An elementary proof of Cohen-Gabber theorem in the equal characteristic $p&gt;0$ case

Let K denote an n-dimensional local field. The aim of this expository paper is to survey the basic arithmetic theory of the n-dimensional local field K together with its Milnor Ktheory and Parshin topological K-theory; to review Kato's ramification theory for finite abelian extensions of the n-dimensional local field K, and to state the local abelian higherdimensional K-theoretic generalization of local abelian class field theory of Hasse, which is developed by Kato and Parshin. The paper is geared toward non-abelian generalization of this theory.

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“…The main references for this section are [33] and the excellent reviews [36,37,41,50]. Let K be an n-dimensional local field.…”

Section: N-dimensional Local Fieldsmentioning

confidence: 99%

“…is a prime element of K 1 with respect to ν K 1 . So, following [33] and [42,44,45], n-dimensional local fields can be classified as follows. For the n-dimensional local field K:…”

Section: N-dimensional Local Fieldsmentioning

confidence: 99%

Local abelian Kato-Parshin reciprocity law: A survey

Ikeda

Serbest

2021

Hacettepe Journal of Mathematics and Statistics

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“…This is what we aim to illustrate. More precisely, let (R, m, k) be a reduced complete local F p -algebra of dimension d. 27 Let X be the spectrum of R. By Cohen-Gabber theorem (see [KS18] for an elementary proof), R admits a generically étale Noether normalization, i.e. there is a generically étale finite (sub)extension A := k x 1 , .…”

Section: Trace Transposable Maps Along Noether Normalizations Of Cohe...mentioning

confidence: 99%

On the behavior of $F$-signatures, splitting primes, and test modules under finite covers

Carvajal-Rojas¹,

Stäbler²

2019

Preprint

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We give a comprehensive treatment on how certain fundamental objects in Cartier theory such a F -signatures, splitting primes, splitting ratios, and test modules behave under finite covers. We recover previously known results as particular instances. To this end, we expand on the notion of transposability along a section section of the relative canonical module as first introduced by K. Schwede and K. Tucker.

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“…Proof. Note first that, by the Cohen-Gabber Structure Theorem, every complete local ring admits such a coefficient field and system of parameters; see [KS15] for an elementary proof. Let K = Frac(A) ⊆ L = Frac(R) be the corresponding extension of fraction fields.…”

Section: Preliminariesmentioning

confidence: 99%

F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison

Polstra

Tucker

2018

Alg. Number Th.

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We present a unified approach to the study of F -signature, Hilbert-Kunz multiplicity, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that give vastly simplified proofs of existence, semicontinuity, and positivity. Furthermore, we give an affirmative answer to a question of Watanabe and Yoshida allowing the F -signature to be viewed as the infimum of relative differences in the Hilbert-Kunz multiplicites of the cofinite ideals in a local ring.

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An elementary proof of Cohen-Gabber theorem in the equal characteristic $p>0$ case

Abstract: The aim of this article is to give a new proof of Cohen-Gabber theorem in the equal characteristic p > 0 case.

Cited by 12 publications

References 4 publications

Local abelian Kato-Parshin reciprocity law: A survey

Local abelian Kato-Parshin reciprocity law: A survey

On the behavior of $F$-signatures, splitting primes, and test modules under finite covers

F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison

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