Cohomological Cohen--Macaulayness in Non-Noetherian Rings

Abstract

We study Cohen--Macaulayness, in the sense of Hamilton--Marley, of non-Noetherian rings arising as big Cohen--Macaulay algebras. Motivated by Bhatt's notion of cohomological Cohen--Macaulayness, we call a locally finite-dimensional ring CCM if its structure sheaf satisfies this condition. Our main comparison theorem shows that every locally finite-dimensional CCM ring is locally HMCM. Using this theorem, we prove that if $A$ is Noetherian and $R$ is an integral $A$-algebra that is locally balanced big Cohen--Macaulay over $A$, then $R$ is CCM and hence locally HMCM. In particular, if $A$ is an excellent Noetherian domain, $p$ is a prime, $n\geq1$, and $A/pA\neq0$, then $A^+/p^nA^+$ is CCM and locally HMCM. We also show, using finite-dimensional valuation domains, that CCM is strictly stronger than locally HMCM. Finally, for a Noetherian ring $A$ of characteristic $p>0$ and its perfection $A_{\mathrm{perf}}$, we prove that the following conditions are equivalent: $A$ is locally weakly $F$-nilpotent; $A_{\mathrm{perf}}$ is a locally balanced big Cohen--Macaulay $A$-algebra; and $A_{\mathrm{perf}}$ is CCM. Under these equivalent conditions, $A_{\mathrm{perf}}$ is locally HMCM.