arXiv subjects

Shinnosuke Ishiro

Publications and source records attributed to Shinnosuke Ishiro.

7 recordsLinked to original sources

arXiv2026

An application of Fontaine's monoidal maps to perfectoid towers

To connect arithmetic and ring-theoretic properties of rings of mixed characteristic with those of positive characteristic, we introduce monoidal maps for perfectoid towers. Using these maps, we discuss the almost integrality of perfectoid towers and of their tilts. We also show that the towers constructed by F. Andreatta via ramification theory become perfectoid towers, and we apply the monoidal maps to deduce the normality of their small tilts.

math.AC↗

arXiv2025

{\delta}-rings, perfectoid towers, and lim Cohen-Macaulay sequences

The aim of this article is to study basic structures and interrelations of $\delta$-rings, perfectoid towers, and lim Cohen--Macaulay sequences over Noetherian rings in positive or mixed characteristic. We also discuss the deformation of perfectoid purity via perfectoid towers. In the latter part of this paper, we discuss some methods for constructing perfectoid towers, dealing with $p$-torsion-free and $p$-torsion cases, respectively. Some interesting examples arise as quotients by monomial or binomial ideals or determinantal rings. We also explain a geometric method with a view toward constructing rings with certain singularities.

math.AC↗

arXiv2025

Log-regularity of monoid algebras

Local log-regular rings are Cohen-Macaulay local domains introduced by Kazuya Kato to expand the theory of toric varieties without a base. In this note, we show that local rings of monoid algebras over regular rings are log-regular. As a corollary, we show that monoid algebras are log-regular rings.

math.AC↗

arXiv2022

Local log-regular rings vs. toric rings

Local log-regular rings are a certain class of Cohen-Macaulay local rings that are treated in logarithmic geometry. Our paper aims to provide purely commutative ring theoretic proof of some ring-theoretic properties of local log-regular rings such as an explicit description of a canonical module, and the finite generation of the divisor class group.

math.AC↗

arXiv2022

Perfectoid towers and their tilts : with an application to the \'etale 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 \'etale 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↗

arXiv2021

Surjectivity of some local cohomology map and the second vanishing theorem

The second vanishing theorem has a long history in the theory of local cohomology modules, which connects the vanishing of a complete regular local ring with a topological property of the punctured spectrum of the ring under some conditions. However, the case of complete ramified regular local rings is unresolved. In this paper, we give a partial answer to the second vanishing theorem in the ramified case. Our proof is inspired by the theory of surjective elements in the theory of local cohomology.

math.AC↗

arXiv2020

Another proof of the almost purity theorem for perfectoid valuation rings

The almost purity theorem is central to the geometry of perfectoid spaces and has numerous applications in algebra and geometry. This result is known to have several different proofs in the case that the base ring is a perfectoid valuation ring. We give a new proof by exploiting the behavior of Faltings' normalized length under the Frobenius map.

math.AC↗