arXiv subjects

Kei Nakazato

Publications and source records attributed to Kei Nakazato.

5 recordsLinked to original sources

arXiv2026

Prismatic Kunz's theorem

In this paper, we prove "prismatic Kunz's theorem" which states that a complete Noetherian local ring $R$ of residue characteristic $p$ is a regular local ring if and only if the Frobenius lift on a prismatic complex of (a derived enhancement of) $R$ over a specific prism $(A, I)$ is faithfully flat. This generalizes classical Kunz's theorem from the perspective of extending the "Frobenius map" to mixed characteristic rings. Our approach involves studying the deformation problem of the "regularity" of prisms and demonstrating the faithful flatness of the structure map of the prismatic complex.

math.AC↗

arXiv2025

Perfectoid towers and their tilts : with an application to the étale cohomology groups of local log-regular rings

To initiate a systematic study on the applications of perfectoid methods to Noetherian rings, we introduce the notions of perfectoid towers and their tilts. We mainly show that the tilting operation preserves several homological invariants and finiteness properties. Using this, we also provide a comparison result on étale cohomology groups under the tilting. As an application, we prove finiteness of the prime-to-$p$-torsion subgroup of the divisor class group of a local log-regular ring that appears in logarithmic geometry in the mixed characteristic case.

math.AC↗

arXiv2023

A variant of perfectoid Abhyankar's lemma and almost Cohen-Macaulay algebras

In this paper, we prove that a complete Noetherian local domain of mixed characteristic $p>0$ with perfect residue field has an integral extension that is an integrally closed, almost Cohen-Macaulay domain such that the Frobenius map is surjective modulo $p$. This result is seen as a mixed characteristic analogue of the fact that the perfect closure of a complete local domain in positive characteristic is almost Cohen-Macaulay. To this aim, we carry out a detailed study of decompletion of perfectoid rings and establish the Witt-perfect (decompleted) version of André's perfectoid Abhyankar's lemma and Riemann's extension theorem.

math.AC↗

arXiv2021

Finite étale extension of Tate rings and decompletion of perfectoid algebras

In this paper, we examine the behavior of ideal-adic separatedness and completeness under certain ring extensions using trace map. Then we prove that adic completeness of a base ring is hereditary to its ring extension under reasonable conditions. We aim to give many results on ascent and descent of certain ring theoretic properties under completion. As an application, we give conceptual details to the proof of the almost purity theorem for Witt-perfect rings by Davis and Kedlaya. Witt-perfect rings have the advantage that one does not need to assume that the rings are complete and separated.

math.AC↗

arXiv2018

Counter-examples to non-noetherian Elkik's approximation theorem

Elkik established a remarkable theorem that can be applied for any noetherian henselian ring. For algebraic equations with a formal solution (restricted by some smoothness assumption), this theorem provides a solution adically close to the formal one in the base ring. In this paper, we show that the theorem would fail for some non-noetherian henselian rings. These rings do not satisfy several conditions weaker than noetherianness, such as weak proregularity (due to Grothendieck et al.) of the defining ideal. We describe the resulting pathologies.

math.AC↗