The Stacks project

First proof. Let $K$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$. Let $(K_ n)$ be a Deligne system for $U \to X$ whose restriction to $U$ is constant with value $K$. Of course this means that $(K_ n)$ represents $Rj_!K$ in $\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ X)$. Observe that both $Rj'_!Rg_*K$ and $Rf_*Rj_!K$ restrict to the constant pro-object with value $Rg_*K$ on $V$. This is immediate for the first one and for the second one it follows from the fact that $(Rf_*K_ n)|_ V = Rg_*(K_ n|_ U) = Rg_*K$. By the uniqueness of Deligne systems in Lemma 48.30.3 it suffices to show that $(Rf_*K_ n)$ is pro-isomorphic to a Deligne system. The lemma referenced will also show that the isomorphism we obtain is functorial.

Proof that $(Rf_*K_ n)$ is pro-isomorphic to a Deligne system. First, we observe that the question is independent of the choice of the Deligne system $(K_ n)$ corresponding to $K$ (by the aforementioned uniqueness). By Lemmas 48.30.4 and 48.30.7 if we have a distinguished triangle

in $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$ and the result holds for $K$ and $M$, then the result holds for $L$. Using the distinguished triangles of canonical truncations (Derived Categories, Remark 13.12.4) we reduce to the problem studied in the next paragraph.

Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $\mathcal{J} \subset \mathcal{O}_ Y$ be a quasi-coherent sheaf of ideals cutting out $Y \setminus V$. Denote $\mathcal{J}^ n\mathcal{F}$ the image of $f^*\mathcal{J}^ n \otimes \mathcal{F} \to \mathcal{F}$. We have to show that $(Rf_*(\mathcal{J}^ n\mathcal{F}))$ is a Deligne system. By Lemma 48.30.10 the question is local on $Y$. Thus we may assume $Y = \mathop{\mathrm{Spec}}(A)$ is affine and $\mathcal{J}$ corresponds to an ideal $I \subset A$. By Lemma 48.30.9 it suffices to show that the inverse system of cohomology modules $(H^ p(X, I^ n\mathcal{F}))$ is pro-isomorphic to the inverse system $(I^ n M)$ for some finite $A$-module $M$. This is shown in Cohomology of Schemes, Lemma 30.20.3. $\square$

Second proof. Let $K$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$. Let $L$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$. We will construct a bijection

functorial in $K$ and $L$. Fixing $K$ this will determine an isomorphism of pro-objects $Rf_*Rj_!K \to Rj'_!Rg_*K$ by Categories, Remark 4.22.7 and varying $K$ we obtain that this determines an isomorphism of functors. To actually produce the isomorphism we use the sequence of functorial equalities

The first equality is true by Lemma 48.30.1. The second equality is true because $g$ is proper (as the base change of $f$ to $V$) and hence $g^!$ is the right adjoint of pushforward by construction, see Section 48.16. The third equality holds as $g^!(L|_ V) = f^!L|_ U$ by Lemma 48.17.2. Since $f^!L$ is in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ by Lemma 48.17.6 the fourth equality follows from Lemma 48.30.2. The fifth equality holds again because $f^!$ is the right adjoint to $Rf_*$ as $f$ is proper. $\square$