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.

Abstract. We fix a field k of characteristic p. For a finite group G denote by δ(G) and σ(G) respectively the minimal number d, such that for any finite dimensional representation V of G over k and any v ∈ V G \ {0} or v ∈ V \ {0} respectively, there exists a homogeneous invariant f ∈ k[V ] G of positive degree at most d such that f (v) = 0. Let P be a Sylow-p-subgroup of G (which we take to be trivial if the group order is not divisble by p). We show that δ(G) = |P |., where l is the smallest prime divisor of |G|. These results extend known results in the non-modular case to the modular case.

We explicitly construct a finite set of separating invariants for the basic Ga-actions. These are the finite dimensional indecomposable rational linear representations of the additive group Ga of a field of characteristic zero, and their invariants are the kernel of the Weitzenböck derivation Dn = x 0 ∂ ∂x 1 + . . . + x n−1 ∂ ∂xn .

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