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

Abstract

This paper studies polynomial extensions of HMCM rings and localization phenomena for torsion-free modules. Here 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 under polynomial extensions in general, but is preserved for stably coherent rings of finite weak global dimension. We also revisit polynomial grade, give a counterexample to the ``Moreover'' assertion in [HM07, Proposition 2.7], and characterize when localization preserves torsion-freeness using regular saturation, total rings of fractions, and Krull primes.