hcju/stacksqa huggingface dataset:
Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. To prove (1) we have to show that $Rf_*E$ has quasi-coherent cohomology sheaves. This question is local on $Y$, hence we may assume $Y$ is quasi-compact. Pick $N = N(X, Y, f)$ as in Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-vanishing-higher-direct-images}. Thus $R^pf_*\mathcal{F} = 0$ for all quasi-coherent $\mathcal{O}_X$-modules $\mathcal{F}$ and all $p \geq N$. Moreover $R^pf_*\mathcal{F}$ is quasi-coherent for all $p$ by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-higher-direct-image}. These statements remain true after base change. \medskip\noindent First, assume $E$ is bounded below. We will show (1) and (2) and (3) hold for such $E$ with our choice of $N$. In this case we can for example use the spectral sequence $$ R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E $$ (Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}), the quasi-coherence of $R^pf_*H^q(E)$, and the vanishing of $R^pf_*H^q(E)$ for $p \geq N$ to see that (1), (2), and (3) hold in this case. \medskip\noindent Next we prove (2) and (3). Say $H^m(E) = 0$ for $m > 0$. Let $V$ be an affine object of $Y_\etale$. We have $H^p(V \times_Y X, \mathcal{F}) = 0$ for $p \geq N$, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-quasi-coherence-higher-direct-images-application}. Hence we may apply Lemma \ref{lemma-application-nice-K-injective} to the functor $\Gamma(V \times_Y X, -)$ to see that $$ R\Gamma(V, Rf_*E) = R\Gamma(V \times_Y X, E) $$ has vanishing cohomology in degrees $\geq N$. Since this holds for all $V$ affine in $Y_\etale$ we conclude that $H^m(Rf_*E) = 0$ for $m \geq N$. \medskip\noindent Next, we prove (1) in the general case. Recall that there is a distinguished triangle $$ \tau_{\leq -n - 1}E \to E \to \tau_{\geq -n}E \to (\tau_{\leq -n - 1}E)[1] $$ in $D(\mathcal{O}_X)$, see Derived Categories, Remark \ref{derived-remark-truncation-distinguished-triangle}. By (2) we see that $Rf_*\tau_{\leq -n - 1}E$ has vanishing cohomology sheaves in degrees $\geq -n + N$. Thus, given an integer $q$ we see that $R^qf_*E$ is equal to $R^qf_*\tau_{\geq -n}E$ for some $n$ and the result above applies.
Let $E_i$ be a family of objects of $D_\QCoh(\mathcal{O}_X)$ and set $E = \bigoplus E_i$. We want to show that the map $$ \bigoplus Rf_*E_i \longrightarrow Rf_*E $$ is an isomorphism. We will show it induces an isomorphism on cohomology sheaves in degree $0$ which will imply the lemma. Choose an integer $N$ as in Lemma \ref{lemma-quasi-coherence-direct-image}. Then $R^0f_*E = R^0f_*\tau_{\geq -N}E$ and $R^0f_*E_i = R^0f_*\tau_{\geq -N}E_i$ by the lemma cited. Observe that $\tau_{\geq -N}E = \bigoplus \tau_{\geq -N}E_i$. Thus we may assume all of the $E_i$ have vanishing cohomology sheaves in degrees $< -N$. Next we use the spectral sequences $$ R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E \quad\text{and}\quad R^pf_*H^q(E_i) \Rightarrow R^{p + q}f_*E_i $$ (Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}) to reduce to the case of a direct sum of quasi-coherent sheaves. This case is handled by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-colimit-cohomology}.
The statement means that a morphism $\alpha : E \to F$ of $D_\QCoh(\mathcal{O}_X)$ is an isomorphism if $Rf_*\alpha$ is an isomorphism. We may check this on cohomology sheaves. In particular, the question is \'etale local on $Y$. Hence we may assume $Y$ and therefore $X$ is affine. In this case the problem reduces to the case of schemes (Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-morphism}) via Lemma \ref{lemma-derived-quasi-coherent-small-etale-site} and Remark \ref{remark-match-total-direct-images}.
