F-Crystals, Prismatization, and Shtukas

该项目支持代数几何的研究。数学的这一部分研究代数簇，即，代数方程组的解。现代代数几何也研究代数栈;这些是类似于代数簇的几何对象，其点允许具有非平凡对称性。本项目的主题是利用代数栈来研究棱柱上同调，棱柱上同调是Bhatt和Scholze最近提出的一个非常有前途的p-adic代数簇的上同调理论。该项目还为研究生提供了研究培训的机会。首席研究员计划引入棱镜化函子，并利用它们来构建一个自然的理论，棱镜上同调系数的精神是什么方丹和詹森所谓的F-规范。他还计划发展一个版本的棱柱理论，其中领域的p-adic号码是取代当地领域的特点p和找到一个明确的描述类别的晶体在这方面。另一个目标是描述正式团体的Tannakian类收敛的F-isocrystals在一个完美的领域的特征p.这个奖项反映了NSF的法定使命，并已被认为是值得支持的，通过使用基金会的智力价值和更广泛的影响审查标准进行评估。

The project supports research in algebraic geometry. This part of mathematics studies algebraic varieties, i.e., sets of solutions of systems of algebraic equations. Modern algebraic geometry also studies algebraic stacks; these are geometric objects similar to algebraic varieties, whose points are allowed to have nontrivial symmetries. The main theme of this project is to use algebraic stacks to study prismatic cohomology, which is a very promising cohomology theory of p-adic algebraic varieties introduced recently by Bhatt and Scholze. The project also provides research training opportunities for graduate students.The principal investigator plans to introduce prismatization functors and to use them to construct a natural theory of coefficients for prismatic cohomology in the spirit of what Fontaine and Jannsen call F-gauges. He also plans to develop a version of the prismatic theory in which the field of p-adic numbers is replaced by a local field of characteristic p and to find an explicit description of the category of crystals in this context. Another goal is to describe in terms of formal groups the Tannakian category of convergent F-isocrystals on a smooth variety over a perfect field of characteristic p.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.