Prismatic F-gauges and Fontaine-Laffaille Modules

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 infinity-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 constraint. 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.

If time permits, I will explain a generalization of the above statement to the case of prismatic F-gauges over a smooth p-adic formal scheme.

The talk is based on a joint work with Gleb Terentiuk and Yujie Xu.