Depth and Cohomological Connectivity in Modular Invariant Theory ^{∗}

Peter Fleischmann, Gregor Kemper, and R. James Shank

July 1, 2003

Abstract

Let G be a finite group acting linearly on a finite dimensional vector space V over a field K of characteristic p. Assume that p divides the order of G so that V is a modular representation and let P be a Sylow p-subgroup for G. Define the cohomological connectivity of the symmetric algebra S(V^{∗}) to be the smallest positive integer m such that H^{m}(G, S(V^{∗})) ≠ 0. We show that min {dim_{K}(V^{P}) + m + 1, dim_{K}(V)} is a lower bound for the depth of S(V^{∗})^{G}. We characterize those representations for which the lower bound is sharp and give several examples of representations satisfying the criterion. In particular, we show that if G is p-nilpotent and P is cyclic then, for any modular representation, the depth of S(V^{∗})^{G} is min {dim_{K}(V^{P}) + 2, dim_{K}(V)}.

Introduction

Let G be a finite group acting linearly on a finite dimensional vector space V over a field K of characteristic p, and assume that p divides the group order, |G|, so that V is a modular representation of G. We write R := S(V^{∗}) for the symmetric algebra of the dual of V and R^{G} for the ring of invariants. For an ideal I in R^{G} and an R^{G}-module M, grade(I, M) denotes the maximal length of an M-regular sequence whose elements lie in I. The depth of M is grade(R_{+}^{G}, M) where R_{+}^{G} is the ideal generated by all homogeneous invariants of positive degree. By the Auslander-Buchsbaum formula, dim_{K}(V) − depth(R^{G}) is the projective dimension of R^{G} as a module over a homogeneous system of parameters. Thus depth(R^{G}) determines a measure of the structural complexity of R^{G}. However, despite significant recent progress, computing depth(R^{G}) remains, in general, a difficult problem.

The cohomology of the group G with coefficients R, H^{∗}(G, R), has proven to be a useful tool in determining depth(R^{G}). Let m be the least positive integer such that H^{m}(G, R) ≠ 0. Then we call cc_{G}(R) := m the cohomological connectivity of the G-module R. In Section 1 we observe that

depth(R^{G}) ≥ min {dim_{K}(V^{P}) + cc_{G}(R) + 1, dim_{K}(V)}  (1)

where P denotes a Sylow p-subgroup of G. This gives a stronger lower bound than the one obtained by Ellingsrud and Skjelbred [4]. One might hope that the inequality is actually an equality. In fact, this was proved to be true if p^{2} does not divide |G| (see Kemper [10, Theorem 3.1]) or if G is cyclic (see Ellingsrud and Skjelbred [4] or Campbell et al. [3]). However, the following example shows that equality does not hold in general.

∗This research was supported by EPSRC grant GR/R32055/01

2000 Mathematics Subject Classification: 13A50, 20J06, 13C15.

Example 0.1. Suppose that q is a power of p and that F_{q} ⊆ K. Consider the group G = U_{n}(F_{q}) of all n × n upper triangular matrices with entries in F_{q} and ones on the diagonal. G is a p-group, and for the natural action on V = K^{n} we have dim(V^{G}) = 1. Moreover, H^{1}(G, K) ≠ 0, so cc_{G}(R) = 1, since H^{1}(G, K) is embedded into H^{1}(G, R) via the identification of K with the homogeneous component of degree 0 of R. Thus dim_{K}(V^{G})+cc_{G}(R)+1 = 3. But it is well known that R^{G} is isomorphic to a polynomial ring (see Smith [13, Proposition 5.5.5]). In particular, R^{G} is Cohen-Macaulay, so depth(R^{G}) = n. For n > 3, this is greater than dim_{K}(V^{G}) + cc_{G}(R) + 1. ◁

This paper is primarily concerned with identifying representations for which equality holds. Extending the terminology used in Kemper [10], we will call R^{G} flat if

depth(R^{G}) = min {dim_{K}(V^{P}) + cc_{G}(R) + 1, dim_{K}(V)} .

As a consequence of (1), the ring R^{G} is simultaneously flat and Cohen-Macaulay if and only if cc_{G}(R) + 1 ≥ codim_{K}(V^{P}). Thus, throughout the paper, we will assume cc_{G}(R) + 1 < codim_{K}(V^{P}).

As a matter of convenience we will assume that K is algebraically closed. Since extending K does not change the depth of R^{G}, this assumption does not limit the scope of our results.

In Section 1 we prove (1) and introduce an ideal 𝔦 (see also Fleischmann [6] and Fleischmann and Shank [7]) which plays a crucial role in the paper. In Section 2 we characterize flatness and give two conditions which together guarantee flatness. In Section 3 and Section 4 we explore these two conditions. In Section 5 we apply our results to groups with cyclic Sylow p-subgroup. In particular we prove that if G is p-nilpotent and P is cyclic then

depth(R^{G}) = min {dim_{K}(V^{P}) + 2, dim_{K}(V)} .

In Section 6 we give several examples of flat R^{G} for G with non-cyclic Sylow p-subgroup. In Section 7 we use a spectral sequence, analogous to that used by Ellingsrud and Skjelbred [4], to prove a technical result required in Section 2.

1 The Inequality and the Ideal

For a prime ideal 𝔓 ∈ Spec(R), define the inertia subgroup as

G_{𝔓} := {g ∈ G | g(f) − f ∈ 𝔓 for all f ∈ R}.

Fleischmann and Shank [7, Theorem 2.4(ii)] proved that if p does not divide the index [G : G_{𝔓}] then

depth(R^{G}) = dim(R/𝔓) + grade(𝔭, R^{G}),  (2)

where 𝔭 = R^{G} ∩ 𝔓.

Assume that p does not divide the index [G : G_{𝔓}]. Then there is a Sylow p-subgroup P ≤ G, which is contained in G_{𝔓}. Let V^{P} denote the fixed space of P in V and take ℑ := I(V^{P}) ⊂ R to be the ideal generated by all linear forms in V^{∗} that vanish on V^{P}. Then ℑ is a prime ideal which can be described as follows:

Lemma 1.1. ℑ = ((g − 1)V^{∗} | g ∈ P)R ⊂ 𝔓.

Proof. Suppose g ∈ G. Clearly (g − 1)V^{∗} ⊆ I(V^{g}). Since the rank of a matrix is equal to the rank of its transpose, rank ((g − 1) |_{V^{∗}}) = codim(V^{g}). Thus I(V^{g}) = ((g − 1) V^{∗}) R. Furthermore, V^{P} = ⋂_{g∈P} V^{g}. Thus, using the fact that the ideal associated to an intersection of varieties is the sum of the ideals associated to the varieties, we have I(V^{P}) = (∑_{g∈P} (g − 1)V^{∗})R = ((g − 1)V^{∗} | g ∈ P) R.

On the other hand, obviously P ⊆ G_{ℑ}, so p does not divide [G : G_{ℑ}]. This shows that ℑ is minimal among prime ideals of R, to which the hypothesis of(2) applies. Define 𝔦 := R^{G} ∩ ℑ. Note that, while ℑ may depend on the choice of Sylow p-subgroup, 𝔦 does not. Clearly

ht(ℑ) = dim_{K}(V) − dim_{K}(V^{P}).