Since $f$ is affine the map $f_*\mathcal{O}_X \to Rf_*\mathcal{O}_X$ is an isomorphism (Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-affine-vanishing-higher-direct-images}). There is a canonical map $E \otimes^\mathbf{L} f_*\mathcal{O}_X = E \otimes^\mathbf{L} Rf_*\mathcal{O}_X \to Rf_* Lf^* E$ adjoint to the map $$ Lf^*(E \otimes^\mathbf{L} Rf_*\mathcal{O}_X) = Lf^*E \otimes^\mathbf{L} Lf^*Rf_*\mathcal{O}_X \longrightarrow Lf^* E \otimes^\mathbf{L} \mathcal{O}_X = Lf^* E $$ coming from $1 : Lf^*E \to Lf^*E$ and the canonical map $Lf^*Rf_*\mathcal{O}_X \to \mathcal{O}_X$. To check the map so constructed is an isomorphism we may work locally on $Y$. Hence we may assume $Y$ and therefore $X$ is affine. In this case the problem reduces to the case of schemes (Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-morphism-pull-push}) via Lemma \ref{lemma-derived-quasi-coherent-small-etale-site} and Remark \ref{remark-match-total-direct-images}.
The implications (3) $\Rightarrow$ (1) and (1) $\Rightarrow$ (2) are immediate. Thus it suffices to prove that (2) implies (3). We urge the reader to find their own proof of this fact. Let $Z'$ and $Z''$ be closed subspaces with $T = |Z'| = |Z''|$ such that $Z' \to Y$ is a proper morphism of algebraic spaces. We have to show that $Z'' \to Y$ is proper too. Let $Z''' = Z' \cup Z''$ be the scheme theoretic union, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-scheme-theoretic-intersection-union}. Then $Z'''$ is another closed subspace with $|Z'''| = T$. This follows for example from the description of scheme theoretic unions in Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-scheme-theoretic-union}. Since $Z'' \to Z'''$ is a closed immersion it suffices to prove that $Z''' \to Y$ is proper (see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-closed-immersion-proper} and \ref{spaces-morphisms-lemma-composition-proper}). The morphism $Z' \to Z'''$ is a bijective closed immersion and in particular surjective and universally closed. Then the fact that $Z' \to Y$ is separated implies that $Z''' \to Y$ is separated, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-universally-closed-separated}. Moreover $Z''' \to Y$ is locally of finite type as $X \to Y$ is locally of finite type (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-immersion-locally-finite-type} and \ref{spaces-morphisms-lemma-composition-finite-type}). Since $Z' \to Y$ is quasi-compact and $Z' \to Z'''$ is a universal homeomorphism we see that $Z''' \to Y$ is quasi-compact. Finally, since $Z' \to Y$ is universally closed, we see that the same thing is true for $Z''' \to Y$ by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-proper-is-proper}. This finishes the proof.
Observe that the statement makes sense as $f'$ is locally of finite type by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-finite-type}. Let $Z \subset X$ be the reduced induced closed subspace structure on $T$. Denote $Z' = (g')^{-1}(Z)$ the scheme theoretic inverse image. Then $Z' = X' \times_X Z = (Y' \times_Y X) \times_X Z = Y' \times_Y Z$ is proper over $Y'$ as a base change of $Z$ over $Y$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-proper}). On the other hand, we have $T' = |Z'|$. Hence the lemma holds.
Proof of (1). Assume $Y$ is separated over $B$ and $T \subset |X|$ is a closed subset proper over $B$. Let $Z$ be the reduced induced closed subspace structure on $T$ and apply Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-scheme-theoretic-image-is-proper} to $Z \to Y$ over $B$ to conclude. \medskip\noindent Proof of (2). Assume $f$ is universally closed and $T \subset |X|$ is a closed subset proper over $B$. Let $Z$ be the reduced induced closed subspace structure on $T$ and let $Z'$ be the reduced induced closed subspace structure on $|f|(T)$. We obtain an induced morphism $Z \to Z'$. Denote $Z'' = f^{-1}(Z')$ the scheme theoretic inverse image. Then $Z'' \to Z'$ is universally closed as a base change of $f$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-proper}). Hence $Z \to Z'$ is universally closed as a composition of the closed immersion $Z \to Z''$ and $Z'' \to Z'$ (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-closed-immersion-proper} and \ref{spaces-morphisms-lemma-composition-proper}). We conclude that $Z' \to B$ is separated by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-universally-closed-separated}. Since $Z \to B$ is quasi-compact and $Z \to Z'$ is surjective we see that $Z' \to B$ is quasi-compact. Since $Z' \to B$ is the composition of $Z' \to Y$ and $Y \to B$ we see that $Z' \to B$ is locally of finite type (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-immersion-locally-finite-type} and \ref{spaces-morphisms-lemma-composition-finite-type}). Finally, since $Z \to B$ is universally closed, we see that the same thing is true for $Z' \to B$ by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-proper-is-proper}. This finishes the proof. \medskip\noindent Proof of (3). Assume $f$ is proper and $T \subset |Y|$ is a closed subset proper over $B$. Let $Z$ be the reduced induced closed subspace structure on $T$. Denote $Z' = f^{-1}(Z)$ the scheme theoretic inverse image. Then $Z' \to Z$ is proper as a base change of $f$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-proper}). Whence $Z' \to B$ is proper as the composition of $Z' \to Z$ and $Z \to B$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-proper}). This finishes the proof.
