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For a smooth proper variety $X$ over discrete valuation ring $\mathcal{O}$ of mixed characteristic $(0,p)$, let $X_K$ be the generic fibre over a generic point $\textbf{Spec} K$ and let $X_k$ be the special fibre over a special point $\textbf{Spec} k$. We say that the variety $X_k$ has ordinary reduction iff the cohomology group $H^i(X_k,d\Omega^j)=0$ for any $i,j$.

My question is that p-adic etale cohomology $H^i_{et}(X,\mathbb{Z}_p)$ of $X_K$ knows $X_k$ has ordinary reduction or not.

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    $\begingroup$ What is the meaning of $d$ in that cohomology group $H^i(X_k, d\Omega^j)$? $\endgroup$
    – Will Sawin
    Apr 2 at 0:41
  • $\begingroup$ @WillSawin exact differentials of degree $j+1$, often denoted by $B^{j+1}$. $\endgroup$ Apr 2 at 12:22
  • $\begingroup$ @PiotrAchinger Is $B^{j+1}$ a Zariski sheaf? $\endgroup$
    – Z. M
    Apr 2 at 16:27
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    $\begingroup$ Good point, of course what is meant is the sheaf theoretic image of d, so locally exact differentials. $\endgroup$ Apr 2 at 16:36

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I think you meant the etale cohomology of the geometric generic fiber, equipped with the action of the Galois group of $K$. Moreover, in $p$-adic Hodge theory it is usually necessary to assume that the residue field $k$ is perfect.

In this case, the answer is yes under the additional (customary) assumption that the crystalline and Hodge cohomology groups of $X$ are torsion free (this assumption is needed for the equivalence of two "standard" definitions of ordinarity: the one you gave and the one in terms of Hodge and Newton polygons). This essentially follows from the results of Bernadette Perrin-Riou "Representations p-adiques ordinaires" in Asterisque no. 223, 1994 "Periodes p-adiques" MR 1293973. See Proposition 6.7 in https://arxiv.org/abs/2005.02246 (I claim no originality here, but the statement gives what you want).

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  • $\begingroup$ @Achinger Thank you very much!! I is exactly the thing I wanted. $\endgroup$
    – OOOOOO
    Apr 2 at 14:08

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