The category of prismatic F-gauges on a p-adic scheme X, recently introduced by Bhatt-Lurie and Drinfeld, plays a role similar to the category of Hodge D-modules attached to a variety over the field of complex numbers. It is the input to classifications of geometric objects over X such as p-divisible groups as well as computations in algebraic K-theory. It is also a tool for describing Galois representations associated with schemes over p-adic fields. The category of prismatic F-gauges is very interesting already for X being the spectrum of p-adic integers: the prismatic F-gauge associated to a scheme Y over Z_p remembers the information about prismatic cohomology of Y associated with any prism. In the first part of my talk I review the notion of prismatic F-gauge. Then 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. 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.
