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Let $A$ is an abelian scheme over a base scheme $S$. Let $S \rightarrow S'$ be a thickening defined by an ideal of square zero (for example).

If $p$ is locally nilpotent on $S$, then Serre-Tate and Grothendieck-Messing imply that lifting $A$ to $S'$ is "the same" as lifting the Hodge filtration on the Dieudonné crystal (evaluated on $S$) of the associated $p$-divisible group.

Is there anything similar if the base scheme $S$ is characteristic zero? Mixed characteristic? Say, using the Hodge filtration on algebraic de Rham cohomology?

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    $\begingroup$ I think one needs to use not just algebraic de Rham cohomology but its isomorphism with singular cohomology for this. $\endgroup$
    – Will Sawin
    Commented Oct 11, 2023 at 0:05
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    $\begingroup$ You can make this work with crystalline cohomology of abelian schemes in char 0. Essentially you just have to show that the Kodaira-Spencer map is the same as whet you get from the hodge filtration thing. It is easier than the char p case (especially if you first prove the thing for p-divisible groups which is quite a bit harder as you have to prove unobstructedness first). $\endgroup$
    – Johan
    Commented Oct 11, 2023 at 19:00
  • $\begingroup$ Thanks for the helpful comments. @Johan Do you know whether this is written down somewhere? $\endgroup$
    – 351910953
    Commented Oct 12, 2023 at 14:37

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