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.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/613179 |
| Date | January 2016 |
| Creators | Henniges, Alex Jay |
| Contributors | Cais, Bryden R., Tiep, Pham H., Joshi, Kirti N., Sharifi, Romyar, Cais, Bryden R. |
| Publisher | The University of Arizona. |
| Source Sets | University of Arizona |
| Language | en_US |
| Detected Language | English |
| Type | text, Electronic Dissertation |
| 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. |
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