Existence of almost Cohen–Macaulay algebras implies the existence of big Cohen–Macaulay algebras

Abstract

Abstract. In [1], the dagger closure is extended over finitely generated modules over Noetherian local domain (R, m) and it is proved to be a Dietz closure. In this short note we show that it also satisfies the 'Algebra axiom' of [9] and this leads to the following result of this paper: For a complete Noetherian local domain, if it is contained in an almost Cohen-Macaulay domain, then there exists a balanced big Cohen-Macaulay algebra over it.

Reduction of the Small Cohen–Macaulay Conjecture to excellent unique factorization domains

“…By an argument as in the proof of Theorem 4.4 we can deduce that S is necessarily Cohen-Macaulay regardless of the residual characteristic, because we have a quadratic extension. This fact is asserted in Theorem 4.5 8. Recall that S has mixed characteristic and therefore Frac(S) has characteristic zero.…”

“…This observation shows that E T (T) has at most two direct summands and, as an immediate consequence, T has at most two associated primes. Now if α 1 and α 2 are distinct square roots of g 2 in Frac(S) 8 then it is easily verified that the rules (r, s) → r + sα i , (1 ≤ i ≤ 2) define distinct ring homomorphisms, Ψ i : T → S, with distinct kernels, p 1 , p 2 ∈ Ass(T). Now the claim, easily, follows from the fact that T has at most two associated primes.…”

“…In [27] when R has dimension three and mixed characteristic Hochster constructed a big Cohen-Macaulay algebra using the Heitmann's result regarding the almost Cohen-Macaulayness of R + . Recently, in [8], Bhattacharyya showed that the Big Cohen-Macaulay Algebra Conjecture follows from the existence of almost big Cohen-Macaulay algebras. The question whether R + is always almost Cohen-Macaulay in mixed characteristic is open in dimension greater than or equal to 4.…”