IF[d n,n−1 , . . . , d n,0
] and hence has finite global dimension, so a famous equality of Auslander and Buchsbaum [1] allows us to convert the computation of the homological codimension of IF [V ] G over D * (n) into an equivalent computation of the homological (i.e. projective) dimen-
sion of IF[V ]
G over D * (n). For this we introduce a spectral sequence, which, loosely speaking, is a Koszul-Serre dual to the one used by Ellingsrud and Skjelbred in [8] . We hope in this way to make this circle of ideas available to a larger audience then seems to have been attracted by [8] alone. G . So the question posed at the outset is only of interest in the modular case (for finite groups, i.e., where the characteristic of the ground field divides the group order).
sion of IF[V ]
The following example, due to M.D. Neusel, brings out the subtle nature of the problem. Z /2 can be computed by using either Molien's theorem to compute the Poincaré series over C and the last conclusion of [22] Z /2 , t) = 1 2
sion of IF[V ]
(1 + t) 3 + (1 − t) 3 (1 − t) 3 (1 − t 2 ) 3 = 1 + 3t 2 (1 − t) 3 (1 − t 2 ) 3 = 1 + 3t + 12t 2 + 28t 3 + · · · .
sion of IF[V ]
Therefore the space of invariant linear forms has dimension 3, the space of invariant quadratic forms dimension 12, the space of invariant cubic forms dimension 28, etc. In degrees 1 and 2 it is relatively easy to find bases for the invariant forms:
sion of IF[V ]
is a basis for the invariant linear forms, and the 6 products
sion of IF[V ]
together with the 6 quadratic polynomials 
sion of IF[V ]
is acyclic. For the sake of clarity here is the definition of the Koszul complex and notation that we are using (see [22] §6.2, [21] part II §1).
DEFINITION DEFINITION DEFINITION DEFINITION DEFINITION :
Let A be a graded connected commutative algebra over a field IF and a 1 , . . . , a n ∈ A. The Koszul complex of A with respect to a 1 , . . . , a n ∈ A is the differential graded commutative algebra 
DEFINITION DEFINITION DEFINITION DEFINITION DEFINITION :
and the differential is defined by requiring
DEFINITION DEFINITION DEFINITION DEFINITION DEFINITION :
Introduce the Koszul complex
DEFINITION DEFINITION DEFINITION DEFINITION DEFINITION :
The group G acts on K via the representation ρ on IF[V ] and trivially on As a tool to deal with the problems encountered in the introduction and §1 we set up a spectral sequence 2 relating the cohomology of a G-cocomplex to the cohomology of the fixed cocomplex. In order to keep the discussion as simple as possible 3 we suppose that ρ : G GL(n, IF) is a representation of a finite group G over the field IF and that h 1 , . . . , h k ∈IF [V ] G are invariant polynomials which form a regular sequence in IF [V ] . Let (K, ) denote the Koszul complex [4] or [6] ). Introduce the double complex
DEFINITION DEFINITION DEFINITION DEFINITION DEFINITION :
A bit of care is needed with the gradings to turn this into an acceptable double complex, i.e., one satisfying the standard grading conventions (see e.g. [15] ). The differential d * coming from the projective resolution P(G) appears in the contravariant variable of Hom IF(G) ( , ) and hence the differential d * induced by d on C increases the grading in C coming from P(G), Associated to this double complex are two spectral sequences, [15] chapter XI, which are independent of the choice of the resolution P(G). Using the spectral sequence where we first apply the Koszul differential and then the differential from the projective resolution P(G) we obtain:
DEFINITION DEFINITION DEFINITION DEFINITION DEFINITION :
With the preceding hypotheses and notations the augmentation map of complexes
DEFINITION DEFINITION DEFINITION DEFINITION DEFINITION :
induces an isomorphism
DEFINITION DEFINITION DEFINITION DEFINITION DEFINITION :
.
PROOF PROOF PROOF PROOF PROOF :
The map g is an equivalence of complexes, and hence so is
PROOF PROOF PROOF PROOF PROOF :
from which the desired conclusion follows.