Hence (2) gives

depth(R^{G}) = dim_{K}(V^{P}) + grade(𝔦, R^{G}).  (3)

Using Kemper [9, Theorem 1.4]), for any proper ideal I ⫋ R^{G} we have

grade(I, R^{G}) ≥ min {cc_{G}(R) + 1, ht(I)}  (4)

Applying (3) and (4) with I = 𝔦, yields

depth(R^{G}) ≥ dim_{K}(V^{P}) + min {cc_{G}(R) + 1, ht(𝔦)} .

Observing that ht(𝔦) = ht(ℑ) = dim_{K}(V) − dim_{K}(V^{P}) proves the following.

Theorem 1.2. depth(R^{G}) ≥ min {dim_{K}(V^{P}) + cc_{G}(R) + 1, dim_{K}(V)} .

2 Characterizing Flatness

The following theorem, which combines the scope of Fleischmann and Shank [7] with the generality of Kemper [10, Theorem 1.5], underpins our characterization of flatness.

Theorem 2.1. Let g_{1}, g_{2}, …, g_{d} ∈ R^{G} be such that the images \overline{g_{1}}, \overline{g_{2}}, …, \overline{g_{d}} form a homogeneous system of parameters in the quotient ring R^{G}/𝔦. Let M be a module over the ring End_{KG}(R) such that (g_{1}, g_{2}, …, g_{d})M ≠ M. Then (g_{1}, g_{2}, …, g_{d}) is a regular M-sequence in R^{G}. In particular depth_{R^{G}}(M) ≥ d = dim_{K}(V^{P}).

Proof. First we prove the result for the special case G = P. Let W := (V^{G})^{⊥} = {f ∈ V^{∗} | f|_{V^{G}} = 0}. Note that P acts trivially on the quotient V^{∗}/W. Hence we can apply Kemper [10, Proposition 1.2] with N = G = P, which shows that for each i there is a map Ψ_{i} ∈ End_{KG}(R), commuting with the action of the elements g_{1}, …, g_{i−1}, g_{i+1}, …, g_{d}, and satisfying Ψ_{i}(g_{i}r) = r for every r ∈ R. In other words Ψ_{i} is a left inverse to the multiplication map μ_{g_{i}} : R → R, r ↦ g_{i}r. Suppose (g_{1}, g_{2}, …, g_{s}) is a regular sequence on M with s < d, and suppose m ∈ M with g_{s+1}(m) ∈ (g_{1}, g_{2}, …, g_{s})M. Then m = Ψ_{s+1}(g_{s+1}m) ∈ (g_{1}, g_{2}, …, g_{s})M, proving that (g_{1}, g_{2}, …, g_{s+1}) is M-regular as well. Hence d = dim(R^{G}/𝔦) = dim_{K}(V^{P}) ≤ depth_{R^{P}} (M).

Now let G be arbitrary, P a Sylow p-subgroup and 𝔦_{P} := ℑ ∩ R^{P} with 𝔦 = 𝔦_{P} ∩ R^{G}. Moreover let E := End_{K}(R). Thus End_{KH}(R) = E^{H} for every subgroup H ≤ G. Note that the surjective relative transfer operator Tr_{P}^{G} : E^{P} → E^{G} induces a splitting of E^{G}- modules E^{P} = E^{G} ⊕ C. Hence N := E^{P} ⊗_{E^{G}} M is an E^{P} -module, such that the restricted E^{G}-module has M as a direct summand. Assume that g_{1}, g_{2}, …, g_{d} are elements such that the images \overline{g_{1}}, \overline{g_{2}}, …, \overline{g_{d}} form a homogeneous system of parameters in the quotient ring R^{G}/𝔦. Since R^{P} /𝔦_{P} is a finite extension of R^{G}/𝔦, these images also form a homogeneous system of parameters in R^{P} /𝔦_{P}, and therefore the originals form an N -regular sequence. Since M is a direct summand of N_{|E^{G}}, the sequence (g_{1}, g_{2}, …, g_{d}) is also M-regular.

Corollary 2.2. For every nonzero element α ∈ H^{m}(G, R) one has Ann_{R^{G}}(α) ≤ 𝔦.

Proof. It is obvious from the definition of cohomology, that H^{m}(G, R) is a module over End_{KG}(R). Let x ∈ R^{G} \ 𝔦, then \overline{x} is contained in a homogeneous system of parameters of R^{G}/𝔦 and therefore acts regularly on H^{m}(G, R). Hence x ∉ Ann_{R^{G}}(α).

The following conceptual characterization of flatness in terms of the prime ideal 𝔦 will guide further investigation.

Theorem 2.3. Assume that m := cc_{G}(R) < codim_{K}(V^{P}) − 1. Then the following are equivalent:

(a) R^{G} is flat;

(b) The ideal 𝔦 is an associated prime of the R^{G}-module H^{m}(G, R);

(c) H^{m}(G, R) contains an R^{G}-submodule of Krull dimension dim_{K}(V^{P}).

If these conditions are satisfied then depth_{R^{G}}(H^{m}(G, R)) = dim_{K}(V^{P}).

Proof. (a) ⇒ (b): From Theorem 7.1 we see that grade(𝔦, R^{G}) = m + 1 implies that grade(𝔦, H^{m}(G, R)) = 0, hence 𝔦 consists of zero-divisors on H^{m}(G, R) and therefore lies in some associated prime p := Ann_{R^{G}}(α) with α ∈ H^{m}(G, R) \ {0}. On the other hand, Corollary 2.2 implies that Ann_{R^{G}}(α) ⊆ 𝔦. Hence 𝔦 = p.

(b) ⇒ (c): Take the submodule generated by α.

(c) ⇒ (a): Let Y ≤ H^{m}(G, R) be an R^{G}-submodule of Krull dimension dim_{K}(V^{P}). Then there is an associated prime p of Y such that the Krull dimensions of the modules R^{G}/p, Y and R^{G}/𝔦 coincide. Again Corollary 2.2 shows that p = 𝔦, which implies that grade(𝔦, H^{m}(G, R)) = 0. Hence grade(𝔦, R^{G}) = m + 1, by Theorem 7.1. If 𝔦 is an associated prime of H^{m}(G, R), then the R^{G}-depth of this module is less than or equal to dim R^{G}/𝔦 = dim_{K}(V^{P}) (see Bruns and Herzog [2, Proposition 1.2.13]). On the other hand it follows from Kemper [10, Theorem 2.13]), or from Corollary 2.2, that this depth is also greater than or equal to dim_{K}(V^{P}).

While we are aware of a number of instances were the existence of a suitable positive degree cohomology class provides a sufficient condition determining the depth of R^{G}, we believe that Theorem 2.3 is the first example of a theorem giving a condition on the positive degree cohomology which is necessary as well as sufficient.

Suppose m := cc_{G}(R) and α ∈ H^{m}(G, R) \ {0}. Then the variety in V associated to Ann_{R^{G}}(α) is

V(α) := {x ∈ V | f(x) = 0 for all f ∈ Ann_{R^{G}}(α)}.

Similarly the variety in V associated to the prime ideal 𝔦 = ℑ ∩ R^{G} is

V(𝔦) = ⋃_{P′} V^{P′},  (5)

where the union runs through all Sylow p-subgroups P′ ≤ G. The condition V(α) ⊆ ⋃_{P′} V^{P′} implies 𝔦 ⊆ \sqrt{Ann_{R^{G}}(α)} and therefore, by Corollary 2.2 , we have equality in both inclusions. Using this we obtain the following geometric version of Theorem 2.3.

