A basic question on local cohomology

Question

I had posted this question on stackexchange but did not get any response, hence putting it up on mathoverflow.

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 $H^k_{Z \cap U}(j_*(\mathcal{F}|_U)) \cong H^k_{Z \cap U}(\mathcal{F}|_U)$ for all $k \ge 0$?

N.B. If necessary, one can assume that $Z$ is smooth.

Answer 1

You've actually made way too many assumptions! All you need is the following setup: $f\colon X \to Y $ a map of topological spaces, $Z $ a subspace of $X $, and $U $ an open neighborhood in $X$ of $Z $ in which $Z $ is closed. Let $f'=f|_U $. If $F $ is any sheaf on $X $, recall that $$\Gamma_Z (X, F) := \ker (F (U)\to F (U\setminus Z)). $$ If this definition is unfamiliar, note that it's independent of $U $ (as long as $Z $ is closed in $U $); in the algebraic case, this is the same as the definition using the sheaf of ideals of $Z $ on $U $.

Let's compare $\Gamma_Z(Y, f'_*(F |_U)) $ and $\Gamma_Z(U, F |_U)$. By definition, these are both just $\Gamma_Z(X,F) $. Therefore, the derived functors of $\Gamma_Z(Y, f'_*(\bullet |_U)) $, $\Gamma_Z(U, \bullet|_U)$, and $\Gamma_Z(X,\bullet ) $ are the same, namely local cohomology.