Hochschild cohomology and stratifying ideals

Suppose BB is an algebra with a stratifying ideal BeBBeB generated by an idempotent ee. We will establish long exact sequences relating the Hochschild cohomology groups of the three algebras BB, B/BeBB/BeB and eBeeBe. This provides a common generalization of various known results, all of which extend Happel’s long exact sequence for one-point extensions. Applying one of these sequences to Hochschild cohomology algebras modulo the ideal generated by homogeneous nilpotent elements shows, in some cases, that these algebras are finitely generated.