Cohomology under a closed immersion

If $i:Z\hookrightarrow X$ is a closed immersion of Noetherian schemes, $X$ is separated, and $\mathcal{F}$ is quasi-coherent on $Z$, thenIndeed, an affine cover of $X$ induces an affine cover of $Z$, and the two associated Čech cochain complexes are termwise identical.