Group actions on rings and the Čech complex

We have previously shown that, when a finite group G acts on a polynomial ring S in n variables over a finite field k, only finitely many isomorphism classes of indecomposable kG-modules occur as summands of S. We have also shown that the regularity of the invariant subring SG is at most zero, which has various consequences, for example SG is generated in degrees at most n(|G|-1) (provided n, |G|≥2). Both of these theorems depend on the Structure Theorem of Karagueuzian and the author, which is proved by means of a long and complicated calculation. The aim of this paper is to prove these results using a more conceptual method. © 2013 Elsevier Ltd.