The $\ast$-transforms of Acyclic Complexes

Abstract

Let $R$ be an $n$-dimensional Cohen-Macaulay local ring and $Q$ a parameter ideal of $R$. Suppose that an acyclic complex $(F_{\bullet}, \varphi_{\bullet})$ of length $n$ of finitely generated free $R$-modules is given. We put $M = {\rm Im} \, \varphi_{1}$, which is an $R$-submodule of $F_{0}$. Then $F_{\bullet}$ is an $R$-free resolution of $F_{0}/M$. In this paper, we describe a concrete procedure to get an acyclic complex $^\ast\!{F_{\bullet}}$ of length $n$ that resolves $F_{0}/(M :_{F_{0}} Q)$.