Cohomological Cohen–Macaulayness in Non-Noetherian Rings

Abstract

Hamilton–Marley proved that the absolute integral closure A+ of an excellent Noetherian domain A of positive characteristic is Cohen–Macaulay in their sense (HMCM). However, HMCM is not preserved under localization in general, as shown by an example constructed by the speaker, and it remained unclear whether A+ is locally HMCM. In this talk, motivated by Bhatt’s notion of cohomological Cohen–Macaulayness, we introduce a stronger notion called CCM and show that every locally finite dimensional CCM ring is locally HMCM. As an application, we prove that integral algebras which are locally balanced big Cohen–Macaulay over a Noetherian ring are CCM. In particular, this shows that A+ is locally HMCM, strengthening the result of Hamilton–Marley.