For a proper flat map $f: X \to Y$ (of reasonable schemes) and a closed embedding $i: Y' \to Y$, we know by EGA the base change quasi-isomorphism, where $f'$ and $i'$ are the pullbacks of $f$ and $i$:

$$i^* R f_* \mathcal O \sim R f'_* i'^* \mathcal O$$

Here is my question: is a similar statement true when we replace the $\mathcal O$-linear pullback functor $i^*$ by the ordinary pullback functor (of sheaves of abelian groups): $$i^{-1} R f_* \mathcal O \stackrel{???} \sim R f'_* i^{'-1} \mathcal O$$

Equivalently, writing $j_!$ for the extension by zero functor, where $j$ is the complement of $i$, do we have (where $f''$ is the pullback of $f$ to the open complement)

$$R f''_* j_! \mathcal O \stackrel{???} \sim j'_! R f_* \mathcal O?$$

Thank you!

For a proper flat map $f: X \to Y$ (of reasonable schemes) and a closed embedding $i: Y' \to Y$, we know by EGA the base change quasi-isomorphism, where $f'$ and $i'$ are the pullbacks of $f$ and $i$:

$$L i^* R f_* \mathcal O \sim R f'_* L i'^* \mathcal O$$

Here is my question: is a similar statement true when we replace the $\mathcal O$-linear pullback functor $i^*$ by the ordinary pullback functor (of sheaves of abelian groups): $$i^{-1} R f_* \mathcal O \stackrel{???} \sim R f'_* i^{'-1} \mathcal O$$

Equivalently, writing $j_!$ for the extension by zero functor, where $j$ is the complement of $i$, do we have (where $f''$ is the pullback of $f$ to the open complement)

$$R f''_* j_! \mathcal O \stackrel{???} \sim j'_! R f_* \mathcal O?$$

Thank you!