}R_{p}, where p runs on all height one primes p in C. Let μ : R = S(V) ∗ G → End_{S(V)^{G}}S(V), be the natural ring homomorphism. Clearly μ is injective. Therefore μ(R) = ∩_{p}μ(R)_{p} and since μ(R)_{p} = μ(R_{p}) = End_{S(V)_{p}^{G}}S(V)_{p}, we get that μ(R) = ∩_{p}End_{S(V)_{p}^{G}}S(V)_{p} and thus μ(R) = End_{S(V)^{G}}S(V), settling (2).

Finally S(V)_{p} is a finitely generated projective S(V)_{p}^{G} module and S(V)_{p}^{G} is the center of the Azumaya algebra R_{p} ≅ End_{S(V)_{p}^{G}}(S(V)_{p}). Therefore by the lifting property of Azumaya

10 AMIRAM BRAUN

algebras (e.g. [8, Proposition 2.3, p. 48], S(V)_{p} is a finitely generated projective R_{p}-module. This settles (3). □

We shall now prove Theorem A of the introduction.

Theorem 2.19. Let G ⊂ GL(V) be a finite group, where dim_{F} V is finite and F is a field of arbitrary characteristic. Suppose that G contains no pseudo-reflection of any kind. Then the following are equivalent:

(1) G ⊂ SL(V),

(2) S(V)^{G} is a symmetric C-algebra, for every polynomial ring C such that S(V)^{G}/C is a finite extension and the homogenous generators of C have degrees divisible by |G|.

Proof: Suppose firstly that G ⊂ SL(V). Then, by Corollary 2.9 and Proposition 2.10, ν_{ϕ} = Id, that is Hom_{C}(R, C) = R.ϕ and r.ϕ = ϕ.r for each r ∈ R = S(V) ∗ G. Consequently R is a symmetric C-algebra, and by Corollary 2.18(2) End_{S(V)^{G}}S(V) is a symmetric C-algebra as well. Now by Corollary 2.18(3) S(V) is a height one projective End_{S(V)^{G}}S(V) = μ(R)-module. Thus by Lemma 2.16 End_{μ(R)}S(V) is C-symmetric. But clearly End_{μ(R)}S(V) = S(V)^{G} and so S(V)^{G} is a symmetric C-algebra and (2) is established.

Suppose next that (2) holds. By assumption and Corollary 2.18(2) we have S(V) ∗ G ≅ End_{S(V)^{G}}S(V). Now the fact that S(V) is a height one projective S(V)^{G}-module is deduced from the Dedekind property of S(V)_{p}^{G} and the finiteness of S(V)_{p} over S(V)_{p}^{G}. Therefore by Lemma 2.16 R = S(V) ∗ G ≅ End_{S(V)^{G}}S(V) is a symmetric C-algebra. That is Hom_{C}(R, C) = R.f with ν_{f} = Id. Now by Corollary 2.9 Hom_{C}(R, C) = R.ϕ, and by Remark 2.12(1) ν_{f} and ν_{ϕ} differ by an automorphism which is induced by an invertible element u ∈ R. That is ν_{ϕ}(r) = uru^{−1} for each r ∈ R. Also by remark 2.12(2), ν_{ϕ}(s) = s for each s ∈ S(V). Consequently us = su for each s ∈ S(V) and therefore uy = yu, for each y ∈ Q(S(V)). Thus u is in the centralizer of Q(S(V)) in Q(S(V)) ∗ G ≅ End_{Q(S(V))^{G}}Q(S(V)). Now

[Q(S(V)) : Q(S(V))^{G}]^{2} = |G|^{2} = [(Q(S(V)) ∗ G) : (Q(S(V))^{G}],

implies by [11, p. 105, 3-rd paragraph], that Q(S(V)) is a maximal commutative subalgebra of End_{Q(S(V))^{G}}Q(S(V)), thus showing that u ∈ Q(S(V)). Therefore u^{−1}, u ∈ R∩Q(S(V)) = S(V), and hence u ∈ F. Therefore ν_{ϕ} = Id. Finally for each g ∈ G we have g = Id(g) = ν_{ϕ}(g) = det(g)^{−1}g and so det(g) = 1. □

ON THE GORENSTEIN PROPERTY FOR MODULAR INVARIANTS 11

We next prove Theorem C.

