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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
DOI: https://doi.org/10.1090/tran/7280
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
Email: cais@math.arizona.edu

Tong Liu
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: tongliu@math.purdue.edu

DOI: https://doi.org/10.1090/tran/7280
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