The Cohen–Macaulay property of separating invariants of finite groups

Abstract

Abstract. In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.

Separating invariants and local cohomology

“…The group G is isomorphic to C 4 2 , and generated by reflections (namely those elements where exactly one of the α i 's is non-zero). This is a remarkable example since its invariant ring is not CohenMacaulay (see [10]) and, moreover, neither is any graded separating subalgebra (see [6]) despite the action of G being generated by reflections.…”

On the depth of separating algebras for finite groups

“…The corresponding question of depth for separating algebras has not, to the best of our knowledge, been studied at all. In Dufresne et al (2009) it was shown that many of the criteria which imply a ring of invariants is not Cohen-Macaulay actually preclude the existence of a Cohen-Macaulay geometric separating algebra. This article is in many ways a sequel to Dufresne et al (2009).…”

“…In Dufresne et al (2009) it was shown that many of the criteria which imply a ring of invariants is not Cohen-Macaulay actually preclude the existence of a Cohen-Macaulay geometric separating algebra. This article is in many ways a sequel to Dufresne et al (2009). In the former we obtained lower bounds for the Cohen-Macaulay defect of separating algebras.…”

Explicit separating invariants for cyclic P-groups

“…complete intersection). But some recent results [13] and [16] suggest that, in general, separating subalgebras do not provide substantial improvements in terms of the Cohen-Macaulay defect.…”