Theorem 2.20. Let G ⊂ GL(V) be a finite group which contains no pseudo-reflection of any kind. Suppose that S(V)^{G} is Gorenstein. Then G ⊂ SL(V).

Proof: By Noether normalization there exists a polynomial subring C, with homogenous generators of degrees divisible by |G|, such that S(V)^{G} is a finitely generated C-module. Therefore by [6, Proposition 3.6.11, Proposition 3.6.12] S(V)^{G} ≅ Hom_{C}(S(V)^{G}, C). The required conclusion follows now from Theorem 2.19. □

Our next result is Theorem B.

Theorem 2.21. Let G ⊂ SL(V) be a finite group which contains no pseudo-reflection of any kind. Let q be a prime ideal in S(V)^{G}. Then S(V)_{q}^{G} is Gorenstein if it is Cohen-Macaulay.

Proof: Let p = q ∩ C and let q = q_{1}, ..., q_{t} be all the prime ideals in S(V)^{G} lying above

p. By Going Down K.dimC_{p} = K.dimS(V)_{q}^{G} and consequently K.dimĈ_{p} = K.dim{ \atop S(V)}_{q}^{G}

where { \atop S(V)}_{q}^{G} ≡ A, is the q_{q}-adic completion of S(V)_{q}^{G} and Ĉ_{p} is the p_{p}-adic completion of C_{p}. Moreover Ĉ_{p} and A are both normal domains. Now by [18, Theorem 8.7, Theorem 8.15] we have:

S(V)^{G}⊗_{C} Ĉ_{p} = S(V)_{p}^{G}⊗_{C_{p}} Ĉ_{p} ≅ lim_{←i}S(V)_{p}^{G}/p_{p}^{i}S(V)_{p}^{G} ≅ lim_{←i}S(V)_{q_{1}}^{G}/q_{1_{q_{1}}}^{i} ⊕...⊕lim_{←i}S(V)_{q_{t}}^{G}/q_{t_{q_{t}}}^{i}

Also by Theorem 2.19 Hom_{C}(S(V)^{G}, C) ≅ S(V)^{G} and since Ĉ_{p} is flat over C we have by [17, Lemma 4.1] that HomĈ_{p}(S(V)^{G} ⊗_{C} Ĉ_{p}, Ĉ_{p}) ≅ S(V)^{G} ⊗_{C} Ĉ_{p}. This combined with the previous series of isomorphisms for S(V)^{G} ⊗_{C} Ĉ_{p} and the fact that A is a direct summand of S(V)^{G} ⊗_{C} Ĉ_{p} yields Hom_{C_{p}}(A, Ĉ_{p}) ≅ A (see the proof of [4, Proposition 2.6] for more details). Moreover, since S(V)^{G} ⊗_{C} Ĉ_{p} is a finite Ĉ_{p}-module, the same holds for A. Now, since A is Cohen-Macaulay then by [6, Theorme 3.37(b)] w_{A} ≅ HomĈ_{p}(A, Ĉ_{p}), where w_{A} is the dualizing module of A. Therefore w_{A} ≅ A and by [6, Theorem 3.37(a)] A ≡ { \atop S(V)}_{q}^{G} is Gorenstein. Finally by [6, Proposition 3.1.19(c)] S(V)_{q}^{G} is Gorenstein. □

Corollary 2.22. Let G ⊂ SL(V) be a finite group with no pseudo-reflection and dim_{F} V ≥ 3. Let q be a prime ideal in S(V)^{G} with height(q) ≤ 3. Then S(V)_{q}^{G} is Gorenstein.

Proof: By [20, Proposition 5.6.10] S(V)_{q}^{G} is Cohen-Macaulay, and Theorem 2.21 is applicable. □

The next special case is worth recording.

Corollary 2.23. Let G ⊂ SL(3, F) be a finite group with no transvection. Then S(V)^{G} is Gorenstein.

Proof: By assumption G has no pseudo-reflection. Also by [20, Proposition 5.6.10] S(V)^{G} is Cohen-Macaulay. Consequently by Theorem 2.21, S(V)^{G} is Gorenstein. □

The following result is a consequence of Theorem A and a result of G. Evans and P. Griffith. Recall that K.dimS(V)^{G} denotes the classical Krull dimension of S(V)^{G} and it also equals to K.dimS(V) = dim_{F} V.