Theorem 2.4. Assume that m := cc_{G}(R) < codim_{K}(V^{P}) − 1 and α ∈ H^{m}(G, R) \ {0}. Then the following are equivalent:

(a) V(α) ⊆ V(𝔦);

(b) V(α) = V(𝔦);

(c) \sqrt{Ann_{R^{G}}(α)} = 𝔦.

Furthermore, R^{G} is flat if and only if there exists α ∈ H^{m}(G, R) \ {0} satisfying (a), (b) or (c).

Proof. Using Theorem 2.3 it suffices to note that \sqrt{Ann_{R^{G}}(α)} = 𝔦 implies that 𝔦 is an associated prime of H^{m}(G, R). However this is a standard fact about Noetherian rings and modules.

In order to apply Theorem 2.4 we need to find classes α ∈ H^{m}(G, R) \ {0} satisfying V(α) ⊆ V(𝔦) or, equivalently, 𝔦 ⊆ \sqrt{Ann_{R^{G}}(α)}. By Lorenz and Pathak [12, Lemma 1.3], elements in the image of the relative transfer, Tr_{H}^{G}, annihilate cohomology classes in the kernel of the restriction, res_{G,H}, and, by Fleischmann [6, 12.5], 𝔦 is the radical of a sum of relative transfer ideals.

Recall that for a subgroup H ≤ G the relative transfer is the R^{G}-module homomorphism Tr_{H}^{G} : R^{H} → R^{G}, r ↦ ∑_{g} g(r), where g runs through a set of representatives of H-cosets satisfying G = ⊎ gH. This definition is independent of the choice of representatives and the image Tr_{H}^{G}(R^{H}) is an ideal in R^{G}. For any set χ of subgroups of G we define

𝔦_{χ} := ∑_{X∈χ} Tr_{X}^{G}(R^{X}) ◁ R^{G}.

This is a proper ideal of R^{G} if and only if p divides all indices [G : X] for X ∈ χ. The varieties of relative transfer ideals Tr_{H}^{G}(R^{H}) have been described in Fleischmann [6, Theorem 12.4] and Lorenz and Pathak [12, Lemma 1.1]:

V(Tr_{H}^{G}(R^{H})) = {v ∈ V | p | [G_{v} : G_{v} ∩ ^{g}H] for all g ∈ G}  (6)

where ^{g}H := gHg^{−1}. Note that, if p divides [G : H], then H does not contain a Sylow p-subgroup of G and V(i) ⊆ V(Tr_{H}^{G}(R^{H})). Therefore \sqrt{𝔦_{χ}} ⊆ 𝔦 whenever \sqrt{𝔦_{χ}} ⫋ R^{G}. Furthermore, 𝔦 = \sqrt{𝔦_{p−prop}}, where “p-prop” denotes the set of all proper subgroups of Sylow p-subgroups of G (see Fleischmann [6, 12.5] or Theorem 3.1 below). For a family χ of subgroups of G, define

K_{χ}^{j} := K_{χ}^{j}(G) := ⋂_{X∈χ} ker ((res_{G,X}) |_{H^{j}(G,R)} )

and K_{χ} := ⨁_{j>1} K_{χ}^{j}. The following lemma is an immediate consequence of Lorenz and Pathak [12, Lemma 1.3].

Lemma 2.5. For each family χ of subgroups of G one has

𝔦_{χ} ⊆ Ann_{R^{G}}(K_{χ}).

This leads to the following criterion of flatness.

Theorem 2.6. For m = cc_{G}(R) and χ a family of subgroups of G, if \sqrt{𝔦_{χ}} = 𝔦 and K_{χ}^{m} ≠ 0, then R^{G} is flat.

Proof. Lemma 2.5 and K_{χ}^{m} ≠ 0 imply \sqrt{𝔦_{χ}} ⊆ \sqrt{Ann_{R^{G}}(α)} ⊆ 𝔦. Using the first condition gives \sqrt{Ann_{R^{G}}(α)} = 𝔦. Hence the hypothesis (c) of Theorem 2.4 holds.

3 The Geometry of the Ideal 𝔦

In this section we use the geometry of 𝔦 to investigate families of subgroups, χ, for which \sqrt{𝔦_{χ}} = 𝔦. This is an important first step in the use of Theorem 2.6. Since Tr_{^{g}H}^{G}(R ^{^{g}H}) = Tr_{H}^{G}(R^{H}) for every g ∈ G and Tr_{H}^{G}(Tr_{U}^{H}(R^{U})) = Tr_{U}^{G}(R^{U}) for U ≤ H, the ideal \sqrt{𝔦_{χ}} does not change if we adjoin to χ all subgroups of conjugates of X ∈ χ. Similarly K_{χ} does not change either, hence we can, if we wish, assume that χ is closed under taking conjugates and subgroups. Let χ′_{p} denote the set of p-subgroups of G not in χ.

Theorem 3.1. If χ is closed under taking conjugates and subgroups, then

V(𝔦_{χ}) = {v ∈ V | p | [G_{v} : G_{v} ∩ X] for all X ∈ χ} = ⋃_{Q∈χ′_{p}} V^{Q}.

Proof. The first equality follows from Equation (6) using

V(𝔦_{χ}) = V (∑_{X∈χ} Tr_{X}^{G}(R^{X})) = ⋂_{X∈χ} V (Tr_{X}^{G}(R^{X})) .

We now prove the second equality. To show the first inclusion, suppose v ∈ V(𝔦_{χ}) and let P_{v} denote a Sylow p-subgroup of the isotropy subgroup G_{v}. Since p divides [G_{v} : G_{v} ∩ X] for all X ∈ χ, it is clear that P_{v} ∉ χ. Thus v ∈ V^{P_{v}} ⊆ ⋃_{Q∈χ′_{p}} V^{Q}.

To show the reverse inclusion, let v ∈ V^{Q} for some Q ∈ χ′_{p} and assume, by way of contradiction, that v ∉ V(𝔦_{χ}). Then there exists H ∈ χ such that p does not divide [G_{v} : G_{v} ∩ H]. Thus H contains a Sylow p-subgroup of G_{v}, say P_{v}. Since χ is closed under taking subgroups, P_{v} ∈ χ. However, Q is a p-subgroup of G_{v}. Therefore some conjugate of Q is contained in P_{v}. Again, since χ is closed under conjugation and taking subgroups, we arrive at the contradiction Q ∈ χ.

Corollary 3.2. For j ∈ {1, 2}, let χ_{j} be closed under conjugation and taking subgroups. Then

V(𝔦_{χ_{1}}) ⊆ V(𝔦_{χ_{2}})

if and only if the following holds:

For each p-group Q ∈ χ_{2} \ χ_{1} there exists S ∈ (χ_{2})′_{p}, such that Q ⫋ S and V^{Q} = V^{S}.

