A mixed characteristic analogue of the perfection of rings and its almost Cohen-Macaulay property
Ryo Ishizuka, Kazuma Shimomoto
TL;DR
The paper develops a mixed-characteristic analogue of the perfect closure by constructing a perfectoid-like ring inside the absolute integral closure of a complete Noetherian local domain of mixed characteristic with a perfect residue field, via adjoining compatible systems of $p$-power roots. It establishes that the resulting completed ring is $(pg)^{1/p^\infty}$-almost Cohen-Macaulay over a Cohen-structure base, leveraging André's flatness lemma and a Riemann extension theorem, and connects the construction with notions of perfectoidization from prismatic cohomology. The approach provides an explicit, intrinsic mixed-characteristic object with perfection-like properties, enabling almost-flat and almost-faithful behavior relative to $A$, and yields almost surjectivity/injectivity statements to the absolute integral closure and to tilts. This work bridges perfectoid techniques with classical commutative algebra, offering a concrete tool for transferring positive-characteristic ideas to mixed-characteristic contexts and highlighting the role of uniform completion in obtaining the main results.
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 André'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.