Proposition 2.24. Let G ⊂ SL(V) be a finite group with no pseudo-reflection. Suppose that e.dimS(V)^{G} ≤ K.dimS(V)^{G} + 2. Then S(V)^{G} is a complete intersection.

Proof: By standard results (e.g. [16, Lemma 2.9]), we may dispose off the strict in equality cases. Suppose therefore that e.dimS(V)^{G} = K.dimS(V)^{G} + 2. Let m be the unique maximal graded ideal of S(V)^{G}. Choose, as in Theorem A, C ⊂ S(V)^{G}, a graded polynomial subring, with p its unique graded maximal ideal. Since m is the unique prime ideal in S(V)^{G} which lies over p, we have S(V)_{m}^{G} = S(V)_{p}^{G}. Then by [16, Lemma 2.3] e.dimS(V)_{p}^{G} = K.dimS(V)_{p}^{G} + 2. Now, Theorem A implies that Hom_{C_{p}}(S(V)_{p}^{G}, C_{p}) is a cyclic S(V)_{p}^{G} = S(V)_{m}^{G}-module. Consequently by [9, Theorem 4.7] S(V)_{m}^{G} is a complete intersection. Now by [16, Proposition 2.16] the property of being a complete intersection is a local one, thus by [16, Definition 1.4(L2)] S(V)^{G} is also a complete intersection. □

3. Polynomial Tensor Exterior Algebras Invariants

Let G ⊂ GL(V) be a finite group and V is F -finite dimensional with charF ≠ 2. We shall show here how the proof of Theorem D is obtained by modifying the proofs of Theorems A and B. An interesting feature is the separation of the argument according to the even versus odd nature of dim_{F} V.

It is proven in [2, Theorem 5.3.2], assuming (charF, |G|) = 1, that D ≡ (S(V)⊗_{F} Λ(V))^{G} is Gorenstein iff it is Cohen-Macaulay. Benson’s definition for being Gorenstein requires the following:

(1) D is a finitely generated free module over a polynomial subring C ⊆ D,

(2) Hom_{C}(D, C) ≅ D, as left (or right) D-modules.

In fact item (2) is mentioned in [2, p. 64], where for its analogous proof one is referred to [2, Proposition 4.6.1]and the resulting isomorphism is as one sided D-modules.

ON THE GORENSTEIN PROPERTY FOR MODULAR INVARIANTS 13

This definition for a non-commutative ring to be Gorenstein is also known as being a free Frobenius extension of C (see remark 2.12). Another possibility for the Gorenstein property to hold is to require that the isomorphism in (2) should be as D-bimodules. Namely D is C-symmetric (as in Definition 2.13).

Our proof will show that, if dim_{F} V is odd, then D is Gorenstein in the bimodule sense. However, if dim_{F} V is even, then D is almost never C-symmetric and only the weaker, one sided Gorenstein property, always holds.

We shall need the following few results.

Lemma 3.1. Let A be a ring and H ⊂ AutA, a finite group such that Σ_{g∈H}g(a) = 1 for some a ∈ A. Then A is a projective A ∗ H module.

Proof: See for example [1]. □

Corollary 3.2. Set A ≡ S(V) ⊗_{F} Λ(V) and let G be as above. Let C ⊂ S(V)^{G} be a polynomial ring as described in Theorem A. Suppost that G has no transvections. Then A_{p} is a projective A_{p} ∗ G-module, for each height one prime ideal p in C.

Proof: It is well known that the assumption on G implies that the extension S(V)_{p}/S(V)_{p}^{G} is a tamely ramified one. In particular there exists a_{p} ∈ S(V)_{p} satisfying Σ_{g}g(a_{p}) = 1. Now apply the previous Lemma. □

We are now able to prove, provided dim_{F} V is odd, an extension to Theorem A.

Proposition 3.3. Suppose that dim_{F} V is odd, G has no transvections and charF ≠ 2. Then (S(V) ⊗_{F} Λ(V))^{G} is C-symmetric, where C is chosen as in Theorem A.

