Structure of irreducible homomorphisms to/from free modules

TL;DR

This paper studies the structure of certain irreducible module homomorphisms over noetherian local rings and shows that specific vanishing conditions imply the ring's regularity.

Contribution

It characterizes the structure of irreducible monomorphisms and epimorphisms to/from free modules and links Ext and Tor vanishing to ring regularity.

Findings

Irreducible homomorphisms have a specific structure over noetherian local rings.

Self-vanishing of Ext and Tor for modules with such homomorphisms implies the ring is regular.

The results connect module homomorphism properties to the regularity of the underlying ring.

Abstract

The primary goal of this paper is to investigate the structure of irreducible monomorphisms to and irreducible epimorphisms from finitely generated free modules over a noetherian local ring. Then we show that over such a ring, self-vanishing of Ext and Tor for a finitely generated module admitting such an irreducible homomorphism forces the ring to be regular.