# A basic question on local cohomology

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.

• @SándorKovács: Look at Exp. I in SGA2 for a thorough discussion of local cohomology using locally closed sets (such as $Z \cap U$ inside $X$ above). – grghxy Oct 3 '15 at 7:37
• @grghxy: Thanks. (Perhaps by now you could get rid of that "air of mystery" you prefer. It's so much nicer to thank a real person...) – Sándor Kovács Oct 4 '15 at 20:24

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.