*-transforms of acyclic complexes

TL;DR

This paper presents a concrete method to construct acyclic complexes resolving the colon ideal of a module with respect to a parameter ideal in a Cohen-Macaulay local ring, enhancing understanding of resolutions in commutative algebra.

Contribution

It introduces a specific procedure to derive acyclic complexes resolving F_{0}/(M : Q) from given complexes, expanding tools for module resolution in Cohen-Macaulay rings.

Findings

Provides a concrete construction method for acyclic complexes

Resolves modules of the form F_{0}/(M : Q)

Enhances techniques for module resolution in Cohen-Macaulay rings

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 = 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 of length n that resolves F_{0}/(M : Q).