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typos corrected
user25309
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Are the Eigenvalues of the Frobenius on Crystalline cohomology bounded by degree?

Excuse me if this question is trivial or trivially false, or not at this sites level. Lets work over an algebraically closed field of characteristic $p>0$, say $k$. The action of the Frobenius on $H^r_{crys}(X/k)$ gives us a sequence of positive rational numbers $a_i$ by taking the order of the eigenvalues.

Now if $X$ has a good lift to $W(k)$, $Y$, say with torsion free $H^p(Y, \Omega_Y/W)$, I believe we can decompose the Frobenius morphism using Hodge decomposition to obtain an inequality $a_i\le r$.

My question is whether this is true in general. This may be trivial, but I am a bit new to Crystalline cohomology so I can't figure it out.

Pax
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