Breuil–Kisin modules via crystalline cohomology

Cais, Bryden; Liu, Tong

doi:10.1090/tran/7280

Breuil–Kisin modules via crystalline cohomology
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by Bryden Cais and Tong Liu PDF

Trans. Amer. Math. Soc. 371 (2019), 1199-1230 Request permission

Corrigendum: Trans. Amer. Math. Soc. 373 (2020), 2251-2252.

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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