Totally Reflexive Modules and Poincaré Series

Abstract

We study Cohen-Macaulay non-Gorenstein local rings $(R,\mathfrak{m},k)$ admitting certain totally reflexive modules. More precisely, we give a description of the Poincaré series of $k$ by using the Poincaré series of a non-zero totally reflexive module with minimal multiplicity. Our results generalize a result of Yoshino to higher-dimensional Cohen-Macaulay local rings. Moreover, from a quasi-Gorenstein ideal satisfying some conditions, we construct a family of non-isomorphic indecomposable totally reflexive modules having an arbitrarily large minimal number of generators.

arXiv commentFinal version. To appear in J. Algebra