Group actions on rings and the Čech complex

We have previously shown that, when a finite group GG acts on a polynomial ring SS in nn variables over a finite field kk, only finitely many isomorphism classes of indecomposable kGkG-modules occur as summands of SS. We have also shown that the regularity of the invariant subring SGSG is at most zero, which has various consequences, for example SGSG is generated in degrees at most n(|G|−1)n(|G|−1) (provided n,|G|≥2n,|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.