Finiteness and Duality for the Cohomology of Prismatic Crystals

Prismatic site of a p-adic formal scheme was introduced in the recent pioneer work of Bhatt—Scholze. It provides a uniform framework for various p-adic cohomology theories. Prismatic crystals are natural analogues of classical crystalline crystals on prismatic sites. In this talk, after reviewing some basic definitions of the prismatic site, I will discuss some basic properties of the cohomology of prismatic crystals on smooth p-adic formal schemes. The key ingredient is an explicit local description of (reduced) prismatic crystals in terms of Higgs modules.