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 \(i\).


14F30 \(p\)-adic cohomology, crystalline cohomology
14G22 Rigid analytic geometry
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