Cohomological Cohen–Macaulayness in Non-Noetherian Rings

Comment

This paper studies 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, and prove a comparison theorem showing that every locally finite-dimensional CCM ring is locally HMCM. Using this theorem, we show 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≥1, and A/pA≠0, then A⁺/pⁿA⁺ 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_perf, we prove that the following conditions are equivalent: A is locally weakly F-nilpotent; A_perf is a locally balanced big Cohen–Macaulay A-algebra; and A_perf is CCM. Under these equivalent conditions, A_perf is locally HMCM.