# Modules with Pure Resolutions

## TL;DR

This paper characterizes Cohen-Macaulay properties of standard graded algebras and modules through pure resolutions and Betti number relations, providing new criteria for Cohen-Macaulayness and module classification.

## Contribution

It introduces a characterization of Cohen-Macaulay algebras via pure Cohen-Macaulay modules and establishes Herzog-Kuhl equations relating Betti numbers for pure modules.

## Key findings

- Cohen-Macaulay property characterized by pure Cohen-Macaulay modules.
- Herzog-Kuhl equations relate Betti numbers for pure modules.
- Characterization of Cohen-Macaulay modules when the algebra is Cohen-Macaulay.

## Abstract

We show that the property of a standard graded algebra R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module corresponding to any degree sequence of length at most depth(R). We also give a relation in terms of graded Betti numbers, called the Herzog-Kuhl equations, for a pure R-module M to satisfy the condition dim(R) - depth(R) = dim(M) - depth(M). When R is Cohen-Macaulay, we prove an analogous result characterizing all graded Cohen-Macaulay R-modules.