Représentations Cristallines et $F$-Cristaux: le Cas d'un Corps Résiduel Imparfait

Brinon, Olivier; Trihan, Fabien

doi:10.4171/rsmup/119-4

Cited by 6 publications

(24 citation statements)

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“…Remark 4.6. Analogous statements hold when T is a crystalline G R -representation with Hodge-Tate weights in [0, r] for the case er < p − 1, since [BT08] constructs more generally a functor from crystalline representations with Hodge-Tate weights in [0, r] to Kisin modules of height r when the base is a complete discrete valuation field whose residue field has a finite p-basis.…”

Section: Construction Of Kisin Modulesmentioning

confidence: 83%

“…is an example of a complete discrete valuation field with a residue field having a finite p-basis, studied in [BT08]. On the other hand, for each maximal ideal q ∈ mSpecR 0 , let R 0,q be the q-adic completion of R 0,q .…”

Section: Base Ring and Crystalline Period Ring In The Relative Casementioning

confidence: 99%

“…There are three major ingredients for the proof of Theorem 1.2. Firstly, Brinon and Trihan proved in [BT08] the generalization of Theorem 1.1 for the case when the base is a complete discrete valuation ring whose residue field has a finite p-basis. We use this result together with the fact that the p-adic completion of R 0,(p) is an example of such rings studied loc.…”

Section: Introductionmentioning

confidence: 99%

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Relative crystalline representations and $p$-divisible groups in the small ramification case

Liu,

Moon

2019

Preprint

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Let k be a perfect field of characteristic p > 2, and let K be a finite totally ramified extension over W (k)[ 1 p ] of ramification degree e. Let R 0 be a relative base ring over W (k) t ±1 1 , . . . , t ±1 m satisfying some mild conditions, and let R = R 0 ⊗ W (k) O K . We show that if e < p − 1, then every crystalline representation of π ét 1 (SpecR[ 1 p ]) with Hodge-Tate weights in [0, 1] arises from a p-divisible group over R.

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Section: Construction Of Kisin Modulesmentioning

confidence: 83%

Section: Base Ring and Crystalline Period Ring In The Relative Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Relative crystalline representations and $p$-divisible groups in the small ramification case

Liu,

Moon

2019

Preprint

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“…Theorem 1 was first proved by Breuil [Bre00] and all the remaining statements were proved by Kisin [Kis06]. Brinon and Trihan [BT08] generalised the results of Kisin [Kis06] and Faltings [Fal99,§6] to p-divisible groups over a p-adic discrete valuation ring with imperfect residue field admitting finite p-basis.…”

mentioning

confidence: 98%

“…We then prove Proposition 2 by generalising [CL09, §2.2]. This allows us to avoid the rigid analytic construction (cf., [Kis06,§1] and [BT08,§4]), which is hard to generalise to the relative setting. To prove Theorem 5, we generalise the strategy of [Kis06,§2.3].…”

mentioning

confidence: 99%

The Relative Breuil–Kisin Classification ofp-Divisible Groups and Finite Flat Group Schemes

Kim

2014

Int Math Res Notices

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Assume that p > 2, and let O K be a p-adic discrete valuation ring with residue field admitting a finite p-basis, and let R be a formally smooth formally finite-type O K -algebra. (Indeed, we allow slightly more general rings R.) We construct an anti-equivalence of categories between the categories of p-divisible groups over R and certain semi-linear algebra objects which generalise (ϕ, S)-modules of height 1 (or Kisin modules). A similar classification result for p-power order finite flat group schemes is deduced from the classification of p-divisible groups. We also show compatibility of various construction of (Zp-lattice or torsion) Galois representations, including the relative version of Faltings' integral comparison theorem for p-divisible groups. We obtain partial results when p = 2.Ki S (ϕ, ∇ 0 ) of certain "torsion Kisin S-modules" (Definition 9.2.1). Furthermore, we have a natural G R∞ -equivariant isomorphism H(R) ∼ = T * (M * (H)).Let us review previous results when p > 2. When R = O K with perfect residue field, all our main results are known already. Theorem 4(1) is proved by Faltings [Fal99, §6], while the first assertion was already proved by Fontaine [Fon82, 1 Indeed, we will work with slightly more general rings than Brinon [Bri06,Bri08]. See §4.4 for more details.2 In this paper, we only consider commutative finite locally free group schemes, so we will often suppress the adjective "commutative". Definition 6.1.6) as follows:Lemma 6.2.2. Assume that p = 2 and R satisfies the p-basis condition ( §2.2.1). Then the functors Mod S (ϕ) ϕ nilp → MF S (ϕ) ϕ nilp and Mod S (ϕ) ψ nilp → MF S (ϕ) ψ nilp , defined by S ⊗ ϕ,S (·), are fully faithful. The same statement holds for the full subcategories of ϕand ψnilpotent objects in Mod S (ϕ, ∇) and Mod KiThe same proof of Lemma 6.2 shows that it suffices to prove the lemma when R is a p-adic discrete valuation ring with perfect residue field, which follows from combining Proposition 1.1.9 and Theorem 1.2.8 in [Kis09]. Proposition 6.3. Assume that R satisfies the p-basis condition ( §2.2.1). Then the functors Mod, are essentially surjective. In particular, they are equivalences of categories if p > 2 or if they are restricted to ϕand ψnilpotent objects, and are equivalence of categories up to isogeny if p = 2.We prove the proposition later in §6.4. Let us record some interesting corollaries. The following is immediate from Corollary 6.3.1. Assume that R satisfies the p-basis condition ( §2.2.1). If p > 2 then there exists an exact contravariant functorsatisfies the normality assumption ( §2.2.4), and an anti-equivalence of categories if R satisfies the formally finite-type assumption ( §2.2.2). If p = 2 then we have an exact contravariant functor M * [ 1 p ] : G → M * (G)[ 1 p ] on the isogeny categories {p-divisible groups over R}[1/p] → Mod S (ϕ, ∇)[1/p], which is fully faithful if R satisfies the formally finite-type assumption ( §2.2.2).When p = 2, we have the following strengthening of Corollary 6.3.1 for ϕand ψnilpotent objects: Corollary 6.3.2. Let ...

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Completed prismatic F-crystals and crystalline Z_p-local systems

Du,

Liu,

Moon

et al. 2024

Compositio Math.

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We introduce the notion of completed $F$ -crystals on the absolute prismatic site of a smooth $p$ -adic formal scheme. We define a functor from the category of completed prismatic $F$ -crystals to that of crystalline étale $\mathbf {Z}_p$ -local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a mixed characteristic complete discrete valuation ring with perfect residue field.

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Représentations Cristallines et $F$-Cristaux: le Cas d'un Corps Résiduel Imparfait

Cited by 6 publications

References 5 publications

Relative crystalline representations and $p$-divisible groups in the small ramification case

Relative crystalline representations and $p$-divisible groups in the small ramification case

The Relative Breuil–Kisin Classification ofp-Divisible Groups and Finite Flat Group Schemes

Completed prismatic F-crystals and crystalline Z_p-local systems

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Représentations Cristallines et $F$-Cristaux: le Cas d'un Corps Résiduel Imparfait

Cited by 6 publications

References 5 publications

Relative crystalline representations and $p$-divisible groups in the small ramification case

Relative crystalline representations and $p$-divisible groups in the small ramification case

The Relative Breuil–Kisin Classification ofp-Divisible Groups and Finite Flat Group Schemes

Completed prismatic F-crystals and crystalline Zp-local systems

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Completed prismatic F-crystals and crystalline Z_p-local systems