THE RELATIVE TRACE IDEAL AND THE DEPTH OF MODULAR RINGS OF INVARIANTS

PETER FLEISCHMANN AND R. JAMES SHANK

Abstract. We prove that for a modular representation, the depth of the ring of invariants is the sum of the dimension of the fixed point space of the p-Sylow subgroup and the grade of the relative trace ideal. We also determine which of the Dickson invariants lie in the radical of the relative trace ideal and we describe how to use the Dickson invariants to compute the grade of the relative trace ideal.

1. Preliminaries

Let G be a finite group, p a prime divisor of the order of G, k a field of characteristic p and V a finitely generated kG-module. The action of G on V induces an action on the dual, V^{∗}, which extends to an action by algebra automorphisms on the symmetric algebra A := Sym(V^{∗}). The ring of invariants, A^{G} := {f ∈ A | g(f) = f, ∀g ∈ G}, is a finitely generated subring of A. The depth of A^{G} is the length of a maximal regular sequence in A^{G}. Since V is a modular representation of G, the ring A^{G} may fail to be Cohen-Macaulay. In other words the depth might be less than the Krull dimension. This difference is measured by the Cohen-Macaulay defect, def(A^{G}) := dim(A^{G}) − depth(A^{G}). By the Auslander-Buchsbaum formula, def(A^{G}) is equal to the projective dimension of A^{G} as a module over a homogeneous system of parameters. Thus def(A^{G}) is a measure of the structural complexity of A^{G}. Despite significant recent progress (see, for example, [5], [11], [12], [13], [14], [15], [16], [18]), much of which refers in one way or another to the ground breaking work of Ellingsrud and Skjelbred [7], computing the depth of A^{G} remains, in general, quite difficult.

For a subgroup H ≤ G the relative trace (also called the relative transfer) is the map of A^{G}-modules defined by

\begin{aligned}t_{H}^{G} : & A^{H} & ⟶ & A^{G} \\ & f & ⟼ & ∑_{g∈G/H} g(f). \end{aligned}

The image of t_{H}^{G} is an ideal in the ring A^{G}. Choose a p-Sylow subgroup P ≤ G and define the relative trace ideal by A_{<P}^{G} := ∑_{Q<P} t_{Q}^{G}(A^{Q}). The relative trace ideal is the largest ideal of A^{G} which can be constructed using relative trace maps from proper subgroups of P. Ideals of this form were studied in [8] where

it was shown that the radical of the ideal, \sqrt{A_{<P}^{G}}, is always a prime ideal of height codim_{k}(V^{P}). Here V^{P} denotes the subspace of P -fixed points in V. If k is algebraically closed, the ideal A_{<P}^{G} has a very natural geometric interpretation: its variety, V, in the orbit space V/G, consists of all those G-orbits such that a suitable stabilizer contains P or, in other words,

Date: September 7, 2001.

1991 Mathematics Subject Classification. 13A50.

V = {v^{G} ∈ V/G | p does not divide |v^{G}|}
= {v^{G} ∈ V/G | v ∈ V^{P} }

(see [8, Proposition 12.5]). In particular the ideal \sqrt{A_{<P}^{P}} ≤ A^{P} is the ideal corresponding to the linear subvariety V^{P} ≤ V/P, the space of P -fixed points in the space of P -orbits.

It is well known that dim(V^{P}) is a lower bound for the depth of A^{G} (see, for example, [5, Theorem 2]). In Section 2 we prove that the ‘missing part’ of the depth is provided by regular elements in the relative trace ideal, A_{<P}^{G}. As a consequence, computing the depth of A^{G} reduces to computing the grade of A_{<P}^{G}.

In Section 3 we describe how the Dickson invariants, the generators of A^{GL_{n}(F_{q})}, can be used, when k = F_{q}, to compute the grade of the relative trace ideal.

We recommend [6] and [4] as basic references for the required commutative algebra and [1] and [17] as basic references for the invariant theory of finite groups.

