Hochschild cohomology of algebras with homological ideals

Abstract

Let $\varphi: A \to B$ be a homological epimorphism of $k$-algebras. We investigate the relationship of the Hochschild cohomologies $H^{i}(A)$ and $H^{i}(B)$ of $A$ and $B$, and show that they can be connected by a long exact sequence. In particular, if $A$ is a quasihereditary algebra and $B$ is the quotient of $A$ by a minimal heredity ideal, then the long exact sequence provides information on $H^{i}(A)$, $H^{i}(B)$ and the extension groups between costandard modules and standard modules, thus one can actually compute $H^{i}(A)$ inductively. As a consequence, we obtain the Hochschild cohomology of all nonsemisimple Temperley-Lieb algebras and representation-finite Schur algebras.