Let $X$ be a smooth, projective variety, $i:X \hookrightarrow \mathbb{P}^n$ a closed immersion for some $n>0$, $U \subset X$ an open subset and $Z \subset X$ a local complete intersection subscheme. Denote by $j:U \to \mathbb{P}^n$ the natural immersion. Let $\mathcal{F}$ be a locally free sheaf on $X$. Is $H^i_{Z \cap U}(j_*(\mathcal{F}|_U)) \cong H^i_{Z \cap U}(\mathcal{F}|_U)$?
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How are you defining $H^i_{Z \cap U} (j_*(\mathcal F|_U))$ ? $Z\cap U$ need not be a closed subset of $\mathbb P^n$ ... – user102248 Jul 09 '19 at 15:13