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. ...

In this paper we study the local cohomology of all finitely generated bigraded modules over a standard bigraded polynomial ring which have only one nonvanishing local cohomology with respect to one of the irrelevant bigraded ideals.