## Unipotent $$F$$-isocrystals. ($$F$$-isocristaux unipotents.)(French)Zbl 0936.14017

Let $$K$$ be a complete ultrametric field of characteristic zero, with ring of integers $$\mathcal V$$ and residue field $$k$$. Let $$P$$ be a formal $$\mathcal V$$-scheme and let $$Y$$ be a closed $$k$$-scheme of $$P$$ with open $$X\subset Y$$. $$P$$ is supposed to be smooth in the neighbourhood of $$X$$. Write $$]X[_P$$ and $$]Y[_P$$ for the tubes of $$X$$ and $$Y$$ in $$P_K$$, respectively. For a strict neighbourhood $$V$$ of $$]X[_P$$ in $$]Y[_P$$, and an $$\mathcal O_V$$-module $$\mathcal E$$ one writes $$j^{\dagger}\mathcal E :=j_*\lim_{\rightarrow}j'_*{j'}^*\mathcal E$$, where $$j:V\hookrightarrow ]Y[_P$$ is the inclusion, and where $$j'$$ runs over the inclusions of strict neighbourhoods $$V'$$ of $$]X[_P$$ into $$V$$. Thus $$j^{\dagger}$$ is an exact functor from the category of coherent $$\mathcal O_V$$-modules with integrable connection into the category of coherent $$j^{\dagger}\mathcal O_{]Y[}$$-modules with integrable connection. Actually one has an equivalence between the inductive limit category of coherent $$\mathcal O_V$$-modules with integrable connection and the category of coherent $$j^{\dagger}\mathcal O_{]Y[}$$-modules with integrable connection. An integrable connection on a coherent $$j^{\dagger}\mathcal O_{]Y[}$$-module is called overconvergent (along the complement of $$X$$ in $$Y$$) if the so-called Taylor isomorphism comes from an isomorphism on a strict neighbourhood of the tube of the diagonal. The notion of overconvergence is stable under extensions.
For a separated scheme $$X$$ of finite type over $$k$$, one may define the category of overconvergent isocrystals on $$X$$ via their realizations. The construction amounts to an equivalence of this category with the category of coherent $$j^{\dagger}\mathcal O_{]Y[}$$-modules with overconvergent integrable connection, where $$X\subset Y$$ is open in the proper $$k$$-scheme $$Y$$, etc. For an overconvergent isocrystal $$E$$ on $$X$$, one defines the rigid cohomology $$H^i_{\text{rig}}(X,E):=H^i_{\text{dR}}(]Y[_P,E_P)$$, independently of the ‘realization’ $$E_P$$ of $$E$$. $$\mathcal O^{\dagger}_X$$ will denote the trivial isocrystal with realization $$j^{\dagger}\mathcal O_{]Y[}$$. For an overconvergent isocrystal $$E$$ on a smooth connected curve $$X$$ over $$k$$ one has a Gysin sequence for the rigid cohomology $$H_{\text{rig}}(X,E)$$. Several other properties of overconvergent isocrystals on various kinds of $$X$$ can be derived. An overconvergent isocrystal is said to be unipotent if it is an iterated extension of $$\mathcal O^{\dagger}_X$$. Unipotent overconvergent isocrystals are stable under extensions, subobjects, quotients, tensor products, internal Hom’s and inverse images. For connected $$X$$ an overconvergent isocrystal $$E$$ on $$X$$ is unipotent iff it admits an increasing, exhaustive, separated filtration with constant gradings. There is a unique such filtration with $$\text{Gr}_iE=H^0_{\text{rig}}(X,E/\text{Fil}_{i-1}E)\otimes_K\mathcal O^{\dagger}_X$$ which is functorial. Moreover, unipotent overconvergent isocrystals satisfy remarkable properties of algebraicity and Frobenius descent.
Assume from now on that $$k$$ is algebraically closed of characteristic $$p>0$$. One has the notion of Frobenius $$F$$, and, by extension, of $$F$$-isocrystals. One can extend several properties of overconvergent isocrystals to overconvergent $$F$$-isocrystals. An overconvergent $$F$$-isocrystal $$E$$ on $$X$$ is unipotent if the underlying isocrystal is unipotent. Such isocrystals on connected $$X$$ are characterized by their natural, $$F$$-stable, increasing, exhaustive and separated filtration with constant gradings. Furthermore, an overconvergent $$F$$-isocrystal on connected $$X$$ is unipotent iff it is an extension of ‘twisted’ $$\mathcal O^{\dagger}_X$$. A unipotent overconvergent $$F$$-isocrystal on a curve is determined by its restriction to a non-empty open subset. There is the notion of slope for overconvergent $$F$$-isocrystals. A unit $$F$$-isocrystal is an $$F$$-isocrystal with all slopes equal to zero. One has the result: On $$U\subset{\mathbb A}^1$$ open, every unit unipotent $$F$$-isocrystal is trivial. Let $$E$$ be a unipotent overconvergent $$F$$-isocrystal on $$U\subset{\mathbb A}^1$$ open. Then $$E$$ is the direct sum of ‘twists’ of overconvergent $$F$$-isocrystals with integer slopes. Moreover, $$E$$ has a unique finite, exhaustive, separated functorial $${\mathbb Q}$$-filtration such that $$\text{Gr}_{\lambda}E\simeq\mathcal O^{\dagger}_X(-\lambda)^{n_{\lambda}}$$. This is called the slope filtration. The slope filtration is finer than the natural filtration. One may ask whether a unipotent overconvergent isocrystal $$E$$ on $$U$$ admits a Frobenius. The answer to this question turns out to be negative.

### MSC:

 14F30 $$p$$-adic cohomology, crystalline cohomology 14G15 Finite ground fields in algebraic geometry 19E20 Relations of $$K$$-theory with cohomology theories
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