Cohomology under a closed immersion
= Cohomology under a closed immersion
If $i:Z\hookrightarrow X$ is a <closed immersion> of <Noetherian scheme>[Noetherian schemes], $X$ is separated, and $\mathcal F$ is <quasi-coherent sheaf>[quasi-coherent] on $Z$, then
$$
H^q(X,i_*\mathcal F)\cong H^q(Z,\mathcal F).
$$
Indeed, an affine cover of $X$ induces an affine cover of $Z$, and the two associated <Čech cochain complex>[Čech cochain complexes] are termwise identical.