Proof. “⇒”: Suppose V(𝔦_{χ_{1}}) ⊆ V(𝔦_{χ_{2}}) and Q is a p-group in χ_{2}\ χ_{1}. Applying Theorem 3.1 gives V^{Q} ⊆ V(𝔦_{χ_{1}}). Thus V^{Q} ⊆ V(𝔦_{χ_{2}}). Hence for each v ∈ V^{Q}, p divides [G_{v} : G_{v} ∩ Q] and there is a Sylow p-subgroup of G_{v}, say P_{v}, with Q ⫋ P_{v}. As v ∈ V(𝔦_{χ_{2}}), using the first equality of Theorem 3.1, we have P_{v} ∉ χ_{2}. Clearly for any v ∈ V we have V^{P_{v}} ⊂ V^{Q}. Furthermore V^{Q} ⊆ ⋃_{v∈V^{Q}} V^{P_{v}}. Thus, since the finite group G has only a finite number of subgroups, V^{Q} = ⋃_{v∈S} V^{P_{v}} for some finite S ⊆ V^{Q}. Since K is algebraically closed and therefore an infinite field, we conclude V^{Q} = V^{P_{v′}} for some Q ⫋ P_{v′} ∈ (χ_{2})′_{p}. “⇐”: From Theorem 3.1,

V(𝔦_{χ_{1}}) = ⋃_{Q∈(χ_{1})′_{p}} V^{Q}.

However, (χ_{1})′_{p} is the disjoint union of (χ_{1})′_{p}∩(χ_{2})′_{p} and (χ_{1})′_{p}∩χ_{2}. If Q ∈ (χ_{1})′_{p}∩χ_{2}, then by hypothesis V^{Q} = V^{S} for some S ∈ (χ_{2})′_{p}. Therefore, for any Q ∈ (χ_{1})′_{p}, V^{Q} ⊆ ⋃_{S∈(χ_{2})′_{p}} V^{S} and the result follows from Theorem 3.1.

Lemma 3.3. Let P be a Sylow p-subgroup of G and let Q be a proper subgroup of P. Define G_{V^{Q}} := {g ∈ G | gw = w ∀w ∈ V^{Q}}. Then V^{P} ⫋ V^{Q} if and only if p | [G : G_{V^{Q}}].

Proof. First observe that V^{P} ⊆ V^{Q} and Q ≤ G_{V^{Q}}.

Suppose p does not divide [G : G_{V^{Q}}]. Then G_{V^{Q}} contains a Sylow p-subgroup of G, say P′. Thus V^{P} ⊆ V^{Q} ⊆ V^{P′}. However, P and P′ are conjugate so dim(V^{P}) = dim(V^{P′}). Therefore V^{P} = V^{Q}.

Suppose p | [G : G_{V^{Q}}]. Then P \ G_{V^{Q}} is non-empty and there exists g ∈ P and v ∈ V^{Q} with gv ≠ v. Therefore V^{P} ⫋ V^{Q}.

In the following we adopt the notation X ≤_{G} H ( X <_{G} H ) to mean ^{g}X ≤ H ≤ G (^{g}X < H ) for some g ∈ G.

Proposition 3.4. Let H_{1}, H_{2}, …, H_{k} ≤ G with p | [G : H_{i}] for all i and let P be a fixed Sylow p-subgroup of G. Then the following are equivalent:

(a) 𝔦 = \sqrt{𝔦_{\{H_{1}, H_{2}, ..., H_{k}\}}};

(b) ∀Q < P with V^{P} ⫋ V^{Q}: Q ≤_{G} H_{i} for some i ∈ {1, …, k};

(c) ∀Q < P with p | [G : G_{V^{Q}}]: Q ≤_{G} H_{i} for some i ∈ {1, …, k}.

Proof. Let χ_{1} := {X ≤ G |∃ i ∈ {1, …, k} : X ≤_{G} H_{i}} and χ_{2} := {Q ≤ G | Q <_{G} P }. From Theorem 3.1 and Equation (5) we see that V(𝔦_{χ_{2}}) = V(𝔦). From Theorem 3.1 and Equation (6) we see that V(𝔦_{χ_{1}}) = V (𝔦_{{H_{1}, H_{2}, ..., H_{k}}}). Moreover, since all H_{i}’s have index divisible by p, we know that V(𝔦) ⊆ V(𝔦_{χ_{1}}). From Corollary 3.2 and Lemma 3.3 we see, that the reverse inclusion is equivalent to either one of (b) or (c). Now the statement follows from Hilbert’s Nullstellensatz.

Remark 3.5.

(a) For any subgroup H ≤ G we have 𝔦 = \sqrt{𝔦_{H}} if and only if every p-subgroup Q ≤ G with p | [G : G_{V^{Q}}] satisfies Q ≤_{G} H. If H is a normal subgroup then this condition is equivalent to the fact that H contains all p-elements g ∈ G such that p divides the index [G : G_{V^{g}}], where G_{V^{g}} is the pointwise stabilizer of the fixed point space of g. In particular the subgroup M := ⟨g ∈ G | g a p − element with p | [G : G_{V^{g}}]⟩ ◁ G is the unique minimal normal subgroup of G with 𝔦 = \sqrt{𝔦_{M}}.

(b) Let P be a fixed Sylow p-subgroup of G. Without loss of generality, we can assume that the subgroups H_{1}, H_{2}, …, H_{k} appearing in Proposition 3.4 are proper subgroups of P. Indeed, by Mackey’s formula we have for each r ∈ R^{H_{i}}: Tr_{H_{i}}^{G}(r) = ∑_{g} Tr_{P ∩ ^{g}H_{i}}^{P}(gr), where g runs through a system of representatives of double cosets P gH_{i} in G. Hence

[G : P] Tr_{H_{i}}^{G}(r) = Tr_{P}^{G}(Tr_{H_{i}}^{G}(r)) = ∑_{g} Tr_{P ∩ ^{g}H_{i}}^{G}(gr)

and we see that

\sqrt{𝔦_{\{H_{1}, H_{2}, ..., H_{k}\}}} = \sqrt{𝔦_{\{P ∩ ^{g}H_{i} | g∈G\}}}.

Example 3.6. Suppose that we are in the “mildly modular” case, i.e., p^{2} does not divide |G|. Take a non-identity p-element g ∈ G. Then ⟨g⟩ ≤ G has order p and therefore is a Sylow p-subgroup. But we certainly have ⟨g⟩ ≤ G_{V^{g}}, hence the index [G : G_{V^{g}}] is not divisible by p. Therefore, by Proposition 3.4, we can take χ = {1} in Theorem 2.6. Since the restriction is non-injective , K_{χ}^{m} ≠ 0. Thus R^{G} is flat and we have recovered Theorem 3.1 of Kemper [10]. ◁

4 Restricting Cohomology

Our primary tool for identifying flat representations is Theorem 2.6. Proposition 3.4 allows us to identify suitable families of subgroups. In this section we address the problem of identifying a suitable non-zero cohomology class. It is convenient to assume that the elements of χ = {H_{1}, …, H_{k}} are proper subgroups of one Sylow p-subgroup, P, of G. We can then attempt to determine the existence of α ∈ K_{χ}^{m} \ {0} locally, i.e. on the level of P. In principle this can be done, using the fact that the restriction map res_{G,P} : H^{∗}(G, R) → H^{∗}(P, R) is injective and its image consists of the stable cohomology

H^{∗}(P, R)_{G−stab} := {η ∈ H^{∗}(P, R) | res_{P,P ∩gP g^{−1}}(η) = res _{^{g}P,P ∩gP g^{−1}}(gη), g ∈ P \G/P },

where P \G/P denotes a system of P −P double coset representatives (see Benson [1, Proposition 3.8.2] or Evens [5, Corollary 4.2.7]). Note also, that the transfer (or corestriction) Tr_{P}^{G} from H^{m}(P, R) to H^{m}(G, R) induces an isomorphism H^{m}(P, R)_{G−stab} ≅ H^{m}(G, R).

