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

Finiteness and Duality for the cohomology of prismatic crystals