Totally reflexive modules and Poincaré series

Abstract

We study Cohen-Macaulay non-Gorenstein local rings (R, 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.2010 Mathematics Subject Classification. 13C13, 13D40.

Some criteria for detecting large homomorphisms of local rings

“…Since φ is large, g i is injective and therefore f i = 0 as desired. We remark that Corollary 3.6 fails if R is not Gorenstein; see [10,Example 3.12]. The following example shows that Corollary 3.6 also fails over Artinian Gorenstein rings of higher socle degree.…”