Stable equivalences of adjoint type.

In this paper we define a class of stable equivalences, namely, the stable equivalences of adjoint type, and study the Hochschild cohomology groups of algebras that are linked by a stable equivalence of adjoint type. This notion of adjoint type is a special case of Morita type, covers the stable equivalence of Morita type for self-injective algebras, and thus includes the case where Brou's conjecture was made (see for instance [Brou M.: Equivalences of blocks of group algberas. In: Finite dimensional algebras and related topics (V. Dlab and L. L. Scott eds.). Kluwer, 1994, 126]). The main results in this paper are: Let A and B be two artin k-algebras such that A and B are projective over k, and let Hn( A) and Hn( B) be the n-th Hochschild cohomology groups of A and B, respectively. (1) If A and B are stably equivalent of adjoint type, then Hn( A) Hn( B) for all n 1. (2) If A and B are stably equivalent of Morita type, then the absolute values of Cartan determinants of A and B are equal. In particular, two cellular algebras over a field have the same Cartan determinant if they are stably equivalent of Morita type. 2000 Mathematics Subject Classification: 16G10, 16E30; 16G70, 18G05, 20J05. [ABSTRACT FROM AUTHOR]