Let $Z_i$ be the reduced induced closed subscheme structure on $T_i$. The morphism $$ Z_1 \amalg \ldots \amalg Z_n \longrightarrow X $$ is finite by Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-closed-immersion-finite} and \ref{spaces-morphisms-lemma-finite-union-finite}. As finite morphisms are universally closed (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-proper}) and since $Z_1 \amalg \ldots \amalg Z_n$ is proper over $S$ we conclude by Lemma \ref{lemma-functoriality-closed-proper-over-base} part (2) that the image $Z_1 \cup \ldots \cup Z_n$ is proper over $S$.
The support $\text{Supp}(\mathcal{F})$ of $\mathcal{F}$ is a closed subset of $|X|$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-support-finite-type}. Hence we can apply Definition \ref{definition-proper-over-base}. Since the scheme theoretic support of $\mathcal{F}$ is a closed subspace whose underlying closed subset is $\text{Supp}(\mathcal{F})$ we see that (1) and (2) are equivalent by Definition \ref{definition-proper-over-base}. It is clear that (2) implies (3). Conversely, if (3) is true, then $\text{Supp}(\mathcal{F}) \subset |Z|$ and hence $\text{Supp}(\mathcal{F})$ is proper over $Y$ for example by Lemma \ref{lemma-closed-closed-proper-over-base}.
Observe that the statement makes sense because $(g')*\mathcal{F}$ is of finite type by Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback}. We have $\text{Supp}((g')^*\mathcal{F}) = |g'|^{-1}(\text{Supp}(\mathcal{F}))$ by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-support-finite-type}. Thus the lemma follows from Lemma \ref{lemma-base-change-closed-proper-over-base}.
Proof of (1). Let $T$, $T'$ be the support of $\mathcal{F}$ and $\mathcal{G}$. Then all the sheaves mentioned in (1) have support contained in $T \cup T'$. Thus the assertion itself is clear from Lemmas \ref{lemma-closed-closed-proper-over-base} and \ref{lemma-union-closed-proper-over-base} provided we check that these sheaves are finite type and quasi-coherent. For quasi-coherence we refer the reader to Properties of Spaces, Section \ref{spaces-properties-section-quasi-coherent}. For ``finite type'' we refer the reader to Properties of Spaces, Section \ref{spaces-properties-section-properties-modules}. \medskip\noindent Proof of (2). The proof is the same as the proof of (1). Note that the assertions make sense as $X$ is locally Noetherian by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian} and by the description of the category of coherent modules in Cohomology of Spaces, Section \ref{spaces-cohomology-section-coherent}.
By Lemma \ref{lemma-module-support-proper-over-base} there exists a closed immersion $i : Z \to X$ with $g = f \circ i : Z \to Y$ proper and $\mathcal{F} = i_*\mathcal{G}$ for some coherent module $\mathcal{G}$ on $Z$. We see that $R^pg_*\mathcal{G}$ is coherent on $S$ by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-proper-pushforward-coherent}. On the other hand, $R^qi_*\mathcal{G} = 0$ for $q > 0$ (Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-finite-pushforward-coherent}). By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-relative-Leray} we get $R^pf_*\mathcal{F} = R^pg_*\mathcal{G}$ and the lemma follows.
Consider the spectral sequence $$ R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E $$ see Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. By assumption and Lemma \ref{lemma-support-proper-over-base-pushforward} the sheaves $R^pf_*H^q(E)$ are coherent. Hence $R^{p + q}f_*E$ is coherent, i.e., $E \in D_{\textit{Coh}}(\mathcal{O}_Y)$. Boundedness from below is trivial. Boundedness from above follows from Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-vanishing-higher-direct-images} or from Lemma \ref{lemma-quasi-coherence-direct-image}.
