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Breuil $ \mathcal{O}$-windows and $ \pi$-divisible $ \mathcal{O}$-modules


Author: Chuangxun Cheng
Journal: Trans. Amer. Math. Soc. 370 (2018), 695-726
MSC (2010): Primary 14L05, 14K10
DOI: https://doi.org/10.1090/tran/7019
Published electronically: August 3, 2017
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Abstract: Let $ p>2$ be a prime number. Let $ \mathcal {O}$ be the ring of integers of a finite extension of $ \mathbb{Q}_p$ and let $ \pi $ be a uniformizer of $ \mathcal {O}$. We prove that, for any complete Noetherian regular local $ \mathcal {O}$-algebra $ R$ with perfect residue field of characteristic $ p$, the category of Breuil $ \mathcal {O}$-windows over $ R$ is equivalent to the category of $ \pi $-divisible $ \mathcal {O}$-modules over $ R$. We also prove that the category of Breuil $ \mathcal {O}$-modules over $ R$ is equivalent to the category of commutative finite flat $ \mathcal {O}$-group schemes over $ R$ which are kernels of isogenies of $ \pi $-divisible $ \mathcal {O}$-modules. As an application of these equivalences, we then prove a boundedness result on Barsotti-Tate groups and deduce some corollaries. The results generalize some earlier results of Zink, Vasiu-Zink, and Lau.


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Additional Information

Chuangxun Cheng
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: cxcheng@nju.edu.cn

DOI: https://doi.org/10.1090/tran/7019
Keywords: $\mathcal{O}$-frames, $\mathcal{O}$-windows, Breuil modules, $p$-divisible groups, $\pi$-divisible $\mathcal{O}$-modules, group schemes, regular rings
Received by editor(s): January 13, 2015
Received by editor(s) in revised form: September 15, 2015, and May 2, 2016
Published electronically: August 3, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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