This talk will be about the conjecture that if a finite group G acts on a smooth projective variety X over a field k, then only finitely many isomorphism classes of kG modules occur in an (infinite) direct sum decomposition of the homogeneous coordinate ring of X into indecomposable kG-modules. I'll discuss a proof of this for X of dimension 1 (joint with F. Bleher) and the proof by D. Karagueuzian and P. Symonds that it is true if X is a projective space.