Slopes in rigid cohomology and unipotent F-isocrystals (Q1965134)

In this article, the authors give some general results concerning the slopes of the Frobenius acting on rigid cohomology. They also establish the existence of a slope filtration for unipotent \(F\)-isocrystals on a smooth variety over an algebraically closed field. Let us recall that a unipotent \(F\)-isocrystal is a successive extension of the trivial isocrystal that is endowed with a Frobenius structure. The authors already published an article concerning these questions [\textit{B.~Chiarellotto} and \textit{B. Le Stum}, Compos. Math. 116, No 1, 81-110 (1999; Zbl 0936.14017)], where they established the existence of the slope filtration for unipotent \(F\)-isocrystals in the case of an open subset of the affine line. In the first two parts of the article, the authors give some technical stuff concerning rigid cohomology. In particular, they prove that the Poincaré duality morphism, the Gysin morphism, and the trace morphism are compatible with Frobenius. Denote by \(K=W(k)[p^{1/e}]\), where \(e\) is a nonnegative integer. The statement concerning the slopes of rigid cohomology is the following. Let \(k\) be an algebraically closed field of characteristic \(p\), and \(X\) an algebraic variety of dimension \(d\) over \(k\). Then the slopes of the rigid cohomology with compact support \(H^i_{\text{rig},c}(X)\) are \(\geq 0\) and \(\leq d\). Moreover these numbers are \(\geq i-d\) and \(\leq i\). The results are the same for the groups \(H^i_{\text{rig}}(X)\), if \(X\) is smooth. The key point to define the existence of the slope filtration of unipotent F-isocrystals is to use a tannakian interpretation of unipotent isocrystals. Suppose that \(X\) is connected and has a rational point \(x\). Then the category of overconvergent unipotent isocrystals over \(X\) is a tannakian category that is neutral (the fiber functor is given by \(E\mapsto E_x\)). Let \(G\) be its fundamental group, \(\mathfrak{g}\) its Lie algebra, \({\mathbb{U}}\) the enveloping algebra of \(\mathfrak{g}\) and \(\widehat{\mathbb{U}}(X,x)\) the completion of \({\mathbb{U}}\) relatively to the augmentation ideal. An \(F\)-\(\widehat{\mathbb{U}}(X,x)\)-isocrystal over \(K\) is a \(\widehat{\mathbb{U}}(X,x)\)-module of finite type, endowed with a semi-linear automorphism \(F\). The fiber functor induces an equivalence between the category of overconvergent unipotent \(F\)-isocrystals over \(X/K\) and the category of \(F\)-\(\widehat{\mathbb{U}}(X,x)\)-isocrystals over \(K\). The existence of slope filtration with pure associated graduated modules is first proven for \(F\)-\(\widehat{\mathbb{U}}(X,x)\)-isocrystals over \(K\) (if \(x\) is a closed point), as a consequence of the Dieudonné-Manin theory, and then proven for overconvergent unipotent \(F\)-isocrystals over \(X/K\). The final result is the following theorem: Let \(X\) be a smooth variety over an algebraically closed field \(k\), and \(E\) a unipotent overconvergent \(F\)-isocrystal over \(X/K\). Then there exists a filtration \(\text{Fil}_{\lambda}E\), such that \(\text{Gr}_{\lambda}E\) is pure of slope \(\lambda \). This article is very interesting and illustrates how powerful a tannakian point of view can be.