PROOF PROOF PROOF PROOF PROOF :
The spectral sequence where we first apply the projective resolution differential (e.g. bar construction differential) and then the Koszul complex differential will become our main technical tool. If we denote this spectral sequence by E r , d r then as a consequence of 2.1 (=⇒ denotes converges to)
PROOF PROOF PROOF PROOF PROOF :
.
PROOF PROOF PROOF PROOF PROOF :
the term E 1 of the spectral sequence takes the
PROOF PROOF PROOF PROOF PROOF :
Therefore the term E 2 may be identified with the cohomology of yet another Koszul complex, namely for the elements
PROOF PROOF PROOF PROOF PROOF :
G are algebraically independent, we see
PROOF PROOF PROOF PROOF PROOF :
where
PROOF PROOF PROOF PROOF PROOF :
and the product in group cohomology. 
PROOF PROOF PROOF PROOF PROOF :
We need only remark that convergence is a consequence of the fact that E s,t
PROOF PROOF PROOF PROOF PROOF :
This spectral sequence is a precursor of Grothendieck's local cohomology spectral sequence [11] .
PROOF PROOF PROOF PROOF PROOF :
As a simple application of this spectral sequence we reprove a result of Landweber and Stong [14] , that serves as a model for further applications. ) is zero in negative degrees, and hence E s,t is also zero in negative degrees. Therefore
PROOF PROOF PROOF PROOF PROOF :
and the result follows. §3. Vector Invariants of Z /2
PROOF PROOF PROOF PROOF PROOF :
In this section we will apply the spectral sequence of 2.2 to rings of invariants in characteristic 2. To see why this case is particularly amenable we begin with a number of general remarks.
PROOF PROOF PROOF PROOF PROOF :
If ρ : G GL(n, IF) is a representation of a finite group G over the field IF of characteristic
PROOF PROOF PROOF PROOF PROOF :
So to establish lower bounds for the homological codimension we may restrict ourselves to the case of p-groups, and henceforth we assume G = P to be a finite p-group. We also assume that IF is a Galois field with q = p m elements. The Dickson algebra (see e.g. [22] 
PROOF PROOF PROOF PROOF PROOF :
in turn allows us to convert the original homological codimension computation to one of the
PROOF PROOF PROOF PROOF PROOF :
It is here that the spectral sequence FIGURE FIGURE FIGURE FIGURE FIGURE 3.1: The vanishing area s + t < 0 for E the terms E s,t for s + t < 0 are all zero. This is a key fact that will be exploited in computations. The terms on the border of the vanishing area, i.e., where s + t = 0 (referred to as the vanishing line) are connected with the stable invariants introduced in [12] and studied in [16] .
PROOF PROOF PROOF PROOF PROOF :
To work with this spectral sequence we need to obtain information about H 
PROOF PROOF PROOF PROOF PROOF :
where: to choose a basis x 1 , . . . , x m , y 1 , . . . , y m , z 1 , . . . , z We are going to study the homological codimension of the ring of invariants IF [X ] Z / p , where IF is a field of characteristic p, and show
PROOF PROOF PROOF PROOF PROOF :
confirming a formula of [8] . Our strategy is the same as in §3: we use the AuslanderBuchsbaum equality to convert a homological codimension computation into the computation of the homological dimension of IF[X ] Z / p as a module over the Dickson algebra D(X ), which we do with the aid of the spectral sequence 2.2
PROOF PROOF PROOF PROOF PROOF :
There are algebra inclusions With the preceding hypotheses and notations the maps , Tr 
PROOF PROOF PROOF PROOF PROOF :
From these equations it is an easy matter to deduce the following criteria:
PROOF PROOF PROOF PROOF PROOF :
free and b w = 0 for all w ∈SIP
PROOF PROOF PROOF PROOF PROOF :
free and b w = 0 for
PROOF PROOF PROOF PROOF PROOF :
Only (ii) requires any comment at all: here one just needs to notice that the system of linear equations 
PROOF PROOF PROOF PROOF PROOF :
for all j ≥ 0. 
PROOF PROOF PROOF PROOF PROOF :
we conclude that 
Regular Sequences in Invariant Rings of Permutation Representations
The computations of sections §3 and 4 combined with the tools of §2 may be used to reexamine the motivational problem posed in §1 to good advantage in the case of permutation representations. Let us begin by examining the case of a representation ρ : Z /2 GL(n, IF). 

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