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

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.