Closure operations that induce big Cohen-Macaulay algebras (Q1703599)

The paper under review studies the concept of tight closure and its alternatives for commutative Noetheiran rings with unity, in hope of extending the definition of tight closure to a theory in mixed characteristic with similar success to that in the equal characteristic case. The focus is on the concept of a Dietz closure introduced in [\textit{G. D. Dietz}, Proc. Am. Math. Soc. 138, No. 11, 3849--3862 (2010; Zbl 1206.13017)] and, in particular, a question raised by Dietz on whether one can add an additional axiom to the Dietz closure axioms to characterize the existence of big Cohen-Macaulay algebras. The main result of the paper accomplishes this. The author introduces an axiom callled the Algebra Axiom and then shows the following result: {Theorem}. 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 conjuction with this result, the author investigates various types of closures, including a class of closures denoted \(cl_S\) that can be associated to a family \(S\) of \(R\)-algebras. When \(B\) is a big Cohen-Macaulay algebra, it is shown that \(cl_B\) is a Dietz closure that satisfies the Algebra Axiom. In the paper it is also shown that there exist Dietz closures that do not satisfy the Algebra Axiom. The results of the paper are especially relevant now that big Cohen-Macaulay algebras are proved to exist in mixed characteristic (in addition to their already established existence in equal characteristic) as a consequence of work of \textit{Y. André} [Publ. Math., Inst. Hautes Étud. Sci. 127, 71--93 (2018; Zbl 1419.13029)].