Independent sequences and freeness criteria

TL;DR

This paper introduces new criteria for module freeness over Noetherian local rings using $M$-independent sequences and Koszul complexes, with applications in linkage theory and patching methods.

Contribution

It provides a novel characterization of $M$-independence via Koszul complexes and new freeness criteria inspired by linkage theory and patching techniques.

Findings

New characterization of $M$-independence using Koszul complexes

Freeness criterion based on strongly $M$-independent sequences

Application of criteria to linkage theory and patching methods

Abstract

Let $M$ be a module over a Noetherian local ring $A$. We study $M$-independent sequences of elements of $\mathfrak{m}_A$ in the sense of Lech and Hanes. The main tool is a new characterization of the $M$-independence of a sequence in terms of the associated Koszul complex. As applications, we give a result in linkage theory, a freeness criterion for $M$ in terms of the existence of a strongly $M$-independent sequence of length $edim(A)$, and another freeness criterion inspired from the patching method of Calegari and Geraghty for balanced modules in their 2018 paper.