New methods for $$(\varphi, \Gamma)$$ ( φ , Γ ) -modules

Kedlaya, Kiran S.

doi:10.1186/s40687-015-0031-z

Cited by 9 publications

(6 citation statements)

References 16 publications

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“…Proof. We reproduce here [26,Corollary 1.4.14]. By Proposition 1.14(iii), there exists x ∈ o K such that y := θ(x)/p is a unit in o K .…”

Section: Tilting Untilting and Witt Vectorsmentioning

confidence: 95%

“…Proof. (The following argument is extracted from [26,Lemma 1.5.4]; see also [44, To this end, we construct a sequence z 0 , z 1 , . .…”

mentioning

confidence: 99%

“…Proof. See [26,Lemma 1.3.3] or [44,Lemma 3.4]. (While the basic construction described here was known to Fontaine, the term tilt, and the notations and , were introduced by Scholze in [44].)…”

mentioning

confidence: 99%

“…Proof of Proposition 1.16. We follow the proof of [26,Theorem 1.5.6]. See [44,Theorem 3.7] for a somewhat different approach (using almost ring theory in place of Witt vectors).…”

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confidence: 99%

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Perfectoid Spaces: An Annotated Bibliography

Kedlaya

2022

Infosys Science Foundation Series

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This annotated bibliography was prepared as part of a five-lecture series at the summer school on perfectoid spaces held at the International Centre for Theoretical Sciences (ICTS), Bengaluru, September 9-13, 2019. It is not intended to be a freestanding reference, although we do include a few short proofs and some sketches of longer proofs; instead, I have attempted to give some complements to my Arizona Winter School 2017 lecture notes [28], which provide a far more complete version of the story.Throughout, fix a prime number p.

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“…Proof. We reproduce here [26,Corollary 1.4.14]. By Proposition 1.14(iii), there exists x ∈ o K such that y := θ(x)/p is a unit in o K .…”

Section: Tilting Untilting and Witt Vectorsmentioning

confidence: 95%

“…Proof. (The following argument is extracted from [26,Lemma 1.5.4]; see also [44, To this end, we construct a sequence z 0 , z 1 , . .…”

mentioning

confidence: 99%

mentioning

confidence: 99%

“…Proof of Proposition 1.16. We follow the proof of [26,Theorem 1.5.6]. See [44,Theorem 3.7] for a somewhat different approach (using almost ring theory in place of Witt vectors).…”

mentioning

confidence: 99%

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Perfectoid Spaces: An Annotated Bibliography

Kedlaya

2022

Infosys Science Foundation Series

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“…It has become clear that essentially all of p-adic Hodge theory can be formulated in terms of (ϕ, Γ)-modules; moreover, this formulation has driven much recent progress in the subject and powered some notable applications in arithmetic geometry [17]. See [31] for a quick introduction to this circle of ideas or [43] for a more in-depth treatment. Multivariable (ϕ, Γ)-modules are also related [19,32] to Scholze's theory of perfectoid spaces.…”

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confidence: 99%

Multivariable $(\varphi ,\Gamma )$-modules and representations of products of Galois groups: The case of the imperfect residue field

Ray¹,

Wei²,

Zábrádi³

2021

Bul. Soc. Math. France

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Let K be a complete discretely valued field with mixed characteristic (0, p) and imperfect residue field k α . Let ∆ be a finite set. We construct an equivalence of categories between finite dimensional F p -representations of the product of ∆ copies of the absolute Galois group of K and multivariable étale (ϕ, Γ)-modules over a multivariable Laurent series ring over k α .Résumé ((ϕ, Γ)-modules multivariables et représentations du produit du groupe de Galois: le cas des corps résiduels imparfaits) Soit K un corps discrètement valué à charactéristique mixte (0, p) et un corps résiduel imparfait k α . Soit ∆ un ensemble fini. Nous établissons une équivalence de catégories entre des représentations de dimensions finies sur F p du produit de ∆ copies du groupe absolu de Galois de K et des (ϕ, Γ)-modules étales multivariables sur un anneau multivariable des séries Laurent sur k α .

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Compairing Categories of Lubin–Tate $$(\varphi _L,\Gamma _L)$$-Modules

Schneider,

Venjakob

2023

Results Math

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In the Lubin–Tate setting we compare different categories of $$(\varphi _L,\Gamma )$$ ( φ L , Γ ) -modules over various perfect or imperfect coefficient rings. Moreover, we study their associated Herr-complexes. Finally, we show that a Lubin Tate extension gives rise to a weakly decompleting, but not decompleting tower in the sense of Kedlaya and Liu.

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New methods for $$(\varphi, \Gamma)$$ ( φ , Γ ) -modules

Cited by 9 publications

References 16 publications

Perfectoid Spaces: An Annotated Bibliography

Perfectoid Spaces: An Annotated Bibliography

Multivariable $(\varphi ,\Gamma )$-modules and representations of products of Galois groups: The case of the imperfect residue field

Compairing Categories of Lubin–Tate $$(\varphi _L,\Gamma _L)$$-Modules

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