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Prismatic F-gauges and Fontaine-Laffaille Modules - Vadim Vologodsky
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https://m.youtube.com/watch?v=tRmjTo4alkc Special Year Workshop on p-adic Arithmetic Geometry Topic: Prismatic F-gauges and Fontaine-Laffaille Modules Speaker: Vadim Vologodsky Affiliation: HSE/Princeton University Date: November 16, 2023 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.
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