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On purity of crystalline cohomology. (Sur la pureté de la cohomologie cristalline.) (French) Zbl 0936.14016
Let $$p$$ be a prime number, $$q=p^a$$, and $$W=W(\mathbb{F}_q)$$, the ring of Witt vectors with fraction field $$K$$. For a proper smooth variety $$X$$ over $$\mathbb{F}_q$$, one may consider its crystalline cohomology $$H_{\text{crys}}^i(X/K)\otimes_WK$$. This is a Weil cohomology endowed with a Frobenius, giving rise to the notions of weights, mixedness and purity. One may also consider the rigid cohomology $$H_{\text{rig}}^i(X/K)$$. These are defined for $$X$$ smooth over $$\mathbb{F}_q$$, and then they are finite dimensional $$K$$-vector spaces, mixed of weight $$w$$ with $$i\leq w\leq 2i$$. For $$X$$ proper and smooth they coincide with the $$H^i_{\text{crys}}(X/K)\otimes_WK$$. One also has rigid cohomology with compact support, $$H^i_{\text{rig,c}}(X/K)$$. It is shown that $$H^i_{\text{rig,c}}(X/K)$$ is mixed of weights $$w$$ with $$2(i-d)\leq w\leq i$$, where $$d$$ is the dimension of $$X$$. As a corollary one obtains a generalization of a result for smooth projective varieties $$X$$ over $$\mathbb{F}_q$$ (due to Katz and Messing):
Let $$X$$ be proper and smooth over $$\mathbb{F}_q$$, then the $$H_{\text{crys}}^i(X/K)\otimes_WK$$ are pure of weight $$i$$.

MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14G22 Rigid analytic geometry
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