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Weights in rigid cohomology applications to unipotent \(F\)-isocrystals. (English) Zbl 0933.14008

Let \(X\) be a smooth scheme defined over a finite field \(k=\mathbb{F}_{p^{a}}\) of characteristic \(p>0\). To \(X\) and for any complete discretely valued field \( K \) whose residue field is \(k\) one can associate the rigid cohomology \(H^{\bullet}_{\text{rig}}(X/K)\). Actually the \( H^{i}_{\text{rig}}(X/K) \) are \(K\)-vector spaces that have been recently proved to be finite dimensional. Moreover, the action of the \(a\)-th iterate of the absolute Frobenius on \(X\) induces a \(K\)-linear map \(F\) on \(H^{i}_{\text{rig}}(X/K)\). The main result of this paper is that this action endows \(H^{i}_{\text{rig}}(X/K)\) with a “mixed with integral weights \( F\)-\(K\)-isocrystal structure”, namely all the roots of the characteristic polynomial of \(F\) have archimedean absolute value equal to \(p^{aj/2}\) with \(j\) an integer (and \(i\leq j\leq 2i\)). The proof relies on de Jong alterations, on a particular case of the Gysin isomorphism that respects the Frobenius structure and on the existence of an adaptated Frobenius in the Monsky-Washnitzer setting.
In the second part the author transposes constructions made by J. Wildeshaus [“Realizations of polylogarithms”, Lect. Notes Math. 1650 (1997; Zbl 0877.11001)] to the \(p\)-adic setting for the unipotent fundamental group in the de Rham and \(\ell\)-adic cases.
In the third part he defines the category of “mixed unipotent overconvergent \(F\)-\(K\)-isocrystals on \(X\)”. The central result there is that an unipotent overconvergent \( F\)-\(K\)-isocrystal on \( X \) is mixed with integral weights if and only if its fiber at some closed point of \( X \) is a mixed \( F\)-\(K\)-isocrystal with integral weights.
Reviewer: G.Christol (Paris)

MSC:

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

Citations:

Zbl 0877.11001
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