HOCHSCHILD COHOMOLOGY OF ALGEBRAS WITH HOMOLOGICAL IDEALS

Let phi : A -> 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 quasi-hereditary 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 non-semisimple Temperley-Lieb algebras and representation-finite Schur algebras.