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变体，即代数方程组的解的集合。现代代数几何也研究代数堆栈；它们是类似于代数簇的几何对象，其点被允许具有非平凡的对称性。这个项目的主题是利用代数堆栈来研究棱柱上同调，这是Bhatt和Scholze最近提出的一种很有前途的p-进代数簇的上同调理论。该项目还为研究生提供了研究培训机会。首席研究员计划引入棱柱化函子，并利用它们来构建棱柱上同调系数的自然理论，这是方丹和詹森所称的F-规范的精神。他还计划发展一种棱镜理论的版本，在该理论中，用特征为p的局域场代替p进数场，并在此背景下找到晶体类别的明确描述。另一个目标是以正式群的形式描述在完美特征域上的光滑簇上收敛的F-同晶的Tannakian范畴。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.