The proof is the same as the proof of Lemma \ref{lemma-direct-image-coherent}. You can also deduce it from Lemma \ref{lemma-direct-image-coherent} by considering what the exact functor $Rf_*$ does to the distinguished triangles $\tau_{\leq a}E \to E \to \tau_{\geq a + 1}E \to \tau_{\leq a}E[1]$.
We can check whether an object of $D(\mathcal{O}_X)$ is in $D_{\textit{Coh}}(\mathcal{O}_X)$ \'etale locally on $X$, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-coherent-Noetherian}. Hence this lemma follows from the case of schemes, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-coherent-internal-hom}.
Recall that $$ \Ext_{\mathcal{O}_X}^n(K, L) = H^n(X, R\SheafHom_{\mathcal{O}_X}(K, L)) = H^n(\Spec(A), Rf_*R\SheafHom_{\mathcal{O}_X}(K, L)) $$ see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-section-RHom-over-U} and Cohomology on Sites, Section \ref{sites-cohomology-section-leray}. Thus the result follows from Lemmas \ref{lemma-coherent-internal-hom} and \ref{lemma-direct-image-coherent-bdd-below}.
We first claim that $P$ holds for every representable quasi-compact and quasi-separated object of $X_{spaces, \etale}$. Namely, suppose that $U \to X$ is \'etale and $U$ is a quasi-compact and quasi-separated scheme. By assumption (1) property $P$ holds for every affine open of $U$. Moreover, if $W, V \subset U$ are quasi-compact open with $V$ affine and $P$ holds for  $W$, $V$, and $W \cap V$, then $P$ holds for $W \cup V$ by (2) (as the pair $(W \subset W \cup V, V \to W \cup V)$ is an elementary distinguished square). Thus $P$ holds for $U$ by the induction principle for schemes, see Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}. \medskip\noindent To finish the proof it suffices to prove $P$ holds for $X$ (because we can simply replace $X$ by any quasi-compact and quasi-separated object of $X_{spaces, \etale}$ we want to prove the result for). We will use the filtration $$ \emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X $$ and the morphisms $f_p : V_p \to U_p$ of Decent Spaces, Lemma \ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated}. We will prove that $P$ holds for $U_p$ by descending induction on $p$. Note that $P$ holds for $U_{n + 1}$ by (1) as an empty algebraic space is affine. Assume $P$ holds for $U_{p + 1}$. Note that $(U_{p + 1} \subset U_p, f_p : V_p \to U_p)$ is an elementary distinguished square, but (2) may not apply as $V_p$ may not be affine. However, as $V_p$ is a quasi-compact scheme we may choose a finite affine open covering $V_p = V_{p, 1} \cup \ldots \cup V_{p, m}$. Set $W_{p, 0} = U_{p + 1}$ and $$ W_{p, i} = U_{p + 1} \cup f_p(V_{p, 1} \cup \ldots \cup V_{p, i}) $$ for $i = 1, \ldots, m$. These are quasi-compact open subspaces of $X$. Then we have $$ U_{p + 1} = W_{p, 0} \subset W_{p, 1} \subset \ldots \subset W_{p, m} = U_p $$ and the pairs $$ (W_{p, 0} \subset W_{p, 1}, f_p|_{V_{p, 1}}), (W_{p, 1} \subset W_{p, 2}, f_p|_{V_{p, 2}}),\ldots, (W_{p, m - 1} \subset W_{p, m}, f_p|_{V_{p, m}}) $$ are elementary distinguished squares by Lemma \ref{lemma-make-more-elementary-distinguished-squares}. Note that $P$ holds for each $V_{p, 1}$ (as affine schemes) and for $W_{p, i} \times_{W_{p, i + 1}} V_{p, i + 1}$ as this is a quasi-compact open of $V_{p, i + 1}$ and hence $P$ holds for it by the first paragraph of this proof. Thus (2) applies to each of these and we inductively conclude $P$ holds for $W_{p, 1}, \ldots, W_{p, m} = U_p$.
