Closure operations that induce big Cohen–Macaulay algebras

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

“…It is also proved that a local domain R has a Dietz closure that satisfies the Algebra Axiom if and only if R has a big Cohen-Macaulay algebra. In this short note we show that dagger closure as in [1] also satisfies the 'Algebra axiom' of [9]. This leads us to the following result of this paper.…”

“…Consider the dagger closure operation as defined in Definition 2.2 of the previous section and from Theorem 2.4, we know that it is a Dietz closure. Now, if we can show that this dagger closure also satisfies 'Algebra axiom' as stated in [9], then using Theorem 3.3 of [9], we can conclude.…”

“…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. …”

“…To see that dagger closure satisfies the 'Algebra axiom' of [9], we assume that injective map R α → M is dag-phantom, where M is a finitely generated R-module. From Lemma 3.1, this is equivalent to the fact that for every ǫ > 0, there exist an element a ∈ A with v(a) < ǫ and an A-module homomorphism id A ⊗ R γ ǫ : A ⊗ R M → A, such that (id A ⊗ R γ ǫ ) • (id A ⊗ R α) = a(id A ⊗ R id R ).…”

“…Recently, in [9], an additional axiom is introduced, which is known as the 'Algebra Axiom', and it is shown there that many closure operations satisfy this axiom. It is also proved that a local domain R has a Dietz closure that satisfies the Algebra Axiom if and only if R has a big Cohen-Macaulay algebra.…”