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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 \(K\otimes Mor_{cris}(E,F)\).

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