Liu, YumingZhou, GuodongZimmermann, Alexander2012-04-192012-04-192012-04-19201210.1007/s00209-010-0825-zhttps://infoscience.epfl.ch/handle/20.500.14299/79511WOS:000301580000010Let 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.Auslander-Reiten conjectureHigman idealProjective centerStable equivalence of Morita typeStable Hochschild homologyTransfer mapGeneralized Reynolds IdealsMorita TypeAlgebrasEquivalencesCategoriesConstructionsInvarianceFunctorstext::journal::journal article::research article