Finiteness and duality for the cohomology of prismatic crystals

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