Proof: By the standard conventions each non-zero element of V has degree 2 in S(V) and their differentials images in Λ(V) have degree 1. Choose C = F [f_{1}, ..., f_{n}], a polynomial subring of S(V) with degf_{i} = 2k_{i} and |G| divides k_{i}, for i = 1, ..., n. Denote e = Σ_{i=1}^{n}(2k_{i} − 2), d = Σ_{i=1}^{n}(2k_{i} − 1) and m = (f_{1}, ...f_{n}). Set A ≡ S(V) ⊗_{F} Λ(V).

The computations in [2, p.63] show that \overline{A} ≡ A/mA is graded, having a maximal nonzero term in degree d and dim_{F} \overline{A}_{d} = 1. Similarly \overline{S(V)} ≡ S(V)/mS(V) has a maximal non-zero term in degree e and dim_{F} \overline{S(V)}_{e} = 1. Observe that \overline{S(V})⊗_{F} Λ(V) = \overline{A}. Therefore if x_{1}, ...x_{n} is a basis of V, one can choose a basis at S(V), b_{1}, ..., b_{s} = β̃, so that their set of images is an homogenous basis of \overline{S(V)} and \overline{b}_{s} = β is a basis element of \overline{S(V)}_{e}. Moreover we choose them so that deg(b_{i}) = deg\overline{b}_{i} for i = 1, ..., s and degb_{i} ≤ degb_{j} if i < j. Now

14 AMIRAM BRAUN

by tensoring with the standard basis of Λ(V) we obtain a basis of A {a_{1}, ...a_{r} = α̃}, where \{b_{i}\} is a subset, deg(a_{i}) = deg\overline{a}_{i}, α ≡ βdx_{1} ∧ ⋯ ∧ dx_{n} is a basis element of \overline{A}_{d}, and deg(a_{i}) ≤ dega_{j} if i < j. Clearly degβ̃ = degβ = e and degα̃ = deg α = d.

As in Definition 2.6 we next define:

ϕ(a_{i}h) = \{ \begin{matrix}0, if i < r or h ≠ 1_{G} \\ 1, if i=r and h = 1_{G} \end{matrix} , and extend to R ≡ A ∗ G by C-linearity.

Following the proof of Lemma 2.8 we want to show that \overline{R}.\overline{ϕ} = Hom_{F} (\overline{R}, \overline{F}), where \overline{R} ≡ R/mR, and \overline{ϕ} is the obvious projection of ϕ on \overline{R}. To this end we need the following property of \overline{A} (extending Lemma 2.1(3)):

(3.1)  For each 0 ≠ a ∈ \overline{A}_{i} there exists b ∈ \overline{A}_{d−i} such that ab = uα with 0 ≠ u ∈ F.

To show this, denote by dx^{λ} ≡ dx_{λ_{1}} ∧ ... ∧ dx_{λ_{s}}, where λ ≡ (λ_{1}, ..., λ_{s}). Now a = Σ_{λ}a_{λ}dx^{λ}, with dega_{λ} + degdx^{λ} = i. Choose δ so that a_{δ} ≠ 0 and j ≡ dega_{δ} is minimal amongst all dega_{λ}’s. Let μ_{δ} = {1, ..., n} ∖ δ. Then clearly dx^{δ} ∧ dx^{μ_{δ}} = ±dx_{1} ∧ ... ∧ dx_{n}. Pick by Lemma 2.1(3) b_{δ} ∈ \overline{S(V)}_{e−j} so that a_{δ}b_{δ} = uβ, with 0 ≠ u ∈ F. If λ ≠ δ with dega_{λ} > dega_{δ} then we get that a_{λ}b_{δ} = 0. Now if λ ≠ δ and dega_{λ} = dega_{δ} then clearly dx^{λ} ∧ dx^{μ_{δ}} = 0. All in all a(b_{δ}dx^{μ_{δ})}) = ±uα, 0 ≠ u ∈ F, as needed.

