On The Existence Of Totally Reflexive Modules

Abstract

In this manuscript, we investigate the existence of non-free totally reflexive modules over two classes of commutative local (Noetherian) rings.First, we demonstrate existence over a class of local rings which are defined by a Gorenstein homomorphism. Among the corollaries to this result, we recover a theorem of Avramov, Gasharov, and Peeva (1997) concerning the existence of non-free totally reflexive modules over local rings with embedded deformations. We also give a general construction for a class of local rings which satisfy the hypotheses of our theorem, and we show it is able to produce rings without embedded deformations.The second focus of this work is to give necessary conditions for the existence of a non-free totally reflexive module with a Koszul syzygy over a local ring for which the fourth power of the maximal ideal vanishes. We characterize the Hilbert series of such a ring in terms of the Betti sequence of the module. These characterizations extend similar results of Yoshino (2003) concerning the same existence question over local rings for which the cube of the maximal ideal is zero. In particular, we consider necessary conditions for the existence of certain asymmetric complete resolutions, which are known to exist by work of Jorgensen and & Scedil;ega (2005).