The rank of syzygies under the action by a finite group

Abstract

Let S be a Noetherian local ring with maximal ideal J and k the residue field of S. Let G be a finite group of order n and suppose that G acts on S as automorphisms. Let R = SG and I = JG. We denote by S[G] (resp. R[G]) the twisted group ring of G over S (resp. the group algebra of G over R). Recall that the multiplication of S[G] is defined as follows : sg · th = sg(t) · gh for s, t ∊ S and g, h ∊ G. Let tG(S) = {Σg ∈ Gg(s)/s ∈ S} and call it the trace ideal of S. Note that tG(S) = R if n is a unit of S. We say that S has a normal basis if S ≅ R[G] as R[G]-modules. This condition says that there is an element s of S so that {g(s)}g ∈ G forms an R-free basis of S.