Sheaf cohomology under a closed inclusion ($H^i(Y,i_{\ast }\mathcal{F})\cong H^i(X,\mathcal{F})$)

For a closed inclusion $i:X\hookrightarrow Y$, direct image sheaf formation is exact: stalks on $X$ are unchanged and stalks outside $X$ are zero. It also preserves flasque sheaves. Pushing forward a flasque resolution thus gives a resolution computing cohomology on $Y$, and its complex of global sections is identical to the original one on $X$. This proves the displayed isomorphism in every nonnegative degree.