Lemma 4.1. Let P be a Sylow p-subgroup of G with normalizer quotient N := N_{G}(P)/P, let χ := {Q_{1}, Q_{2}, …, Q_{k}} be a family of proper subgroups of P and let

χ̂ := χ ∪ {P ∩^{g} P | g ∈ G \ N_{G}(P)}.

Then

H^{m}(P, R)^{N} ∩ Kχ̂(P) = H^{m}(P, R)_{G−stab} ∩ Kχ̂(P) ⊆ H^{m}(P, R)_{G−stab} ∩ K_{χ}(P) ≅ K_{χ}^{m}(G),

where the last isomorphism is induced by transfer and restriction.

Proof. Since H^{m}(P, R)_{G−stab} ≅ H^{m}(G, R) and G acts trivially on H^{m}(G, R), we have H^{m}(P, R)_{G−stab} ⊆ H^{m}(P, R)^{N} and H^{m}(P, R)_{G−stab} ∩ Kχ̂(P) ⊆ H^{m}(P, R)^{N} ∩ Kχ̂(P). Suppose η ∈ H^{m}(P, R)^{N} ∩ Kχ̂(P) and g ∈ G. If g ∈ N_{G}(P) then the stability condition becomes η = gη and is satisfied because η ∈ H^{m}(P, R)^{N}. If g ∈ G \ N_{G}(P), then, since η ∈ Kχ̂(P), we have res_{P,P ∩gP g^{−1}}(η) = 0 and res _{^{g}P,P ∩gP g^{−1}}(gη) = g(res_{P,g^{−1}P g∩P} (η)) = 0. Thus again the stability condition is satisfied. Therefore η ∈ H^{m}(P, R)_{G−stab} and H^{m}(P, R)^{N} ∩ Kχ̂(P) = H^{m}(P, R)_{G−stab} ∩ Kχ̂(P).

Since χ ⊂ χ̂, we have Kχ̂(P) ⊆ K_{χ}(P). Therefore

H^{m}(P, R)_{G−stab} ∩ Kχ̂(P) ⊆ H^{m}(P, R)_{G−stab} ∩ K_{χ}(P).

Suppose ρ ∈ H^{m}(P, R)_{G−stab} ∩ K_{χ}(P). Using the Mackey formula in cohomology (see Benson [1, Proposition 3.6.16]), we get

res_{G,Q_{i}} (Tr_{P}^{G}(ρ)) = ∑_{g∈Q_{i}\G/P} Tr_{Q_{i}∩ ^{g}P}^{Q_{i}} (res _{^{g}P,Q_{i}∩ ^{g}P} (gρ)) = 0,

because

res _{^{g}P,Q_{i}∩ ^{g}P} (gρ) = res_{P ∩ ^{g}P,Q_{i}∩ ^{g}P} (res _{^{g}P,P ∩ ^{g}P} (gρ))
= res_{P ∩ ^{g}P,Q_{i}∩ ^{g}P} (res_{P,P ∩ ^{g}P} (ρ))
= res_{P,Q_{i}∩ ^{g}P} (ρ) = 0.

The fact that res_{G,P} maps K_{χ}^{m}(G) onto H^{m}(P, R)_{G−stab}∩K_{χ}(P) follows from the transitivity of restriction.

We now describe a way to construct suitable N-stable elements in H^{m}(P, R)^{N} ∩Kχ̂(P), whenever the restricted P -module R_{|P} contains a suitable direct summand of ‘induced type’.

Lemma 4.2. Keep the notation from Lemma 4.1 and let W be a KP -module. Suppose that τ ∈ H^{m}(P, W) \ {0} with res_{P,X}(τ) = 0 for all proper subgroups X < P. Assume that the restricted KP -module R_{|P} has a direct summand, isomorphic to the restricted induced module (W^{↑N_{G}(P)})_{|P}. Then there exists a stable element ρ ∈ H^{m}(P, R)^{N} ∩ Kχ̂(P) \ {0}.

Proof. Let i and j denote the embeddings of the direct summands

H^{m}(P, W) ↪ H^{m}(P, W^{↑N_{G}(P)}|P) and H^{m}(P, W^{↑N_{G}(P)}|P) ↪ H^{m}(P, R),

respectively and define ρ := j ∘ res_{N_{G}(P),P} ∘ Tr_{P}^{N_{G}(P)} ∘i(τ). Note that the map

Tr_{P}^{N_{G}(P)} ∘i : H^{m}(P, W) → H^{m}(N_{G}(P), W^{↑N_{G}(P)})

is an isomorphism, due to the Eckmann-Shapiro lemma. Since N_{G}(P)/P has order prime to p, the restriction

res_{N_{G}(P),P} : H^{m}(N_{G}(P), W^{↑N_{G}(P)}) → H^{m}(P, W^{↑N_{G}(P)})^{N}

is an isomorphism (see Evens [5, Exercise 7.2.3. pg 77]) and j in injective. This shows that ρ lies in H^{m}(P, R)^{N} and is non-zero. Moreover

res_{P,X}(ρ) = res_{N_{G}(P),X} Tr_{P}^{N_{G}(P)}(i(τ)) = ∑_{n∈N} res_{P,X}(nτ)
= ∑_{n∈N} ^{n}(res_{P, ^{n^{−1}}X}(τ)) = 0.

Thus ρ ∈ H^{m}(P, R)^{N} ∩ Kχ̂(P) \ {0}.

Proposition 4.3. Let P be a Sylow p-subgroup of G, let m = cc_{G}(R), and let W be a KP -module. Suppose that τ ∈ H^{m}(P, W) \ {0} with res_{P,X}(τ) = 0 for all proper subgroups X < P. Suppose that the restricted KP -module R_{|P} has a direct summand isomorphic to the restricted induced module (W^{↑N_{G}(P)})_{|P}. Then R^{G} is flat.

Proof. Take χ to be the set of all proper subgroups of P. Then by Proposition 3.4, the first condition in Theorem 2.6 is satisfied. The second condition follows from Lemma 4.2 and Lemma 4.1.

5 Cyclic Sylow p-subgroups

In this section we apply our techniques to groups with cyclic Sylow p-subgroup. We start by recording two known facts about group cohomology:

Theorem 5.1. Let P = ⟨σ⟩ be a cyclic group of order z, let α ∈ Aut(P) be an automorphism and let M be a P >◁⟨α⟩-module. Then

H^{i}(P, M) = \begin{cases}M^{P} / Tr^{P} (M) & if i > 0 is even, \\ ker(Tr^{P} |_{M})/(σ − 1)M & if i is odd, \end{cases}

and the induced (right) action of α on H^{∗}(P, M) is given by ρ^{α} = α^{−1}(k^{i} · ρ), where α(σ) = σ^{k} with gcd(k, z) = 1 and deg(ρ) ∈ {2i, 2i − 1}.

Proof. See Evens [5, p. 6].

Theorem 5.2. Let G be any group and K a field of characteristic p. Then

H^{1}(G, K) ≅ Hom(G, Z/p) ⊗_{F_{p}} K,

where F_{p} denotes the prime field of characteristic p. In particular H^{1}(G, K) ≠ 0 if and only if G has a normal subgroup of index p.

