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

Persistent Link:
http://hdl.handle.net/10150/613179
Title:
Kisin-Ren Classification of ϖ-divisible O-modules via the Dieudonné Crystal
Author:
Henniges, Alex Jay
Issue Date:
2016
Publisher:
The University of Arizona.
Rights:
Copyright © 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.
Abstract:
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.
Type:
text; Electronic Dissertation
Keywords:
crystal; divided powers; divisible modules; p-divisible group; Mathematics; Breuil modules
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Cais, Bryden R.

Full metadata record

DC FieldValue Language
dc.language.isoen_USen
dc.titleKisin-Ren Classification of ϖ-divisible O-modules via the Dieudonné Crystalen_US
dc.creatorHenniges, Alex Jayen
dc.contributor.authorHenniges, Alex Jayen
dc.date.issued2016-
dc.publisherThe University of Arizona.en
dc.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.en
dc.description.abstractLet 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.en
dc.typetexten
dc.typeElectronic Dissertationen
dc.subjectcrystalen
dc.subjectdivided powersen
dc.subjectdivisible modulesen
dc.subjectp-divisible groupen
dc.subjectMathematicsen
dc.subjectBreuil modulesen
thesis.degree.namePh.D.en
thesis.degree.leveldoctoralen
thesis.degree.disciplineGraduate Collegeen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorUniversity of Arizonaen
dc.contributor.advisorCais, Bryden R.en
dc.contributor.committeememberTiep, Pham H.en
dc.contributor.committeememberJoshi, Kirti N.en
dc.contributor.committeememberSharifi, Romyaren
dc.contributor.committeememberCais, Bryden R.en
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