We can deduce this from Lemma \ref{lemma-induction-principle-separated}, but instead we will give a direct argument by explicitly redoing the proof of Lemma \ref{lemma-induction-principle}. We will use the filtration $$ \emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X $$ and the morphisms $f_p : V_p \to U_p$ of Decent Spaces, Lemma \ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated}. We will prove that $P$ holds for $W_p = W \cup U_p$ by descending induction on $p$. This will finish the proof as $W_1 = X$. Note that $P$ holds for $W_{n + 1} = W \cap U_{n + 1} = W$ by (1). Assume $P$ holds for $W_{p + 1}$. Observe that $W_p \setminus W_{p + 1}$ (with reduced induced subspace structure) is a closed subspace of $U_p \setminus U_{p + 1}$. Since $(U_{p + 1} \subset U_p, f_p : V_p \to U_p)$ is an elementary distinguished square, the same is true for $(W_{p + 1} \subset W_p, f_p : V_p \to W_p)$. However (2) may not apply as $V_p$ may not be affine. However, as $V_p$ is a quasi-compact scheme we may choose a finite affine open covering $V_p = V_{p, 1} \cup \ldots \cup V_{p, m}$. Set $W_{p, 0} = W_{p + 1}$ and $$ W_{p, i} = W_{p + 1} \cup f_p(V_{p, 1} \cup \ldots \cup V_{p, i}) $$ for $i = 1, \ldots, m$. These are quasi-compact open subspaces of $X$ containing $W$. Then we have $$ W_{p + 1} = W_{p, 0} \subset W_{p, 1} \subset \ldots \subset W_{p, m} = W_p $$ and the pairs $$ (W_{p, 0} \subset W_{p, 1}, f_p|_{V_{p, 1}}), (W_{p, 1} \subset W_{p, 2}, f_p|_{V_{p, 2}}),\ldots, (W_{p, m - 1} \subset W_{p, m}, f_p|_{V_{p, m}}) $$ are elementary distinguished squares by Lemma \ref{lemma-make-more-elementary-distinguished-squares}. Now (2) applies to each of these and we inductively conclude $P$ holds for $W_{p, 1}, \ldots, W_{p, m} = W_p$.
To show the sequence of (1) is exact we may check on stalks at geometric points by Properties of Spaces, Theorem \ref{spaces-properties-theorem-exactness-stalks}. Let $\overline{x}$ be a geometric point of $X$. By Equations (\ref{equation-stalk-restriction}) and (\ref{equation-stalk-j-shriek}) taking stalks at $\overline{x}$ we obtain the sequence $$ 0 \to \bigoplus\nolimits_{(\overline{u}, \overline{v})} \mathcal{F}_{\overline{x}} \to \bigoplus\nolimits_{\overline{u}} \mathcal{F}_{\overline{x}} \oplus \bigoplus\nolimits_{\overline{v}} \mathcal{F}_{\overline{x}} \to \mathcal{F}_{\overline{x}} \to 0 $$ This sequence is exact because for every $\overline{x}$ there either is exactly one $\overline{u}$ mapping to $\overline{x}$, or there is no $\overline{u}$ and exactly one $\overline{v}$ mapping to $\overline{x}$. \medskip\noindent Proof of (2). We have seen in Cohomology on Sites, Section \ref{sites-cohomology-section-properties-K-injective} that the restriction functors and the extension by zero functors on derived categories are computed by just applying the functor to any complex. Let $\mathcal{E}^\bullet$ be a complex of $\mathcal{O}_X$-modules representing $E$. The distinguished triangle of the lemma is the distinguished triangle associated (by Derived Categories, Section \ref{derived-section-canonical-delta-functor} and especially Lemma \ref{derived-lemma-derived-canonical-delta-functor}) to the short exact sequence of complexes of $\mathcal{O}_X$-modules $$ 0 \to j_{U \times_X V!}\mathcal{E}^\bullet|_{U \times_X V} \to j_{U!}\mathcal{E}^\bullet|_U \oplus j_{V!}\mathcal{E}^\bullet|_V \to \mathcal{E}^\bullet \to 0 $$ which is short exact by (1).