Proof. See Landrock [11, Corollary 10.13].

Throughout the rest of this section we assume that P is a cyclic Sylow p-subgroup of G. We will use N to denote the normalizer quotient N_{G}(P)/P. Recall that a finite group G is called p-nilpotent if G has a normal subgroup H of order not divisible by p such that G/H is a p-group.

Theorem 5.3. Let G be a finite group with cyclic Sylow p-subgroup P. Then the following are equivalent:

(a) H^{1}(G, K) ≠ 0;

(b) G is p-nilpotent;

(c) N_{G}(P) = C_{G}(P).

If G acts linearly on a finite-dimensional vector space V over a field K of characteristic p and one of (a), (b) or (c) holds, then R^{G} is flat and

depth (R^{G}) = min {dim_{K}(V^{P}) + 2, dim_{K}(V)} .  (7)

Proof. By Theorem 5.2, (a) implies that there is M ◁ G with G/M of order p. Therefore M ∩ P = Φ(P), the Frattini group of P and M is p-nilpotent by Huppert [8, Chap IV, 4.7]. Hence M = (M ∩ P) · H with normal p′-subgroup H, which in fact is characteristic in M and hence normal in G. It follows that G = P · H is p-nilpotent. On the other hand, if G = P ·H is p-nilpotent, then G/H ≅ P has a factor group of order p, hence H^{1}(G, K) ≠ 0, by Theorem 5.2. So (a) and (b) are equivalent. “(b) ⇒ (c)”: Let G = P · H with normal p′-complement H. Then N_{G}(P) = P N_{H}(P) = P × N_{H}(P) = P × C_{H}(P) = C_{G}(P). “(c) ⇒ (a)”: This implication is Burnside’s theorem (see Huppert [8, Chap IV, 2.6]), which is valid for abelian P.

Suppose that G is p-nilpotent. We apply Lemma 4.1 with χ̂ = χ = {Φ(P)} and m = 1. Notice that 0 ≠ H^{1}(G, K) ↪ H^{1}(G, R), where the embedding is induced from the embedding K ↪ R = K ⊕ R_{+}. From the Lyndon-Hochschild-Serre spectral sequence we get the exact sequence

0 → K ≅ H^{1}(P/Φ(P), K) → H^{1}(P, K) → H^{1}(Φ(P), K)^{P/Φ(P)},

where the first map is inflation and the second one is restriction. Let η denote a non-zero element of H^{1}(P/Φ(P), K). Since N_{G}(P) = C_{G}(P), the element ρ := inf(η) ∈ H^{1}(P, K) \ {0} is N_{G}(P)/P -stable. Furthermore res_{P,Φ(P)}(ρ) = 0. Thus, by Lemma 4.1, the image of ρ is a non-zero element in K_{χ}^{1}. Clearly χ satisfies condition (b) of Proposition 3.4. Thus an application of Theorem 2.6 finishes the proof.

Remark 5.4.

(a) There are few examples in the literature where purely group theoretic properties of G allow the computation of the depth of the ring of invariants. The first such instance is the result of Ellingsrud and Skjelbred [4], who showed that (7) holds if G is a cyclic p-group. This was extended by Campbell et al. [3] to the case where G is abelian and has a cyclic Sylow p-subgroup (see Theorems 1 and 9 in [3]). Corollary 5.3 clearly generalizes this latter result.

(b) A pair (G, V) of a finite group G and a finitely generated KG-module V was called “flat” by Kemper [10] if (7) holds, or in our notation here, if cc_{G}(R) = 1 and R^{G} is flat. Thus we have proved that for a p-nilpotent group with cyclic Sylow p-subgroup, (G, V) is flat for every representation V in characteristic p.

(c) For p = 2, condition (c) from Theorem 5.3 is always satisfied, since C_{G}(H) contains P and therefore N_{G}(H)/C_{G}(H) has odd order, but on the other hand Aut(H) is a 2-group since H is a cyclic 2-group. Thus for p = 2 we can omit the condition of p-nilpotency in Theorem 5.3. ◁

If G is not p-nilpotent but P is cyclic, we can use Lemma 4.1 and Proposition 3.4 to obtain the following.

Proposition 5.5. Let Q := Φ(P) be the maximal subgroup of P and m := cc_{G}(R). Assume that there is non-zero ρ ∈ H^{m}(P, R)^{N} with res_{P,Q}(ρ) = 0. Then R^{G} is flat.

Corollary 5.6. Suppose W is a KP -module such that there exists ρ ∈ H^{1}(P, W^{∗}) \ {0} with res_{P,Q}(ρ) = 0. Then the invariant ring with respect to the induced module V := W^{↑G} is flat with:

depth (S(V^{∗})^{G}) = min {dim_{K}(V^{P}) + 2, dim_{K}(V)} .

Proof. Note that V^{∗} ≅ W^{∗↑G}, so (W^{∗↑N_{G}(P)})_{|P} is a summand of the module R_{|P}. Hence the claim follows from Proposition 4.3.

Alternatively we can use Proposition 5.5 and the following explicit construction of a suitable element in H^{1}(P, W^{∗↑N_{G}(P)})^{N}: For each n ∈ N define k_{n} ∈ (Z/p)^{∗} by ^{n}σ = σ^{k_{n}}. Then η := ∑_{n∈N} n⊗k_{n} ·ρ ∈ H^{1}(P, W^{∗↑N_{G}(P)})^{N}, which is embedded in H^{1}(P, V^{∗})^{N}. Since res_{P,Q}(η) = ∑_{n∈N} n ⊗ k_{n} · res_{P,Q}(ρ) = 0, we see from Lemma 4.1, that m = cc_{G}(R) = 1 and the result follows from Proposition 5.5.

Example 5.7. Let G be the finite simple group SL_{2}(8) and p = 3. Then |G| = 7 · 8 · 9, the Sylow 3-group P is cyclic with P ∩ ^{g}P = 1 for each g ∉ N and N_{G}(P) is dihedral of order 18. Let V be the permutation module V := (K_{↓P})^{↑G}. Then V_{↓P} ≅ K^{(2)} ⊕ K[P]^{(6)}. Hence dim_{K}(V^{P}) = 8 and

depth (S(V^{∗})^{G}) = min {dim_{K}(V^{P}) + 2, dim_{K}(V)} = 10 < dim_{K}(V) = 56.
◁

6 Further Applications

The purpose of this section is to provide several examples where R^{G} is flat and P is non-cyclic. Throughout we take χ to be the set of all proper subgroups of P. Thus, by Proposition 3.4, the first condition of Theorem 2.6 is satisfied.

Proposition 6.1. Suppose that W is P -module and there exists ρ ∈ H^{1}(P, W) \ {0} which restricts to zero in H^{1}(M, W) for all maximal subgroups M ≤ P.

(a) If W is P -module summand of R, then R^{P} is flat.

(b) If W^{↑G} is a summand of R, then R^{G} is flat and cc_{G}(R) = 1.

Proof. Part (a) follows immediately from Theorem 2.6 and part (b) is a consequence of Proposition 4.3.

