Proof of the Stratification Theorem

Abstract

Recall that part (a) of Theorem II.2.13 is already proved. To prove part (b), we need an `H-form’ of Goldie’s Theorem: starting with a suitable H-primek-algebra A, we want to show that the setEof regular H-eigenvectors in A forms a denominator set, and that the localization A[E<sup>-1</sup>] is H-simple (as well as`Hartinian’ in an obvious sense). The action ofHis, of course, extended to an action on A[E^-1] by k-algebra automorphisms in the obvious manner. In view of Lemma II.2.11, it suffices to prove an analogous graded version of Goldie’s Theorem in the context of X(H)-graded rings, and we proceed in that direction.