Cohomological comparison theorem

If f   is an idempotent in a ring Λ, then we find sufficient conditions which imply that the cohomology rings ⊕n≥0ExtΛn(Λ/r,Λ/r) and ⊕n≥0ExtfΛfn(fΛf/frf,fΛf/frf) are eventually isomorphic. This result allows us to compare finite generation and Gelfand–Kirillov dimensions of the cohomology rings of Λ and fΛf. We are also able to compare the global dimensions of Λ and fΛf.