AI 中文总结
本文推广 Cartier 变换到对数情形，通过引入索引结构细化拓扑斯与晶体，利用对数微分算子索引代数的 Azumaya 性质，建立索引晶体范畴的等价，解决 Frobenius 不平坦导致的本质满射问题。
AI 中文摘要
本文是三篇系列文章中的第三篇，该系列文章的目标是将 Ogus 和 Vologodsky 的 Cartier 变换推广到对数情形。在第二篇文章中，我们推广了由 Oyama 提出的该变换的拓扑斯理论版本。设 $W$ 为特征为 $p$ 的完美域上的 Witt 向量环，并赋予 $\operatorname{Spf}W$ 平凡对数结构。设 $\mathfrak{S}$ 为一个 fs 对数 $p$-adic 形式概形，在 $\operatorname{Spf}W$ 上局部有限型且对数平坦，并记其特殊纤维为 $S$。对于对数概形间的对数光滑态射 $X \rightarrow S$，我们构造了类似晶体状的环式拓扑斯 $\mathcal{E}'$ 和 $\underline{\mathcal{E}}$，以及拟凝聚模的晶体子范畴 $\mathcal{C}'$ 和 $\underline{\mathcal{C}}$，在某个提升假设下，它们分别等价于满足一定幂零条件的 Higgs 场和可积连接，并且存在一个拓扑斯态射 $\underline{\mathcal{E}} \rightarrow \mathcal{E}'$。我们证明了该拓扑斯态射的拉回函子保持拟凝聚晶体，并诱导一个完全忠实的函子 $\mathcal{C}' \rightarrow \underline{\mathcal{C}}$。由于在对数光滑情形下 Frobenius 态射一般不平坦，该函子是否本质满射并不清楚。为了解决这个问题，我们通过赋予上述拓扑斯和晶体一个索引结构来对其进行细化，该结构受 Lorenzon 将 Cartier 下降推广到光滑对数概形的索引扩展的启发。利用对数微分算子索引代数的 Azumaya 性质，我们获得了相应索引晶体范畴之间的等价，从而推广了 Cartier 变换。
英文摘要
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.
Comments92 pages. This is the third part of my PhD thesis arXiv:2512.11660