Now the arguments for the proofs of Lemma 2.8, Corollary 2.9 and Proposition 2.10 go through, showing that R.ϕ = Hom_{C}(R, C) and g.ϕ = ϕ.g, for each g ∈ G. To complete the proof we need to show that x.ϕ = ϕ.x for each x ∈ R. To see this we shall make use of the fact that dim_{F} V is odd. Using g.ϕ = ϕ.g, we may clearly assume that x is one of the basis elements in {a_{1}, ..., a_{r}}. Let b ∈ A be another such basis element. If degx + degb < d then by the definition of ϕ we have ϕ(xb) = ϕ(bx) = 0 and consequently (x.ϕ)(b) = (ϕ.x)(b) = 0. Now if degx + degb = d then one of {degx, degb} is even and so xb = bx and again (x.ϕ)(b) = (ϕ.x)(b). For b = a_{i}h with h ≠ 1_{G} we have (x.ϕ)(b) = ϕ(bx) = 0 = ϕ(xb) = (ϕ.x)(b). Therefore R = A ∗ G is C-symmetric and by Lemma 2.16, using Corollary 3.2, End_{R}A is C-summetric. Finally by standard results End_{A∗G}(A) = A^{G}, and we conclude that (S(V) ⊗_{F} Λ(V))^{G} is C-symmetric. □

Proposition 3.4. Suppose that G ⊂ GL(V) has no transvections, dim_{F} V is even and charF ≠ 2. Let C ⊂ S(V)^{G} be chosen as in Proposition 3.3. Then (S(V) ⊗_{F} Λ(V))^{G}[t] ≅ Hom_{C[t]}((S(V) ⊗ Λ(V))^{G}, C[t]) as one sided (S(V) ⊗ Λ(V)^{G})[t]-modules for some variable t. However (S(V) ⊗ Λ(V))^{G} is not C-symmetric unless it solely consists of sums of even elements.

ON THE GORENSTEIN PROPERTY FOR MODULAR INVARIANTS 15

Proof: Let W = V ⊕ F t and extend the action of G to W by requiring g(t) = t, for all g ∈ G. Clearly G ⊂ GL(W) and still has no transvections. Therefore by Proposition 3.3, (S(W) ⊗ Λ(W))^{G} = (S(V) ⊗ Λ(V))^{G}[t] ⊗_{F} dt is C[t]-symmetric.

Let B ≡ (S(V)⊗Λ(V))^{G}[t]. Then (S(W)⊗Λ(W))^{G} = B⊕Bdt is a free B-module of rank 2. Moreover by the C[t]-symmetry we have B⊕Bdt ≅ Hom_{C[t]}(B+Bdt, C[t]) ≡ (B+Bdt).ϕ, by the map b + b′dt → (b + b′dt).ϕ, where b, b′ ∈ B. Let ϕ_{1} ≡ ϕ_{|B}, ϕ_{2} = ϕ_{|Bdt}. Then ϕ = ϕ_{1} + ϕ_{2}. Moreover Bdt.ϕ_{1} = 0.

We have that (B + Bdt).ϕ = (B.ϕ_{1} + Bdt.ϕ_{2}) + B.ϕ_{2}. Clearly B.ϕ_{1} + Bdt.ϕ_{2} ⊆ Hom_{C[t]}(B, C[t]) and B.ϕ_{2} ⊆ Hom_{C[t]}(Bdt, C[t]). Consequently, using

(B + Bdt).ϕ = Hom_{C[T]}(B + Bdt, C[t]) = Hom_{C[t]}(B, C[t]) ⊕ Hom_{C[t]}(Bdt, C[t])

We get:

(3.2)  Hom_{C[t]}(B, C[t]) = B.ϕ_{1} + Bdt.ϕ_{2} and B.ϕ_{2} = Hom_{C[t]}(Bdt, C[t]).

Clearly (b + b_{1}dt).(ϕ_{1} + ϕ_{2}) = (ϕ_{1} + ϕ_{2})(b + b_{1}dt) for b, b_{1} ∈ B, is translated using Bdt.ϕ_{1} = ϕ_{1}.Bdt into:

(3.3)  (b.ϕ_{1} − ϕ_{1}.b) + b_{1}dt.ϕ_{2} − ϕ_{2}.b_{1}dt = ϕ_{2}.b − b.ϕ_{2}.

Now since the left (respectively right) hand side of (3.3) belongs to Hom_{C[t]}(B, C[t]) (respectively Hom_{C[t]}(Bdt, C[t])), we get that both ends of (3.3) are 0. Hence b.ϕ_{2} = ϕ_{2}.b for b ∈ B. Consequently ϕ_{2}.B = B.ϕ_{2} = Hom_{C[t]}(Bdt, C[t]). Moreover b.ϕ_{1} − ϕ_{1}.b = ϕ_{2}.b_{1}dt − b_{1}dt.ϕ_{2} shows that 0 = ϕ_{2}.b_{1}dt − b_{1}dt.ϕ_{2}, for b_{1} ∈ B. In particular ϕ_{2}.dt = dt.ϕ_{2}.

