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arXiv · 2607.16655

On generalization of two results of Foxby

Tony J. Puthenpurakal
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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.

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Tony J. Puthenpurakal. 2026-07-18. On generalization of two results of Foxby. https://arxiv.org/abs/2607.16655

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