拓扑循环同调与棱镜上同调

代数K理论是数学中的核心研究对象之一。对于局部域上的概型，拓扑循环同调是研究其代数K理论的重要工具。近年来，Bhatt-Morrow-Sholze通过motivic滤子的构造，揭示出拓扑循环同调与p-进霍奇理论的深刻联系。申请人利用相对Hochschild同调构建了计算拓扑循环同调的新方法-下降谱序列，并利用下降谱序列完全解决了局部域的拓扑循环同调的计算问题。
.项目计划使用下降谱序列方法，进一步计算局部域上局部完全交概型的拓扑循环同调与代数K理论，研究motivic滤子的结构，以及揭示这些计算和构造与棱镜上同调以及p-进霍奇理论的深刻联系。

Algebraic K-theory is a key object to study in mathematics. For schemes over local fields, topological cyclic homology is an important tool to study their algebraic K-theory. Recent works of Bhatt-Morrow-Scholze has revealed profound connection between topological cyclic homology and p-adic Hodge theory through the construction of motivic filtrations. More recently, the applicants introduced a new approach to determining the structure of topological cyclic homology by means of a descent spectral sequence. Using this new method, the applicants successfully determined the structure of topological cyclic homology for all p-adic local fields. ..The project will apply the technique of descent spectral sequence to study topological cyclic homology and algebraic K-theory, as well as the structure of corresponding motivic filtrations, for schemes of locally complete intersections over local fields. In addition, the applicants hope to unveil the hidden connections between our construction and the prismatic cohomology introduced by Scholze-Bhatt, and the more general p-adic Hodge theory.