## Algebraic Geometry Seminar

Please see the following for more related talks:

**Upcoming Seminars:**
- Wednesday, December 6, 2023, 1:30pm, Gross Hall 304B, Algebraic Geometry Seminar

*Prismatic F-gauges and Fontaine-Laffaille modules*

Vadim Vologodsky
- 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.

- Wednesday, January 10, 2024, 1:30pm, Zoom link, Algebraic Geometry Seminar

*TBA*

Alberto San Miguel Malaney (U Texas at Austin, Mathematics)
- Wednesday, February 14, 2024, 1:30pm, Zoom link, Algebraic Geometry Seminar

*TBA*

Daniel Greb (Universität Duisburg-Essen)
- Wednesday, March 27, 2024, 1:30pm, Zoom link, Algebraic Geometry Seminar

*TBA*

David Urbanik (IHES)

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