Derived direct image of sheaves ($R{f}_{\ast }$)

For a continuous map $f:X\to Y$, the derived direct image $R{f}_{\ast }$ is obtained by applying $f_{\ast }$ to an injective resolution. Composition with derived global sections gives $R\Gamma (Y,R{f}_{\ast }E)\cong R\Gamma (X,E)$.

For a proper map of locally compact spaces and a constructible complex $E$, the derived stalk satisfies $(R{f}_{\ast }E{)}_y\cong R\Gamma (f^{-1}(y),E{|}_{f^{-1}(y)})$. In particular $(R^i{f}_{\ast }\mathbb{Q}{)}_y\cong H^i(f^{-1}(y);\mathbb{Q})$. Properness is what prevents fibre cohomology from missing nearby sections.