On the existence of non-free totally reflexive modules

Abstract

For a standard graded Cohen-Macaulay ring $R$, if the quotient $R/(\underline {x})$ admits non-free totally reflexive modules, where $\underline {x}$ is a system of parameters consisting of elements of degree one, then so does the ring $R$. A non-constructive proof of this statement was given by Takahashi. We will give an explicit construction of the totaly reflexive modules ove $R$ obtained from those over $R/(\underline{x})$. We consider the question of which Stanley-Reisner rings of graphs admit non-free totally reflexive modules and discuss some examples. For an Artinian local ring $(R, \mathfrak m)$ with $\mathfrak m^3 =0$ and containing the complex numbers, we describe an explicit construction of uncountably many non-isomorphic indecomposable totally reflexive modules, under the assumption that at least one such non-free module exists.