Lifting the Cartier transform of Ogus-Vologodsky modulo $p^n$

Abstract

Let $W$ be the ring of the Witt vectors of a perfect field of characteristic $p$, $\mathfrak{X}$ a smooth formal scheme over $W$, ${\mathfrak{X}}^{\prime }$ the base change of $\mathfrak{X}$ by the Frobenius morphism of $W$, ${\mathfrak{X}}_2^{\prime }$ the reduction modulo $p^2$ of ${\mathfrak{X}}^{\prime }$ and $X$ the special fiber of $\mathfrak{X}$. We lift the Cartier transform of Ogus-Vologodsky defined by ${\mathfrak{X}}_2^{\prime }$ modulo $p^n$. More precisely, we construct a functor from the category of $p^n$-torsion ${\mathcal{O}}_{{\mathfrak{X}}^{\prime }}$-modules with integrable $p$-connection to the category of $p^n$-torsion ${\mathcal{O}}_{\mathfrak{X}}$-modules with integrable connection, each subject to suitable nilpotence conditions. Our construction is based on Oyama's reformulation of the Cartier transform of Ogus-Vologodsky in characteristic $p$. If there exists a lifting $F:\mathfrak{X}\to {\mathfrak{X}}^{\prime }$ of the relative Frobenius morphism of $X$, our functor is compatible with a functor constructed by Shiho from $F$. As an application, we give a new interpretation of Faltings' relative Fontaine modules and of the computation of their cohomology.

96 pages, final version, to appear in Mémoires de la Société Mathématique de France

References in corpus (1)

• Localization of modules for a semisimple Lie algebra in prime characteristic

Cited by in corpus (7)

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• Cartier transform and prismatic crystals
• A Simpson correspondence for abelian varieties in characteristic p > 0
• Relative Fontaine-Messing theory over power series rings
• Prismatic crystals for smooth schemes in characteristic $p$ with Frobenius lifting mod $p^2$