Title: On the number of isomorphism classes of totally reflexive modules

Abstract: The category of totally reflexive R-modules has been shown as very useful in detecting the Gorenstein property of the ring. We use this category to recognize hypersurface singularities using work of Buchweitz, Greuel, and Schreyer, Knörrer, Yoshino and others who established remarkable connections between the module theory of R and the character of its singularity.

Let (R,m,k) be a local noetherian ring, and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. The main result of this talk is that if this set is finite then either it has exactly one element (represented by the rank 1 free module) or R is Gorenstein and an isolated singularity.

In presenting this theorem we will use tools from relative homological algebra and we will explore the connections with the category of totally reflexive modules.