An important question in integral p-adic Hodge theory is the study of lattices in crystalline Galois representations. There have been various classifications of such objects, such as Fontaine--Lafaille’s theory, Breuil’s theory of strongly divisible lattices, and Kisin’s theory of Breuil--Kisin modules. Using their prismatic theory, Bhatt--Scholze give a site-theoretic description of such lattices, which has the nice feature that it can specialize to many of the previous classifications by "evaluating" suitably. In this talk, we will recall their result and explain an extension to the Lubin--Tate context.
