对数Cartier变换的拓扑斯理论方法
A Topos-Theoretic Approach to the Logarithmic Cartier Transform
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中文总结 AI 辅助
本文通过拓扑斯理论方法，将对数Cartier变换推广到对数光滑态射，构造晶体范畴并证明其诱导函子忠实满，从而推广了Ogus和Vologodsky的经典结果。
中文摘要 AI 辅助
本文是三篇系列文章中的第二篇，其目标是将Ogus和Vologodsky的Cartier变换推广到对数情形。我们推广了由Oyama提出的该变换的拓扑斯理论版本。设$k$为特征$p>0$的完美域，并赋予$S=\operatorname{Spec}k$平凡对数结构。对于对数概形$X \rightarrow S$的对数光滑态射，我们构造了类晶体环化拓扑斯$\mathcal{E}'$和$\underline{\mathcal{E}}$，以及拟凝聚模的晶体子范畴$\mathcal{C}'$和$\underline{\mathcal{C}}$，在某种提升假设下，它们分别等价于满足一定幂零条件的Higgs场模和可积联络模，并构造了一个拓扑斯态射$\underline{\mathcal{E}} \rightarrow \mathcal{E}'$。然后我们证明该拓扑斯态射的回拉函子保持拟凝聚晶体，从而诱导函子$\mathcal{C}' \rightarrow \underline{\mathcal{C}}$，推广了Cartier变换。最后，我们利用在第一篇文章中证明的态射的对数平坦下降定理，证明该函子是忠实满的。
英文摘要
This article is the second of three articles whose goal is to generalize the Cartier transform of Ogus and Vologodsky to the logarithmic setting. We generalize a topos-theoretic version of this transform, due to Oyama. Let $k$ be a perfect field of positive characteristic $p$ and equip $S=\operatorname{Spec}k$ with the trivial log structure. For a log smooth morphism of logarithmic schemes $X \rightarrow S,$ we construct 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 then prove that the pullback functor of this morphism of topoi preserves quasi-coherent crystals and hence induces a functor $\mathcal{C}' \rightarrow \underline{\mathcal{C}},$ generalizing the Cartier transform. We finally use a log flat descent theorem for morphisms, that we proved in the first article, to prove that this functor is fully faithful.