Suppose that W is a P -module. In order to apply the proposition, we need to identify non-zero elements in H^{1}(P, W) which restrict to zero for all maximal subgroups. Let N and M denote maximal subgroups of P. Note that N and M are normal in P. The inflation map gives an injection from H^{1}(P/N, W^{N}) to H^{1}(P, W) (see, for example, Evens [5, Corollary 7.2.3]). Note that P/N is isomorphic to the cyclic group of order p and W^{N} is a vector space over a field of characteristic p. Thus, unless W^{N} is a projective P/N-module, H^{1}(P/N, W^{N}) is non-zero. Elements in H^{1}(P/N, W^{N}) can be represented by vectors in W^{N} which are in the kernel of the transfer (see Theorem 5.1). Assume W^{N} is not projective and choose u ∈ W^{N} so that the equivalence class, {u}, is non-zero in H^{1}(P/N, W^{N}). Clearly the image of {u} under the inflation map restricts to zero in H^{1}(N, W). We will formulate a necessary and sufficient condition for the inflation of {u} to restrict to zero in H^{1}(M, W). Note that M/(M ∩ N) is also isomorphic to the cyclic group of order p so that elements of H^{1}(M/(M ∩ N), W^{M∩N}) can be represented by vectors in W^{M∩N}.

Lemma 6.2. Suppose that u ∈ W^{N} represents a non-zero element, {u}, in H^{1}(P/N, W^{N}). Then the image of {u} in H^{1}(M, W), under the composition of inflation followed by restriction, is zero if and only if u represents zero in H^{1} (M/(M ∩ N), W^{M∩N}).

Proof. The inflation map is induced by the group epimorphism P → P/N and the KP - module monomorphism W^{N} → W. The group epimorphism factors as P → P/(M ∩ N) → P/N while the KP -module monomorphism factors as W^{N} → W^{M∩N} → W. Thus the inflation map factors as H^{1}(P/N, W^{N}) → H^{1}(P/(M ∩ N), W^{M∩N}) → H^{1}(P, W). The composition H^{1}(P/(M ∩ N), W^{M∩N}) → H^{1}(P, W) → H^{1}(M, W) is induced by the group homomorphism M → P → P/(M ∩N) and the KP -module monomorphism W^{M∩N} → W. The group homomorphism also factors as M → M/(M ∩N) → P/(M ∩N) giving a second factorisation H^{1}(P/(M ∩ N), W^{M∩N}) → H^{1}(M/(M ∩ N), W^{M∩N}) → H^{1}(M, W). Thus the following diagram commutes:

H^{1}(P/N, W^{N}) ⟶  H^{1}(P, W)  ⟶  H^{1}(M, W)
↘  ↑  ↑
H^{1} (P/(M ∩ N), W^{M∩N}) ⟶ H^{1} (M/(M ∩ N), W^{M∩N}) .

Since the inflation map H^{1}(M/(M ∩ N), W^{M∩N}) → H^{1}(M, W) is injective , the kernel of H^{1}(P/N, W^{N}) → H^{1}(M, W) coincides with the kernel of the map from H^{1}(P/N, W^{N}) to H^{1}(M/ (M ∩ N), W^{M∩N}). This map takes the element represented by u in H^{1}(P/N, W^{N}) to the element represented by u in H^{1}(M/ (M ∩ N), W^{M∩N}).

Suppose that P/P′ is an elementary abelian p-group. Choose α ∈ P \ P′. Then there is a maximal subgroup of P, say N, such that P/N = ⟨αN⟩. Since P/P′ is an elementary abelian p-group, we have P/P′ = ⟨αP′⟩ × N/P′. If M is a second maximal subgroup of P, then there exists γ ∈ N such that αγ ∈ M. Since M/M ∩ N is isomorphic to Z/p, we have M/M ∩ N = ⟨αγM ∩ N⟩. Define \overline{α} := αP′ and \overline{γ} := γP′. To apply our technique, we need to find a u ∈ W^{N} which represents a non-zero class in H^{1}(⟨\overline{α}⟩, W^{N}) and represents the zero class in H^{1}(⟨\overline{α} · \overline{γ}⟩, W^{M∩N}) for all M and corresponding γ. If P/P′ is a rank 2 elementary abelian p-group, then M ∩ N = P′ and γ = β^{i} for a fixed β with i = 0, …, p − 1. Example 6.3. Suppose P/P′ is a rank 2 elementary abelian p-group. Choose ω ∈ F_{q} \ F_{p} and let U be a two dimensional vector space over F_{q} with the action of P/P′ given by

\overline{α} = ( \begin{matrix}1 & ω \\ 0 & 1 \end{matrix} ) and \overline{β} = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) .

Take W^{P′} to be S^{ℓ}(U) with ℓ < p. Then W^{P′} is an ℓ + 1 dimensional indecomposable representation of ⟨\overline{αβ^{i}}⟩ ≅ Z/p and W^{N} is the one dimensional fixed point space. Thus any non-zero element of W^{N} represents a non-zero class in H^{1}(⟨\overline{α}⟩, W^{N}) which maps to zero in H^{1}(⟨\overline{αβ^{i}}⟩, W^{P′}).

Various choices for P and W give rise to this action of P/P′ on W^{P′}:

(i) W = W^{P′} and P′ = 1. Then P ≅ Z/p × Z/p. If we take G = GL_{2}(F_{p^{2}}) and V^{∗} = W^{↑G} then, by Proposition 6.1(b), R^{G} is flat with cc_{G}(R) = 1. Thus dim(R^{G}) = (ℓ + 1)(p^{4} − 1)(p^{2} − 1) and depth(R^{G}) = 3. (ii) Take ℓ = 1 and consider the action on W determined by

α = ( \begin{matrix}1 & ω & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) and β = ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) .

In this case, as long as p > 2, P ≅ (Z/p × Z/p) >◁Z/p.

(a) Take V^{∗} to be the direct sum of W and a two dimensional vector space on which β acts trivially and α acts non-trivially, then the action of P on V is generated by bireflections, the dimension of S(V^{∗}) is 5 and the depth of S(V^{∗})^{P} is 4.

(b) Take G to be GL_{3}(F_{p}) and take V^{∗} = W^{↑G}. Then by Proposition 6.1, R^{G} is flat with cc_{G}(R) = 1. Therefore dim(R^{G}) = 3(p^{3} − 1)(p^{2} − 1)(p − 1) and depth(R^{G}) = 3. ◁ Example 6.4. Again suppose P/P′ is a rank 2 elementary abelian p-group. Take p > 2 and consider the action of P/P′ on W^{P′} given by

\overline{α} = ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) and \overline{β} = ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

Then W^{P′} is a three dimensional indecomposable representation of ⟨\overline{αβ^{i}}⟩ and W^{N} is the one dimensional fixed point space. Various choices for P and W give rise to this action of P/P′ on W^{P′}:

(i) W = W^{P′} and P′ = 1. Then P ≅ Z/p × Z/p.

(a) If we take G = GL_{2}(F_{p^{2}}) and V^{∗} = W^{↑G} then, by Proposition 6.1(b), R^{G} is flat with cc_{G}(R) = 1. Thus dim(R^{G}) = 3(p^{4} − 1)(p^{2} − 1) and depth(R^{G}) = 3.

(b) Take V^{∗} to be the direct sum of W and a two dimensional vector space on which α acts trivially and β acts non-trivially, then the action of P on V is generated by bireflections, the dimension of S(V^{∗}) is 5 and the depth of S(V^{∗})^{P} is 4. (ii) Consider the action on W determined by

α = ( \begin{matrix}1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} ) and β = ( \begin{matrix}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) .

