A Note on Dieudonné Theory over Perfectoid Rings

Dieudonné theory over semiperfect rings and perfectoid rings

The Dieudonné crystal of a p-divisible group over a semiperfect ring R can be endowed with a window structure. If R satisfies a boundedness condition, this construction gives an equivalence of categories. As an application one obtains a classification of p-divisible groups and commutative finite locally free p-group schemes over perfectoid rings by Breuil-Kisin-Fargues modules if p>2.

关于解析p-可除群的Dieudonné理论及其在Shimura品种中的应用

A Dieudonné theory for analytic p-divisible groups and applications to Shimura varieties

本文研究了解析p-可除群的Dieudonné理论，并将其应用于Shimura品种，证明了与Hodge-Tate三元组的等价性，并重新解释了Hodge-Tate周期映射。

Moduli of p-divisible groups

We prove several results about p-divisible groups and Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in terms of p-adic Hodge theory. This allows us to formulate and prove duality isomorphisms between basic Rapoport-Zink spaces at infinite level in general. Moreover, we identify the image of the period morphism, reproving results of Faltings. For this, we give a general classification of p-divisible groups over the ring of integers of a complete algebraically closed fi…

平坦上同调的纯净性

Purity for flat cohomology

本文证明完全交 Noether 局部环上有限平坦交换群系数的局部平坦上同调在维数以下消失，并借助 perfectoid 环、arc descent 与棱柱 Dieudonné 理论推广 Gabber–Thomason 纯净性。

晶体与抛物性猜想

Crystals and the parabolicity conjecture

该讲义以阿贝尔簇为例介绍晶体与晶态迪厄多内理论，阐述F-等晶体的抛物性定理及其应用，还提出两个相关研究课题，面向具备基础代数几何知识的读者。

关于$p$进单位圆盘的$K$理论

On the $K$-theory of the $p$-adic unit disk

本研究针对perfectoid环上光滑环的仿射直线$p$完备拓扑循环同调，发现其$\n\b\b{A}^1$不变性失效谱存在优于已有结论的Quillen-Lichtenbaum现象，并借助相关定理得到对应代数$K$理论的描述。