A periodic module of infinite virtual projective dimension

Abstract

A finitely generated module M over a local ring R (with infinite residue field) is said to have finite virtual projective dimension, if the completion R̂ of R can be presented in the form Q/( x), where x is a regular sequence in the local ring Q, and the projective dimension of M̂, viewed as a Q-module, is finite. It is known that if the Betti numbers of M are bounded, then the finiteness of its virtual projective dimension implies that its minimal free resolution eventually becomes periodic of period 2. The purpose of this note is to construct examples which show that the converse does not hold in general.