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Résumé |
The Cartier transform of Ogus and Vologodsky can be seen as a generalization of Cartier descent. It is an equivalence between modules with integrable connections on a smooth scheme over a perfect field of positive characteristic and Higgs modules on the Frobenius base change of this scheme. We discuss a generalization of this transform to log smooth schemes. More precisely, we discuss two generalizations of Shiho's local version and Oyama's crystalline-type version of this transform. For a log smooth scheme $X$ over a perfect field $k$ of positive characteristic, we obtain, under the assumption that the exact relative Frobenius lifts to the Witt vectors, a fully faithful functor from the category of quasi-coherent modules on the base change $X'=X\times_{k,F_k}k$ of $X$ equipped with a quasi-nilpotent Higgs field, to the category of quasi-coherent modules on $X$ equipped with a quasi-nilpotent integrable connection. In another direction and without any lifting assumptions, we construct a crystalline-type interpretation of this functor. To address the issue of essential surjectivity, we refine the topoi and crystals mentioned above by endowing them with an indexed structure, inspired by Lorenzon’s extension of Cartier descent to smooth logarithmic schemes. |
