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arXiv · 2606.25384

Polynomial Extensions and Localization of Non-Noetherian Cohen--Macaulay Rings

Ryoya Ando
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Abstract

This paper studies polynomial extensions and localization of HMCM rings, where HMCM means Cohen--Macaulay in the sense of Hamilton--Marley, a notion for non-Noetherian rings. We show that the HMCM property is not preserved either under polynomial extensions or under localization in general. More precisely, we construct an HMCM ring $A$ such that $A[X]$ is not HMCM, and an HMCM ring $B$ with a prime ideal $\mathfrak q$ such that $B_{\mathfrak q}$ is not HMCM. We also prove a positive result: polynomial rings over stably coherent rings of finite weak global dimension are HMCM. In addition, we revisit polynomial grade, give a counterexample to the ``Moreover'' assertion in [HM07, Proposition 2.7], and study localization of torsion-free modules via regular saturation and Krull primes.

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Ryoya Ando. 2026-06-24. Polynomial Extensions and Localization of Non-Noetherian Cohen--Macaulay Rings. https://arxiv.org/abs/2606.25384

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