Let S,X be schemes and $s \in S$ be a closed point. Let $D(X)$ be the derived category of complexes of sheaves.
Let $$i_s: X \cong {s} \times X \hookrightarrow S \times X$$ be the natural embedding.
Is the sheaf push-forward $i_{S*} \mathcal{O}_X$ the same as derived push-forward $\mathbf{R}i_{s*} \mathcal{O}_X$?
More generally, let $f: Y \to X$ be a scheme morphism, and $E \in D(Y)$. Suppose $F$ is a complex of projective sheaves, and $F \cong E $ in $D(Y)$ (say a locally free resolution of $E$). Then is $f_* F \cong \mathbf{R}f_* E$ in $D(E)$?