局部对数Cartier变换
Local Logarithmic Cartier Transform
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中文总结 AI 辅助
本文推广Ogus-Vologodsky的Cartier变换到对数光滑概形，在Frobenius提升条件下构造了从拟幂零Higgs模到拟幂零可积联络模的完全忠实函子，并证明了对数平坦下降定理。
中文摘要 AI 辅助
本文是三篇系列文章中的第一篇，其目标是将Ogus和Vologodsky的Cartier变换推广到对数情形。在本文中，我们将Shiho所证明的Cartier变换的局部版本推广到对数光滑概形。更精确地，设$k$是一个正特征的完美域，并赋予$\operatorname{Spec}k$平凡的对数结构。对于对数概形的对数光滑态射$X \rightarrow S$，其中$S$在$\operatorname{Spec}k$上是对数平坦且局部有限型的，我们在精确相对Frobenius在$k$的Witt向量上提升为$S$上的对数光滑概形之间的态射的假设下，得到一个从$X$通过$S$的Frobenius $F_S$的基变换$X'=X\times_{S,F_S}S$上配备拟幂零Higgs场的拟凝聚模范畴，到$X$上配备拟幂零可积联络的拟凝聚模范畴的完全忠实函子。为此，我们基于Kato的先前工作，证明了态射的对数平坦下降定理。
英文摘要
This article is the first of three articles whose goal is to generalize the Cartier transform of Ogus and Vologodsky to the logarithmic setting. In this article, we generalize a local version, due to Shiho, of the Cartier transform to log smooth schemes. More precisely, let $k$ be a perfect field of positive characteristic and equip $\operatorname{Spec}k$ with the trivial logarithmic structure. For a log smooth morphism of logarithmic schemes $X \rightarrow S,$ where $S$ is log flat and locally of finite type over $\operatorname{Spec}k,$ we obtain, under the assumption that the exact relative Frobenius lifts over the Witt vectors of $k$ to a morphism between log smooth schemes over $S,$ a fully faithful functor from the category of quasi-coherent modules on the base change $X'=X\times_{S,F_S}S$ of $X$ by the Frobenius $F_S$ of $S,$ equipped with a quasi-nilpotent Higgs field, to the category of quasi-coherent modules on $X$ equipped with a quasi-nilpotent integrable connection. For this, we prove a log flat descent theorem for morphisms, based on previous work by Kato.
