Paper
Logarithmic Cartier Transform
Authors
Sami Fersi
Abstract
We generalize the Cartier transform of Ogus and Vologodsky to log smooth schemes. More precisely, we generalize a local version of this transform, due to Shiho, and a topos-theoretic version, due to Oyama. Let be a perfect field of positive characteristic and equip with the trivial log structure. For a log smooth scheme over 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 of by the Frobenius of equipped with a quasi-nilpotent Higgs field, to the category of quasi-coherent modules on equipped with a quasi-nilpotent integrable connection. In another direction, we construct crystalline-like topoi and subcategories of crystals and equivalent respectively to modules with Higgs fields and integrable connections, and a fully faithful functor Since the Frobenius morphism is not, in general, flat in the log smooth setting, it is not clear that these functors are essentially surjective. To address this issue, 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. Using the Azumaya property of the ring of logarithmic differential operators, we then obtain an equivalence between the corresponding categories of indexed crystals, thereby generalizing the Cartier transform.