Finiteness and Duality for the cohomology of prismatic crystals

Abstract

Let $(A, I)$ be a bounded prism, and $X$ be a smooth $p$-adic formal scheme over $\Spf(A/I)$. We consider the notion of crystals on Bhatt--Scholze's prismatic site $(X/A)_{\prism}$ of $X$ relative to $A$. We prove that if $X$ is proper over $\Spf(A/I)$ of relative dimension $n$, then the cohomology of a prismatic crystal is a perfect complex of $A$-modules with tor-amplitude in degrees $[0,2n]$. We also establish a Poincar\'e duality for the reduced prismatic crystals, i.e. the crystals over the reduced structural sheaf of $(X/A)_{\prism}$. The key ingredient is an explicit local description of reduced prismatic crystals in terms of Higgs modules.