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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Breuil–Kisin modules via crystalline cohomology

Authors: Bryden Cais and Tong Liu
Journal: Trans. Amer. Math. Soc. 371 (2019), 1199-1230
MSC (2010): Primary 14F30; Secondary 11F80
Published electronically: September 20, 2018
Corrigendum: Trans. Amer. Math. Soc. 373 (2020), 2251-2252.
MathSciNet review: 3885176
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Abstract: For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathcal {X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1/p]$, we propose a cohomological construction of the Breuil–Kisin module attached to the $p$-adic étale cohomology $H^i_{\text {\'et}}(X_{\overline {K}},\mathbf {Z}_p)$. We then prove that our proposal works when $p>2$, $i < p-1$, and the crystalline cohomology of the special fiber of $\mathcal {X}$ is torsion-free in degrees $i$ and $i+1$.

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Additional Information

Bryden Cais
Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
MR Author ID: 797038

Tong Liu
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
MR Author ID: 638721

Keywords: Breuil–Kisin modules, crystalline cohomology
Received by editor(s): October 27, 2016
Received by editor(s) in revised form: May 5, 2017
Published electronically: September 20, 2018
Additional Notes: The first author was partially supported by a Simons Foundation Collaboration Grant.
The second author is partially supported by NSF grant DMS-1406926.
Article copyright: © Copyright 2018 American Mathematical Society