Given a smooth proper scheme over a mixed characteristic DVR, we try to understand to what extent the special fiber knows the Hodge numbers of the generic fiber. I'll provide some examples as well as a theorem showing that in "good" situations some numbers defined purely in terms of the special fiber actually give the Hodge numbers of the generic fiber. This naturally leads to consideration of Breuil--Kisin prismatic cohomology, and I'll describe an example which illustrates certain pathological behavior of Hodge--Tate and Hodge--de Rham spectral sequences in mixed characteristic situations.
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