Indexed Logarithmic Cartier Transform

Source abstract

This article is the third of a series of three articles whose goal is to generalize the Cartier transform of Ogus and Vologodsky to the logarithmic setting. In the second one, we generalized a topos-theoretic version of this transform, due to Oyama. Let $W$ be the ring of Witt vectors of a perfect field of positive characteristic $p$ and equip $\operatorname{Spf}W$ with the trivial log structure. Let $\mathfrak{S}$ be an fs log $p$-adic formal scheme, locally of finite type and log flat over $\operatorname{Spf}W$ and denote its special fiber by $S.$ For a log smooth morphism of logarithmic schemes $X \rightarrow S,$ we constructed crystalline-like ringed topoi $\mathcal{E}'$ and $\underline{\mathcal{E}}$ and subcategories of crystals of quasi-coherent modules $\mathcal{C}'$ and $\underline{\mathcal{C}},$ equivalent respectively, under some lifting assumption, to modules with Higgs fields and integrable connections, both satisfying certain nilpotence conditions, and a morphism of topoi $\underline{\mathcal{E}} \rightarrow \mathcal{E}'.$ We proved that the pullback functor of this morphism of topoi preserves quasi-coherent crystals and induces a fully faithful functor $\mathcal{C}' \rightarrow \underline{\mathcal{C}}.$ Since the Frobenius morphism is not, in general, flat in the log smooth setting, it is not clear that this functor is 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 indexed extension of Cartier descent to smooth logarithmic schemes. Using the Azumaya property of the indexed algebra of logarithmic differential operators, we then obtain an equivalence between the corresponding categories of indexed crystals, thereby generalizing the Cartier transform.