In this case P ≅ (Z/p × Z/p) >◁Z/p.

(a) Take V^{∗} to be the direct sum of W and a two dimensional vector space on which α acts trivially and β acts non-trivially, then the dimension of S(V^{∗}) is 6 and the depth of S(V^{∗})^{P} is 4.

(b) Take G to be GL_{3}(F_{p}) and take V^{∗} = W^{↑G}. Then by Proposition 6.1, R^{G} is flat with cc_{G}(R) = 1. Therefore dim(R^{G}) = 4(p^{3} − 1)(p^{2} − 1)(p − 1) and depth(R^{G}) = 3. (iii) Consider the action on W determined by

α = ( \begin{matrix}1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) and β = ( \begin{matrix}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} ) .

In this case, if p > 2, P is a group of order p^{4}. The action of P on V is generated by bireflections, the dimension of S(V^{∗}) is 4 and the depth of S(V^{∗})^{P} is 3. ◁

7 A Spectral Sequence

The purpose of this section is to prove Theorem 7.1 which is used in the proof of Theorem 2.3. Our proof relies on a spectral sequence analogous to the one used by Ellingsrud and Skjelbred [4]. For convenience and completeness, we include a development of the spectral sequence.

Let A be a commutative K-algebra and G a group acting on A via algebra automorphisms. Then A is a KG-module. Furthermore, for each KG-module N and each AG-module M, the group Hom_{KG}(N, M) is an A^{G}-module with (a · α)(x) = a · (α(x)) for a ∈ A^{G} and α ∈ Hom_{KG}(N, M).

Let H : KG − mod → A^{G} − mod be a contravariant left exact functor. Suppose S is an A^{G}-module and N is a KG-module. We choose a projective resolution of S as an A^{G}-module,

P^{∗} = ⋯ P^{2} → P^{1} → P^{0} → S → 0,

and a free resolution of N as a KG-module,

B^{∗} = ⋯ B^{2} → B^{1} → B^{0} → N → 0.

Then we obtain a canonical first quadrant double complex

F^{i,j} := Hom_{A^{G}}(P^{i}, H(B^{j})).

Attached to this double complex are two spectral sequences. The first is constructed using the ‘horizontal filtration’ and satisfies

horE_{1}^{p,q} = H^{q}(Hom_{A^{G}}(P^{∗}, H(B^{p}))) ≅ Ext_{A^{G}}^{q}(S, H(B^{p})),

while the second is constructed using the ‘vertical filtration’ and satisfies

verE_{1}^{p,q} = H^{q}(Hom_{A^{G}}(P^{p}, H(B^{∗}))) ≅ Hom_{A^{G}}(P^{p}, R^{q}H(N))

where R^{q}H denotes the q-th right derived functor of H. It is now straightforward to see that

verE_{2}^{p,q} ≅ Ext_{A^{G}}^{p}(S, R^{q}H(N)).

From elementary properties of first quadrant double complexes, it is clear that both spectral sequences converge to the graded cohomology of the total complex, where the gradings are defined using the horizontal or vertical filtration, respectively. Furthermore, for each of these spectral sequences the r^{th} differential has bidegree (r, 1 − r).

We want to apply this in the situation where S := A^{G}/I for an ideal I ⊴ A^{G}, N ≅ K, the trivial KG-module, and H(X) := Hom_{KG}(X, A), the contravariant Hom-functor. Thus H(B^{j}) = Hom_{KG}(B^{j}, A) ≅ A^{m_{j}} where m_{j} is the rank of the free KG-module B^{j}. Therefore

horE_{1}^{p,q} ≅ Ext_{A^{G}}^{q}(A^{G}/I, A^{m_{p}});
verE_{2}^{p,q} ≅ Ext_{A^{G}}^{p}(A^{G}/I, Ext_{KG}^{q}(K, A));
verE_{2}^{p,0} ≅ Ext_{A^{G}}^{p}(A^{G}/I, A^{G}).

Assume that h := grade(I, A) is finite. (For example, if A is a Cohen-Macaulay ring, and G is finite then h is the height of I.) Then for all q < h we have _{hor}E_{1}^{p,q} = 0 and hence horE_{∞}^{p,q} = 0. Using the horizontal filtration F of the total complex T := tot(F^{i,j}) and the convergence

horE_{r}^{p,q} ⇒ _{hor}E_{∞}^{p,q} ≅ F^{p}(H^{p+q}(T))/F^{p+1}(H^{p+q}(T))

we get for every n = p + q < h:

0 = F^{n+1}(H^{n}(T)) = F^{n}(H^{n}(T)) = F^{n−1}(H^{n}(T)) = ⋯ = F^{0}(H^{n}(T)) = H^{n}(T).

Therefore _{ver}E_{∞}^{p,q} = 0 for p + q < h.

Theorem 7.1. Let m denote the cohomological connectivity cc_{G}(R). Suppose that I is a prime ideal in R^{G} and m + 1 is less than the height, h, of I. Then grade(I, H^{m}(G, R)) = 0 if and only if grade(I, R^{G}) = m + 1.

Proof. We use the spectral sequence _{ver}E_{∗}^{p,q} from the preceding discussion. Since R is Cohen-Macaulay, grade(I, R) = h. Therefore _{ver}E_{∞}^{p,q} = 0 for p + q < h. In particular, verE_{∞}^{0,m} = 0. Furthermore, since H^{i}(G, R) = 0 for 0 < i < m, we have _{ver}E_{2}^{p,q} = 0 for 0 < q < m.

Suppose grade(I, H^{m}(G, R)) = 0. Then _{ver}E_{2}^{0,m} = Ext_{R^{G}}^{0}(R^{G}/I, H^{m}(G, R)) ≠ 0. Suppose α ∈ _{ver}E_{2}^{0,m} \ {0}. Since _{ver}E_{∞}^{0,m} = 0 and the spectral sequence is first quadrant, there is a differential d_{r} such that d_{r}(α) ≠ 0. However, _{ver}E_{2}^{p,q} = 0 for 0 < q < m. Thus r = m + 1 and, since d_{m} has bidegree (m, 1 − m), we conclude _{ver}E_{2}^{m+1,0} ≠ 0. Furthermore, grade(I, R^{G}) is the smallest p such that _{ver}E_{2}^{p,0} ≠ 0. Thus grade(I, R^{G}) ≤ m + 1. However any non-zero element of _{ver}E_{2}^{p,0} for p < m + 1 would survive the spectral sequence giving a non-zero element in _{ver}E_{∞}^{p,0}. Therefore grade(I, R^{G}) = m + 1.

Suppose grade(I, H^{m}(G, R)) > 0. Then _{ver}E_{2}^{0,m} = Ext_{R^{G}}^{0}(R^{G}/I, H^{m}(G, R)) = 0. Thus any non-zero element of _{ver}E_{2}^{p,0} for p ≤ m + 1 < h would survive the spectral sequence giving a non-zero element in _{ver}E_{∞}^{p,0}. Thus grade(I, R^{G}) > m + 1.

References

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P. Fleischmann and R.J. Shank  G. Kemper
Institute of Mathematics and Statistics  Technische Universität München
University of Kent  Zentrum Mathematik - M11
Canterbury, CT2 7NF  Boltzmannstr. 3
UK  85 748 Garching

P.Fleischmann@kent.ac.uk  Germany
R.J.Shank@kent.ac.uk                kemper@ma.tum.de