Now if (dt.ϕ_{2}).z = 0, for some z ∈ B then (dtz).ϕ_{2} = dtz.ϕ_{2} = 0. Also dtz.ϕ_{1} ∈ Bdt.ϕ_{1} = 0. So dtz.ϕ = dtz.(ϕ_{1} + ϕ_{2}) = 0, implying that dtz = 0 and z = 0. Consequently B ≅ (dt.ϕ_{2}).B as right B-modules. Similarly B ≅ ϕ_{2}.B as right B-modules.

Let f ∈ Hom_{C[t]}(B, C[t]), then f̂ ∈ Hom_{C[t]}(Bdt, C[t]) is defined by f̂(bdt) = f(b), for b ∈ B. It is easy to verify that the map f → f̂ induces a right B-modules isomorphism

from Hom_{C[t]}(B, C[t]) onto Hom_{C[t]}(Bdt, C[t]). Also { \atop dt.ϕ_{2}} = ϕ_{2} and therefore B ≅ ϕ_{2}B =

Hom_{C[t]}(Bdt, C[t]) implies that B ≅ (dt.ϕ_{2}).B = Hom_{C[t]}(B, C[t]), as right B-modules. This settles the first part of the proposition.

Now for the second part, we use b.ϕ_{2} = ϕ_{2}.b and dt.ϕ_{2} = ϕ_{2}.dt to get for an homogenous element r ∈ B:

(3.4) r.(dt.ϕ_{2}) = (rdt).ϕ_{2} = (−1)^{degr}(dtr).ϕ_{2} = (−1)^{degr}dt.(r.ϕ_{2}) = (−1)^{degr}(dt.ϕ_{2}).r.

16 AMIRAM BRAUN

Therefore the Nakayama automorphism induced by dt.ϕ_{2} is given by ν_{dt.ϕ_{2}}(r) = (−1)^{degr}r, for each homogenous element r ∈ B. However ν_{dt.ϕ_{2}} is never inner. Indeed let u, u^{−1} ∈ B be such that ν_{dt.ϕ_{2}}(r) = uru^{−1} for r ∈ B. Then after F -normalization, u = 1 + Σ_{i≥1}u_{i} and u^{−1} = 1+Σ_{i≥1}v_{i}, with degu_{i} = degv_{i} = i. Consequently, for an homogenous element r in B we have u^{−1}ru = r+Σ_{k}w_{k}, where degw_{k} > degr, for each k. Hence u^{−1}ru ≠ ν_{dt.ϕ_{2}}(r) = −r, for each odd element r. This shows that B ≡ (S(V) ⊗_{F} Λ(V))^{G}[t] is not C[t] symmetric and consequently S(V) ⊗_{F} Λ(V))^{G} is not C-symmetric. □

Theorem 3.5. Let G ⊂ GL(V) be a finite group with no transvections. Suppose that (S(V) ⊗_{F} Λ(V))^{G} is Cohen-Macaulay, then it is also Gorenstein.

Proof: If dim_{F} V is odd, the result follows directly from Proposition 3.3. Suppose therefore that dim_{F} V is even. Let D ≡ (S(V) ⊗_{F} Λ(V))^{G} and D[t] = B be as in the previous proposition. By assumption D is a finitely generated free C-module and thus D[t] is a finitely generated free C[t]-module. Now by Proposition 3.4 Hom_{C[t]}(D[t], C[t]) ≅ D[t], as right D[t]-modules, thus by the freeness of D[t] over C[t] we have:

Hom_{C[t]/tC[t]}(D[t]/tD[t], C[t]/tC[t]) ≅ D[t]/tD[t], as right D[t]/tD[t] = D − modules.

Now since C[t]/C[t] = C, we get the required result. □

We shall next show that the omission of transvections from G, in Theorem 3.5, is necessary.

Example 3.6. Let G =< σ >, where σ = [\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}] acts linearly on V = F x + F y and

charF = p > 2. Then (S(V) ⊗_{F} Λ(V))^{G} is Cohen-Macaulay, but it is not Gorenstein (even in the one-sided sense).

