Cancellation of homothetic modules
arXiv:1801.02984
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.
Theorem 0.7 in "Cancellation of homothetic modules" is subsumed by Theorem 1.2 in the paper "Cancellation of finite-dimensional Noetherian modules" by the same author