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About the filtration of crystalline cohomology

Suppose $K$ is an finite unramified extension of $\mathbb Q_p$ with residue field $k$, and let $Y$ be an proper smooth variety defined over $k$. We know if $Y$ admits a proper smooth lifting $X/W(k)$ …
7 votes
0 answers
579 views

Is there a cohomology theory wider than crystalline?

We know crystalline cohomology is calculated by taking an inverse limit: $$H_{cris}^i:=\varprojlim_nH_{cris}^i(X/W_n(k))$$ provided $X$ projective smooth over a perfect field $k$ of char $p$. I want t …