Finite étale extensions of Tate rings and decompletion of perfectoid algebras

Nakazato, Kei; Shimomoto, Kazuma

doi:10.1016/j.jalgebra.2021.09.018

Cited by 5 publications

(32 citation statements)

References 16 publications

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“…Indeed, Davis and Kedlaya succeeded in formulating and proving the almost purity theorem for Witt-perfect rings. The present article is seen as a sequel to authors' previous work [44], in which the authors were able to give a conceptual proof to the almost purity theorem by Davis-Kedlaya by analyzing the integral structure of Tate rings under completion. The advantage of working with Witt-perfect rings is that it allows one to take an infinite integral extension over a certain p-adically complete ring to construct an almost Cohen-Macaulay algebra.…”

Section: Introductionmentioning

confidence: 89%

“…Let A 0 be a ring of definition of A and let t ∈ A 0 be a pseudouniformizer of A. As in the proof of [44,Lemma 2.17], we can take a ring of definition B 0 of B that is finitely generated as an A 0 -module and satisfies…”

Section: ) Suppose That a Is Integrally Closed Inmentioning

confidence: 99%

“…Let us recall the almost version of the Witt-perfect condition as introduced in [44]. Definition 4.2 (Almost Witt-perfect ring).…”

Section: A Variant Of Perfectoid Abhyankar's Lemma For Almost Witt-pe...mentioning

confidence: 99%

“…The regularity assumption on the sequence (p, g) ensures that p, g are nonzero divisors on the p-adic completion A; this is due to Lemma 4.5. A detailed study of almost Witt-perfect rings appears in the paper [44]; see also Definition 4.2 below. The functor A → g…”

Section: Introductionmentioning

confidence: 99%

“…The almost purity theorem for Witt-perfect rings is attributed to Davis and Kedlaya; see [17] and [18]. A systematic approach to this important result was carried out in authors' paper [44]. The assumption that A is integral over a Noetherian ring as appearing in Main Theorem 1 is required by the following Witt-perfect Riemann's Extension Theorem; see Theorem 4.15 as well as notation.…”

Section: Introductionmentioning

confidence: 99%

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

Nakazato¹,

Shimomoto²

2020

Preprint

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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 algebra such that the Frobenius map is surjective modulo p. This is seen as a mixed characteristic analogue of the classical result of Hochster-Huneke for the absolute integral closure in characteristic p > 0. To achieve this goal, we establish the Witt-perfect version of André's perfectoid Abhyankar's lemma and Riemann's extension theorem. The advantage of Witt-perfet rings is that they are not assumed to be p-adically complete.

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Section: Introductionmentioning

confidence: 89%

Section: ) Suppose That a Is Integrally Closed Inmentioning

confidence: 99%

“…Let us recall the almost version of the Witt-perfect condition as introduced in [44]. Definition 4.2 (Almost Witt-perfect ring).…”

Section: A Variant Of Perfectoid Abhyankar's Lemma For Almost Witt-pe...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Nakazato¹,

Shimomoto²

2020

Preprint

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Tilting and untilting for ideals in perfectoid rings

Dine,

Ishizuka

2024

Math. Z.

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For a perfectoid ring R of characteristic 0 with tilt $$R^{\flat }$$ R ♭ , we introduce and study a tilting map $$(-)^{\flat }$$ ( - ) ♭ from the set of p-adically closed ideals of R to the set of ideals of $$R^{\flat }$$ R ♭ and an untilting map $$(-)^{\sharp }$$ ( - ) ♯ from the set of radical ideals of $$R^{\flat }$$ R ♭ to the set of ideals of R. The untilting map $$(-)^{\sharp }$$ ( - ) ♯ is defined purely algebraically and generalizes the analytically defined untilting map on closed radical ideals of a perfectoid Tate ring of characteristic p introduced in the first author’s previous work. We prove that the two maps $$\begin{aligned} J\mapsto J^{\flat }~\text {and}~I\mapsto I^{\sharp } \end{aligned}$$ J ↦ J ♭ and I ↦ I ♯ define an inclusion-preserving bijection between the set of ideals J of R such that the quotient R/J is perfectoid and the set of $$p^{\flat }$$ p ♭ -adically closed radical ideals of $$R^{\flat }$$ R ♭ , where $$p^{\flat }\in R^{\flat }$$ p ♭ ∈ R ♭ corresponds to a compatible system of p-power roots of a unit multiple of p in R. Finally, we prove that the maps $$(-)^{\flat }$$ ( - ) ♭ and $$(-)^{\sharp }$$ ( - ) ♯ send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of $${{\,\textrm{Spec}\,}}(R)$$ Spec ( R ) consisting of prime ideals $$\mathfrak {p}$$ p of R such that $$R/\mathfrak {p}$$ R / p is perfectoid and the subspace of $${{\,\textrm{Spec}\,}}(R^{\flat })$$ Spec ( R ♭ ) consisting of $$p^{\flat }$$ p ♭ -adically closed prime ideals of $$R^{\flat }$$ R ♭ . In particular, we obtain a generalization and a new proof of the main result of the first author’s previous work which concerned prime ideals in perfectoid Tate rings.

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A Variant of Perfectoid Abhyankar’s Lemma and Almost Cohen–macaulay Algebras

NAKAZATO,

SHIMOMOTO

2023

Nagoya Math. J.

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In this article, 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 analog 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.

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Finite étale extensions of Tate rings and decompletion of perfectoid algebras

Cited by 5 publications

References 16 publications

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

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

Tilting and untilting for ideals in perfectoid rings

A Variant of Perfectoid Abhyankar’s Lemma and Almost Cohen–macaulay Algebras

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