##
**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.

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.

Reviewer: W.W.J.Hulsbergen (Haarlem)

### MSC:

14F30 | \(p\)-adic cohomology, crystalline cohomology |

14G15 | Finite ground fields in algebraic geometry |

19E20 | Relations of \(K\)-theory with cohomology theories |