2. Depth and Localization

Suppose R is a Noetherian ring, M is a finite R-module, and I is an ideal in R such that IM ≠ M. Then the common length of all maximal M-sequences in I is called the grade of I on M and will be denoted by grade(I, M). If R is a local ring with maximal ideal 𝔪, then grade(𝔪, M) is the depth of M and if R is graded with augmentation ideal R_{+}, then grade(R_{+}, M) is the the depth of M.

Theorem 2.1. (G. Kemper, [12, Theorem 1.5]) Let U ≤ V be a kG-submodule, such that the kernel of the action of G on U has index coprime to p and let M be a non-zero graded module over End_{kG}(A), which is finitely generated over A^{G}. Furthermore let I_{A^{G}}(U) = (U^{⊥})A ∩ A^{G}, then

depth(M) = grade(I_{A^{G}}(U), M) + dim_{k}(U).

The hypotheses of this theorem are satisfied when G = P, M = A^{P} and U = V^{P}. In this case I_{A^{G}}(U) = \sqrt{A_{<P}^{P}}. Therefore, using the fact that an ideal and its radical have the same grade, we have the following corollary.

Corollary 2.2.

depth A^{P} = grade (\sqrt{A_{<P}^{P}} , A^{P} ) + dim V^{P} = grade (A_{<P}^{P} , A^{P} ) + dim V^{P} .

The primary purpose of this section is to extend this corollary to a statement about A^{G}.

For a prime ideal P ∈ Spec(A), let

G_{P} := {g ∈ G | g(f) − f ∈ P, ∀f ∈ A}

and 𝔭 := A^{G} ∩ P. Define the Cohen-Macaulay defect of a graded or local Noetherian ring R by

def(R) := dim(R) − depth(R).

For the local ring A_{𝔭}^{G}, the dimension is the height of the maximal ideal 𝔭. Thus

def(A_{𝔭}^{G}) = ht(𝔭) − depth(A_{𝔭}^{G}).

Since P lies over 𝔭 these ideals have the same height. Furthermore, A is an affine domain so

ht(𝔭) = ht(P) = dim(A) − dim(A/P)

and

(2.2.1) def(A_{𝔭}^{G}) = dim(A) − dim(A/P) − depth(A_{𝔭}^{G}).

Proposition 2.3. Suppose that P ∈ Spec(A) and [G : G_{P}] is relatively prime to p. Then

(i) depth(A^{G}) = depth(A^{G_{P}}), (ii) def(A_{𝔭}^{G}) = def(A^{G_{P}}) = def(A^{G}), and (iii) depth(A^{G}) = dim(A/P) + depth(A_{𝔭}^{G}).

Proof. Since [G : G_{P}] is relatively prime to p, depth(A^{G}) ≥ depth(A^{G_{P}}) (see, for example, [10, Proposition 14]). On the other hand, from [13, Korollar 5.14] or [14, Theorem A (a)], depth(A^{G}) ≤ depth(A^{G_{P}}) and (i) follows. The first equality in (ii) is a clear consequence of either [13, Satz 5.12] or Proposition 2.17 and Theorem 1.9 of [14]. The second equality follows from (i) and the fact that A^{G} and A^{G_{P}} have the same dimension.

Using (ii) gives

depth(A^{G}) = dim(A^{G}) − def(A_{𝔭}^{G}).

Substituting Equation 2.2.1 into this gives

depth(A^{G}) = dim(A^{G}) − (dim(A) − dim(A/P) − depth(A_{𝔭}^{G})).

However, dim(A^{G}) = dim(A) and (iii) follows.

Theorem 2.4. Suppose that P ∈ Spec(A) and [G : G_{P}] is relatively prime to p. Then

(i) depth(A_{𝔭}^{G}) = grade(𝔭, A^{G}), and (ii) depth(A^{G}) = dim(A/P) + grade(𝔭, A^{G}).

