A mixed characteristic analogue of the perfection of rings and its almost Cohen-Macaulay property

Abstract: Over a complete Noetherian local domain of mixed characteristic with perfect residue field, we construct a perfectoid ring which is similar to an explicit representation of a perfect closure in positive characteristic. Then we demonstrate that this perfectoid ring is almost Cohen-Macaulay in the sense of almost ring theory. The proof of this result uses Andr\'e's flatness lemma along with Riemann's extension theorem. We stress that the idea partially originates from the "perfectoidization" in the theory of prismatic cohomology.