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Kisin-Ren Classification of ϖ-divisible O-modules via the Dieudonné Crystal

Let k be a perfect field of characteristic p > 2 and K a totally ramified extension of K₀ = Frac W(k) with uniformizer π. Let F ⊆ K be a subfield with ϖ, ring of integers O, and residue field k(F) ⊆ k with |k(F)| = q. Let W(F) = O⊗(W(k(F))) W(k) and consider the ring 𝔖 = W(F)⟦u⟧ with an endomorphism φ that lifts the q-power Frobenius of k on W(F) and satisfies φ(u) ≡ u^q mod ϖ and φ(u) ≡ 0 mod u. In this dissertation, we use O-divided powers to define the analogue of Breuil-Kisin modules over the rings 𝔖 and S, where S is an O-divided power envelope of the surjection 𝔖 ↠ O(K) sending u to π. We prove that these two module categories are equivalent, generalizing the case when F = Q(p) and ϖ - p. As an application of our theory, we generalize the results of Kisin [17] and Cais-Lau [8] to relate the Faltings Dieudonné crystal of a ϖ-divisible O-module, which gives a Breuil module over S in our sense, to the modules of Kisin-Ren, providing a geometric interpretation to the latter.

Identiferoai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/613179
Date January 2016
CreatorsHenniges, Alex Jay
ContributorsCais, Bryden R., Tiep, Pham H., Joshi, Kirti N., Sharifi, Romyar, Cais, Bryden R.
PublisherThe University of Arizona.
Source SetsUniversity of Arizona
Languageen_US
Detected LanguageEnglish
Typetext, Electronic Dissertation
RightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.

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