The Cohen-Macaulay property of separating invariants of finite groups

Details The Cohen-Macaulay property of separating invariants of finite groups The Cohen-Macaulay property of separating invariants of finite groups

2009-04-07

arxiv.org/abs/0904.1069v2

Mathematics

Commutative Algebra

We removed the conjecture which appeared in previous versions: we give a counter-example. We fixed the proof of Lemma 2.2 (pre

Scientific paper

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 that 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. 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.

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