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Let me upgrade my comment to an answer: The answer is no, the filtration is not independent of the lifting. In fact the relationship between liftings of $Y$ and filtrations lifting the Hodge filtratio …

Yes. A Google search will immediately give you lots of articles in varying generality (see e.g. work of Shiho), but one article I'm particularly fond of is Kim and Hain's "A de Rham-Witt approach to c …

I shall try to stick to the notation in Katz's paper. Let $k$ be a perfect field of characteristic $p>0$. Let $S_{\infty}$ be a $p$-adically complete and separated smooth formal $W(k)$-scheme and $f:X …

This is an old question but since it hasn't received much attention, let me just point out "the next" example beyond that given in the question: Let $k$ be a perfect field of characteristic $p>0$, and …

Let me add a different answer. Let $p$ be a prime with $p>\dim X$. Let $\mathcal{K}_{i}$ denote the higher $K$-sheaf on $X$, and let $S\mathcal{K}_{i}:=\mathrm{im}((\mathcal{O}_{X}^{\times})^{\oplus i …