Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture
Let A and B be two finite dimensional algebras over an algebraically closed field, related to each other by a stable equivalence of Morita type. We prove that A and B have the same number of isomorphism classes of simple modules if and only if their 0-degree Hochschild Homology groups HH (0)(A) and HH (0)(B) have the same dimension. The first of these two equivalent conditions is claimed by the Auslander-Reiten conjecture. For symmetric algebras we will show that the Auslander-Reiten conjecture is equivalent to other dimension equalities, involving the centers and the projective centers of A and B. This motivates our detailed study of the projective center, which now appears to contain the main obstruction to proving the Auslander-Reiten conjecture for symmetric algebras. As a by-product, we get several new invariants of stable equivalences of Morita type.
Keywords: Auslander-Reiten conjecture ; Higman ideal ; Projective center ; Stable equivalence of Morita type ; Stable Hochschild homology ; Transfer map ; Generalized Reynolds Ideals ; Morita Type ; Algebras ; Equivalences ; Categories ; Constructions ; Invariance ; Functors
Record created on 2012-04-19, modified on 2016-08-09