DG module structures and minimal free resolutions modulo an exact zero-divisor

TL;DR

This paper investigates the structure of minimal free resolutions over local rings modulo an exact zero-divisor, revealing a differential graded module structure and providing explicit resolutions in special cases.

Contribution

It introduces a differential graded module structure on minimal free resolutions over rings with zero-divisors and describes resolutions when the pair is exact zero-divisors.

Findings

Resolutions have a differential graded module structure over a specific DG algebra.

Explicit minimal free resolutions are constructed for modules over rings with exact zero-divisors.

The structure simplifies the understanding of resolutions in these algebraic contexts.

Abstract

Let $Q$ be a local ring with maximal ideal $\mathfrak{n}$ and let $f,g\in \mathfrak{n}\smallsetminus\mathfrak{n}^2$ with $fg=0$. When $M$ is a finite $Q$-module with $fM=0$, we show that a minimal free resolution of $M$ over $Q$ has a differential graded module structure over the differential graded algebra $Q\langle y,t\mid \partial(y)=f, \partial(t)=gy\rangle$. When $(f,g)$ is a pair of exact zero divisors, we use this structure to describe a minimal free resolution of $M$ over $Q/(f)$.