Proof. Using [4, Proposition 1.2.10(a)]

grade(𝔭, A^{G}) = inf{depth(A_{𝔮}^{G}) | 𝔭 ⊆ 𝔮, 𝔮 prime}.

Therefore

(2.4.1) grade(𝔭, A^{G}) ≤ depth(A_{𝔭}^{G})

and for some prime 𝔮 with 𝔭 ⊆ 𝔮, grade(𝔭, A^{G}) = depth(A_{𝔮}^{G}). Since A is an integral extension of A^{G}, there is a prime Q lying over 𝔮 with P ⊆ Q. Therefore G_{P} ⊆ G_{Q} and we may apply Proposition 2.3 to the ideal Q. Thus depth(A^{G}) = depth(A^{G_{Q}}) and def(A_{𝔮}^{G}) = def(A^{G}) = def(A_{𝔭}^{G}). However, since 𝔭 ⊆ 𝔮, ht(𝔮) ≥ ht(𝔭) and

grade(𝔭, A^{G}) = depth(A_{𝔮}^{G}) = ht(𝔮) − def(A_{𝔮}^{G}) = ht(𝔮) − def(A_{𝔭}^{G})
≥ ht(𝔭) − def(A_{𝔭}^{G}) = depth(A_{𝔭}^{G}).

Comparing this with Inequality 2.4.1, we have grade(𝔭, A^{G}) = depth(A_{𝔭}^{G}). Substituting into part (iii) of Proposition 2.3 gives

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

as required.

Suppose that P is the prime ideal in A associated to the linear subspace V^{P}. Therefore dim(A/P) = dim(V^{P}). Clearly P ≤ G_{P}. Thus [G : G_{P}] is relatively

prime to p. From [8, Corollary 12.3], the prime P lies over \sqrt{A_{<P}^{G}}. Therefore we

may apply Theorem 2.4 with 𝔭 = \sqrt{A_{<P}^{G}}. Since an ideal and its radical have the same grade (see, for example, [4, Proposition 1.2.10(b)]) we have the following.

Corollary 2.5. depth(A^{G}) = dim(V^{P}) + grade(A_{<P}^{G} , A^{G}).

3. Using the Dickson Invariants

In this section we assume that k = F_{q}, the field with q = p^{r} elements. Let \overline{F_{q}} denote the algebraic closure and define \overline{V} := \overline{F_{q}} ⊗ V. Thus elements of A represent distinct functions on \overline{V}. Consider the usual representation of GL_{n}(F_{q}). Choose a basis, {x_{1}, x_{2}, …, x_{n}}, for V^{∗} and let V_{i}^{∗} = Span_{F_{q}}{x_{1}, …, x_{i}}. Define F_{i}(t) = ∏_{v∈V_{i}^{∗}}(t − v). The ring A^{GL_{n}(F_{q})} is generated by the Dickson invariants, d_{i,n}, which are defined by F_{n}(t) = ∑_{i=0}^{n} d_{i,n}t^{q^{n−i}}. Note that d_{0,n} = 1. We adopt the convention that d_{i,n} = 0 if i > n.

As a consequence of Bourguiba & Zarati’s proof of the Landweber-Stong Conjecture ([3, Theorem 1.1] for r = 1 and [2] or [9] for r > 1) we can use the Dickson invariants as a test sequence to measure depth(A^{G}). In other words, the depth of A^{G} is the largest ℓ such that d_{1,n}, …, d_{ℓ,n} is A^{G}-regular. In this section we show that the Dickson invariants can be used as a test sequence to measure grade(A_{<P}^{G} , A^{G}).

Define h_{i} = F_{i−1}(x_{i}). It follows from the proof of [19, Proposition 1.3b] that

d_{i,n} = d_{i,n−1}^{q} − d_{i−1,n−1}h_{n}^{q−1} (3.0.1)

(Note that there is a typographical error in the statement of [19, Proposition 1.3b].). Using this inductively gives

d_{1,n} = −(h_{1}^{q^{n−1}(q−1)} + h_{2}^{q^{n−2}(q−1)} + ⋯ + h_{n}^{q−1}) (3.0.2)

and

d_{n,n} = (−1)^{n}(h_{1}h_{2} … h_{n})^{q−1}. (3.0.3)

Lemma 3.1. If i > 0 then d_{i,n} ∈ (h_{1}, h_{2}, …, h_{n−i+1})A.

