Cancellation of homothetic modules

TL;DR

This paper extends cancellation theorems for modules over commutative rings, focusing on homothetic modules like canonical modules of Cohen-Macaulay rings, without relying on stable rank conditions.

Contribution

It provides new criteria for cancellation of homothetic modules, generalizing Bass's theorem and applying to canonical modules in Cohen-Macaulay rings.

Findings

Extended Bass's Cancellation Theorem for homothetic modules

Established conditions for split surjective maps in Hom(M,N)

Applied results to canonical modules of Cohen-Macaulay rings

Abstract

Let R be a commutative ring, M an R-module, and N a finitely presented R-module such that the intersection of Max(R) and Supp(N) is finite-dimensional and Noetherian. Suppose also that N is homothetic; in other words, suppose that the natural ring homomorphism from R to the R-endomorphism ring of N is surjective. Working under these conditions, we describe various ways to guarantee the existence of a split surjective map in a specified coset of Hom(M,N). Using these results, we yield an extension of Bass's Cancellation Theorem that gives sufficient conditions for cancelling N or, more generally, a direct summand of a direct sum of finitely many copies of N. Since a canonical module of a Cohen-Macaulay ring is always finitely presented and homothetic, our work reveals a cancellation property of canonical modules of Cohen-Macaulay rings with finite-dimensional maximal spectra. Of note is that our results do not rely on any stable rank conditions.