## The convergent topos in characteristic p.(English)Zbl 0728.14020

The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. III, Prog. Math. 88, 133-162 (1990).
[For the entire collection see Zbl 0717.00010.]
The notion of enlargement (known) and the more general notion of widening of a scheme X over a discrete valuation ring V in characteristic $$p$$ are introduced. Enlargements form a Grothendieck category $$\underline{E}$$ and sheaves on this category form the convergent topos $$(X/V)_{conv}$$. A crystal of $${\mathcal O}_{X/V}$$-modules is a sheaf F in this topos such that for $$g:S'\to S$$ in $$\underline{E}$$ one has an isomorphism $$\rho_ g: g^*_ S(E_ S)\to E_{S'}$$. Some basic results and then the existence of the universal enlargement as a direct limit of enlargements are shown.
A fully faithful functor from widenings to $$(X/V)_{conv}$$ is defined. Some cohomological results related to affine widenings are developed. Convergent cohomology is shown independent of infinitesimal thickenings. If $${\mathcal K}_{X/V}$$ is the tensor product of the field of fractions K of V with $${\mathcal O}_{X/V}$$, the author shows, for instance, that $$H^ i_{conv}(X/V,{\mathcal K}_{X/V})\simeq K\otimes H^ i_{DR}(Y/V)$$ if Y/V is a smooth formal lifting of X/k. Finally, the category of convergent isocrystals on X/V is shown to be a full subcategory of the category of crystals. Here, the category of isocrystals is the category with crystals as objects which are finite type $${\mathcal O}_{X/V}$$-modules and where the morphisms are elements of $$K\otimes Mor_{cris}(E,F)$$.

### MSC:

 14F30 $$p$$-adic cohomology, crystalline cohomology 14F20 Étale and other Grothendieck topologies and (co)homologies 18F10 Grothendieck topologies and Grothendieck topoi

Zbl 0717.00010