Proof: σ(x) = x + y, σ(y) = y, shows that:

N(x) ≡ xσ(x)...σ^{p−1}(x) = x(x + y)...(x + (p − 1)y) = x^{p} − xy^{p−1} is in S(V)^{G}.

So A = F [y, x^{p} − xy^{p−1}] is a polynomial subring of S(V)^{G}. Moreover [Q(S(V)) : Q(A)] = p = [Q(S(V)) : Q(S(V)^{G})], implying that S(V)^{G} = F [y, x^{p} − xy^{p−1}]. Now

S(V) ⊗_{F} Λ(V) = S(V) ⊕ (S(V)dx + S(V)dy) ⊕ S(V)dx ∧ dy,

and G stabilizes each of its summands. Therefore:

(S(V) ⊗_{F} Λ(V)^{G} = S(V)^{G} ⊕ (S(V)dx + S(V)dy)^{G} ⊕ S(V)^{G}dx ∧ dy.

ON THE GORENSTEIN PROPERTY FOR MODULAR INVARIANTS 17

Let fdx+hdy ∈ (S(V)dx+S(V)dy)^{σ}. Then fdx+hdy = f^{σ}(dx+dy)+h^{σ}dy. Hence f = f^{σ} and h − h^{σ} = f ∈ S(V)^{G}. So f ∈ (1 − σ)(S(V)) ∩ S(V)^{G}. Let q = ∑ a_{ij}x^{i}y^{j} ∈ S(V), with a_{ij} ∈ F. Then (1 − σ)(q) = ∑ a_{ij}(x^{i} − (σ(x))^{i})y^{j}. But x^{i} − (σ(x)^{i} = x^{i} − (x + y)^{i} ∈ yS(V) implying that:

f ∈ (1 − σ)(S(V)) ∩ S(V)^{G} ⊆ yS(V) ∩ S(V)^{G} = yS(V)^{G} = (1 − σ)(−xS(V)^{G}).

Therefore f = (1 − σ)(h) = (1 − σ)(a) with a = (−x)s ∈ S(V)^{G}, s ∈ S(V)^{G}. Hence h − a = σ(h − a), showing that h = a + t, t ∈ S(V)^{G}. Consequently:

fdx + hdy = (1 − σ)(−xs)dx + ((−x)s + t)dy = s(ydx − xdy) + tdy, where s, t ∈ S(V)^{G}.

So (S(V)dx + S(V)dy)^{G} ⊆ S(V)^{G}(ydx − xdy) + S(V)^{G}dy and it is easy to verify the opposite inclusion. That is (S(V)dx + S(V)dy)^{G} = S(V)^{G}(ydx − xdy) + S(V)^{G}dy. Now since S(V)^{G}(ydx−xdy)∩S(V)^{G}dy = {0} we conclude that (S(V)dx+S(V)dy)^{G} is a direct sum of two free rank one S(V)^{G}-modules. Consequently, since S(V)^{G} is a polynomial ring we have that:

(S(V) ⊗_{F} Λ(V))^{G} = S(V)^{G} ⊕ S(V)^{G}(ydx − xdy) ⊕ S(V)^{G}dy ⊕ S(V)^{G}dx ∧ dy,

is a Cohen-Macaulay ring.

Now if (S(V) ⊗_{F} Λ(V))^{G} is one sided Gorenstein, then:

E ≡ (S(V) ⊗_{F} Λ(V))^{G}/m(S(V) ⊗_{F} Λ(V))^{G},

is a Frobenious finite dimensional F -algebra, where m = (y, x^{p} − xy^{p−1}). So, if we denote by b̄ the image of b ∈ S(V) in S(V)/mS(V), we get:

E = F ⊕ F (−x̄dy) ⊕ F dy ⊕ F dx ∧ dy.

Let M ≡ F (−x̄dy) ⊕ F dy ⊕ F dx ∧ dy. Then M is the unique maximal ideal in E and M^{2} = 0. Consequently annM = M and dim_{F} (annM) > 1. This shows, by [15, Theorem 221] or [7, Theorem 58.12], that the commutative(!) local ring E is not Gorenstein and in particular it is not Frobenius. □

4. Acknowledgements

Thanks to Yuval Ginosar for stimulating conversations.

18 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 Series 190, Cambridge University Press, Cambridge 1993.

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