Prismatic F-gauges and Fontaine-Laffaille Modules

Vadim Vologodsky

With every bounded prism Bhatt and Scholze associated a cohomology theory of formal p-adic schemes. The prismatic cohomology comes equipped with the Nygaard filtration and the Frobenius endomorphism. The Bhatt-Scholze construction has been advanced further by Drinfeld and Bhatt-Lurie who constructed a cohomology theory with values in a stable  -category of prismatic F-gauges. The new cohomology theory is universal, meaning that, for every bounded prism, the associated prismatic cohomology theory factors through  the category of prismatic F-gauges.  In this talk I will explain how a full subcategory of the category  of prismatic F-gauges formed by objects whose Hodge-Tate weights lie in the interval [0,p-2] is equivalent to the derived category of Fontaine-Laffaille modules with a similar weight constrain. In the geometric context, this means that the prismatic F-gauge associated with a formally smooth  scheme over p-adic integers of dimension < p-1 can be recovered from its Hodge filtered de Rham cohomology equipped with the Nygaard refined crystalline Frobenius endomorphism. The talk is based on a joint work with Gleb Terentiuk and Yujie Xu.

https://www.ias.edu/video/prismatic-f-gauges-and-fontaine-laffaille-modules-0