On generalization of two results of Foxby

Tony J. Puthenpurakal

On generalization of two results of Foxby

Commutative Algebra 2026-07-18 v1

Authors: Tony J. Puthenpurakal

Abstract

Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension $d$ and let $M$ be a finitely generated $A$ -module. Assume $M$ has rank $r > 0$ . We show that if $M$ is NOT Cohen-Macaulay then $\mu_d(\mathfrak{m}, M) > r$ . If further $A$ is unmixed and $\mu_n(\mathfrak{m}, M) \leq 1$ for some $n \geq d$ then we prove $\text{injdim} \ M < \infty$ and $A$ is Cohen-Macaulay.

Keywords

Noetherian local ring Cohen-Macaulay Bass numbers injective dimension Foxby

Cite

@article{arxiv.2607.16655,
  title  = {On generalization of two results of Foxby},
  author = {Tony J. Puthenpurakal},
  journal= {arXiv preprint arXiv:2607.16655},
  year   = {2026}
}