Let $W$ be an object of $X_\etale$. We claim the sequence $$ 0 \to \mathcal{F}(W) \to \mathcal{F}(W \times_X U) \oplus \mathcal{F}(W \times_X V) \to \mathcal{F}(W \times_X U \times_X V) $$ is exact and that an element of the last group can locally on $W$ be lifted to the middle one. By Lemma \ref{lemma-make-more-elementary-distinguished-squares} the pair $(W \times_X U \subset W, V \times_X W \to W)$ is an elementary distinguished square. Thus we may assume $W = X$ and it suffices to prove the same thing for $$ 0 \to \mathcal{F}(X) \to \mathcal{F}(U) \oplus \mathcal{F}(V) \to \mathcal{F}(U \times_X V) $$ We have seen that $$ 0 \to j_{U \times_X V!}\mathcal{O}_{U \times_X V} \to j_{U!}\mathcal{O}_U \oplus j_{V!}\mathcal{O}_V \to \mathcal{O}_X \to 0 $$ is a exact sequence of $\mathcal{O}_X$-modules in Lemma \ref{lemma-exact-sequence-lower-shriek} and applying the right exact functor $\Hom_{\mathcal{O}_X}(- , \mathcal{F})$ gives the sequence above. This also means that the obstruction to lifting $s \in \mathcal{F}(U \times_X V)$ to an element of $\mathcal{F}(U) \oplus \mathcal{F}(V)$ lies in $\Ext^1_{\mathcal{O}_X}(\mathcal{O}_X, \mathcal{F}) = H^1(X, \mathcal{F})$. By locality of cohomology (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-kill-cohomology-class-on-covering}) this obstruction vanishes \'etale locally on $X$ and the proof of (1) is complete. \medskip\noindent Proof of (2). Choose a K-injective complex $\mathcal{I}^\bullet$ representing $E$ whose terms $\mathcal{I}^n$ are injective objects of $\textit{Mod}(\mathcal{O}_X)$, see Injectives, Theorem \ref{injectives-theorem-K-injective-embedding-grothendieck}. Then $\mathcal{I}^\bullet|U$ is a K-injective complex (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-restrict-K-injective-to-open}). Hence $Rj_{U, *}E|_U$ is represented by $j_{U, *}\mathcal{I}^\bullet|_U$. Similarly for $V$ and $U \times_X V$. Hence the distinguished triangle of the lemma is the distinguished triangle associated (by Derived Categories, Section \ref{derived-section-canonical-delta-functor} and especially Lemma \ref{derived-lemma-derived-canonical-delta-functor}) to the short exact sequence of complexes $$ 0 \to \mathcal{I}^\bullet \to j_{U, *}\mathcal{I}^\bullet|_U \oplus j_{V, *}\mathcal{I}^\bullet|_V \to j_{U \times_X V, *}\mathcal{I}^\bullet|_{U \times_X V} \to 0. $$ This sequence is exact by (1).
Choose a K-injective complex $\mathcal{I}^\bullet$ representing $E$. We may assume $\mathcal{I}^n$ is an injective object of $\textit{Mod}(\mathcal{O}_X)$ for all $n$, see Injectives, Theorem \ref{injectives-theorem-K-injective-embedding-grothendieck}. Then $Rf_*E$ is computed by $f_*\mathcal{I}^\bullet$. Similarly for $U$, $V$, and $U \cap V$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-restrict-K-injective-to-open}. Hence the distinguished triangle of the lemma is the distinguished triangle associated (by Derived Categories, Section \ref{derived-section-canonical-delta-functor} and especially Lemma \ref{derived-lemma-derived-canonical-delta-functor}) to the short exact sequence of complexes $$ 0 \to f_*\mathcal{I}^\bullet \to a_*\mathcal{I}^\bullet|_U \oplus b_*\mathcal{I}^\bullet|_V \to c_*\mathcal{I}^\bullet|_{U \times_X V} \to 0. $$ To see this is a short exact sequence of complexes we argue as follows. Pick an injective object $\mathcal{I}$ of $\textit{Mod}(\mathcal{O}_X)$. Apply $f_*$ to the short exact sequence $$ 0 \to \mathcal{I} \to j_{U, *}\mathcal{I}|_U \oplus j_{V, *}\mathcal{I}|_V \to j_{U \times_X V, *}\mathcal{I}|_{U \times_X V} \to 0 $$ of Lemma \ref{lemma-exact-sequence-j-star} and use that $R^1f_*\mathcal{I} = 0$ to get a short exact sequence $$ 0 \to f_*\mathcal{I} \to f_*j_{U, *}\mathcal{I}|_U \oplus f_*j_{V, *}\mathcal{I}|_V \to f_*j_{U \times_X V, *}\mathcal{I}|_{U \times_X V} \to 0 $$ The proof is finished by observing that $a_* = f_*j_{U, *}$ and similarly for $b_*$ and $c_*$.