Proof. The proof is by a double induction first on i and then on n − i.

For i = 1 the result follows from Equation 3.0.2. We now assume that i > 1. For n − i = 0 the result follows from Equation 3.0.3. We now assume that n − i > 0.

Using the secondary induction hypothesis d_{i,n−1} ∈ (h_{1}, h_{2}, … h_{n−i})A. By the primary induction hypothesis d_{i−1,n−1} ∈ (h_{1}, h_{2}, … h_{n−i+1})A. Therefore it follows from Equation 3.0.1 that d_{i,n} ∈ (h_{1}, h_{2}, … h_{n−i+1})A.

Suppose that G ≤ GL_{n}(F_{q}) and dim(V^{P}) = k. Choose a basis for V^{∗} so that \overline{V}^{P} is the variety cut out by x_{1}, …, x_{n−k}. Since h_{1}, …, h_{n−k} are polynomials in x_{1}, …, x_{n−k}, we conclude that h_{1}, …, h_{n−k} are zero on \overline{V}^{P}. Therefore it follows from Lemma 3.1 that, if i > k then d_{i,n} is zero on \overline{V}^{P}.

Corollary 3.2. If i > dim(V^{P}) then d_{i,n} ∈ \sqrt{A_{<P}^{G}}.

Theorem 3.3. Suppose that depth(A^{G}) = ℓ and dim(V^{P}) = k.

(i) If i ≤ k then d_{i,n} ∉ \sqrt{A_{<P}^{G}}, and (ii) d_{k+1,n}, …, d_{ℓ+1,n} is not A^{G}–regular.

Proof. Since the Dickson invariants form a homogeneous system of parameters for A^{G} and dim (\sqrt{A_{<P}^{G}} ) = dim(V^{P}) = k, the ideal \sqrt{A_{<P}^{G}} can contain at most n−k Dickson invariants. Part (i) then follows from Corollary 3.2. From Corollary 2.5,

grade (\sqrt{A_{<P}^{G}} , A^{G}) = ℓ − k. Since d_{k+1,n}, …, d_{ℓ+1,n} is a sequence of ℓ − k + 1

elements from \sqrt{A_{<P}^{G}}, the sequence can not be A^{G}–regular.

Corollary 3.4. Suppose dim(V^{P}) = k. Then

(i) d_{i,n} ∈ \sqrt{A_{<P}^{G}} if and only if i > k and (ii) grade (A_{<P}^{G} , A^{G}) is the largest m such that d_{k+1,n}, …, d_{k+m,n} is A^{G}-regular.

Proof. The first statement follows from Corollary 3.2 and Theorem 3.3(i).

Using Corollary 2.5 and Bourguiba & Zarati/Landweber-Stong, grade (A_{<P}^{G} , A^{G}) is the largest m such that d_{1,n}, …, d_{k+m,n} is A^{G}-regular. Although in general a permutation of a regular sequence need not be a regular sequence, a regular sequence of homogeneous elements in A^{G} is still regular after a permutation (See, for example, [4, Proposition 1.1.6]. The proposition is stated for Noetherian local rings but the proof is valid in the current context.). Therefore, if d_{1,n}, …, d_{k+m,n} is A^{G}-regular, then every subsequence is also A^{G}-regular. In particular, if grade (A_{<P}^{G} , A^{G}) = m, then d_{k+1,n}, …, d_{k+m,n} is A^{G}-regular.

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Institute of Mathematics & Statistics, University of Kent at Canterbury, CT2 7NF, UK

E-mail address: P.Fleischmann@ukc.ac.uk
E-mail address: R.J.Shank@ukc.ac.uk