Existence of unliftable modules

Abstract:

Let $(Q,\operatorname {\mathfrak {n}})$ be a commutative Noetherian local ring, and let $R=Q/(x)$ where $x$ is a non-zerodivisor of $Q$ contained in $\operatorname {\mathfrak {n}}$. Then a finitely generated $R$-module $M$ is said to lift to $Q$ if there exists a finitely generated $Q$-module $M’$ such that $x$ is $M’$-regular and $M \cong M’/xM’$. In this paper we give a general construction of finitely generated $R$-modules of finite projective dimension over $R$ which fail to lift to $Q$ provided $x \in \operatorname {\mathfrak {n}}^{2}$ and the depth of $R$ is at least 2.