Depth and cohomological connectivity in modular invariant theory

Abstract

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 ) .

The Module Structure of a Group Action on a Ring

“…For some classes of groups, for example cyclic ones, the lower bound on the depth in this case is known to be sharp, but this does not always hold: see [11]. In fact, [11] contains a somewhat stronger result that we now generalise. Recall that the depth of a module M over a graded noetherian local ring R can be characterised as the smallest positive integer i such that Ext i (R/m, M) = 0; see [4, 1.2.8], or [4, 9.1.2], [24, proof of 3.14] for the infinitely generated case.…”

“…The case M = k of Theorem 5.2 is a result of Ellingsrud and Skjelbred [7]. For some classes of groups, for example cyclic ones, the lower bound on the depth in this case is known to be sharp, but this does not always hold: see [11]. In fact, [11] contains a somewhat stronger result that we now generalise.…”

On Cohen–Macaulayness and depth of ideals in invariant rings

“…In the modular case, on the other hand, K[V ] G almost always fails to be Cohen-Macaulay, see [12]. The depth of K[V ] G has attracted much attention and has been determined for various families of representations, see for example [3,8,10,13,17]. In this paper we consider ideals of K[V ] G as modules over K[V ] G .…”

On the depth of invariant rings of infinite groups

“…Several papers have dealt with the depth of invariant rings of finite groups, see for example [4,6,7,8,9,10,14,19,21], but there are up to today no quantitative results for the depth of invariants of (infinite) algebraic groups.…”