hcju/stacksqa huggingface dataset:
Use the distinguished triangle of Lemma \ref{lemma-exact-sequence-lower-shriek} to obtain a long exact sequence of $\Hom$'s (from Derived Categories, Lemma \ref{derived-lemma-representable-homological}) and use that $\Hom(j_{U!}E|_U, F) = \Hom(E|_U, F|_U)$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-adjoint-lower-shriek-restrict}.
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}. In the proof of Lemma \ref{lemma-exact-sequence-j-star} we found a short exact sequence of complexes $$ 0 \to \mathcal{I}^\bullet \to j_{U, *}\mathcal{I}^\bullet|_U \oplus j_{V, *}\mathcal{I}^\bullet|_V \to j_{U \times_X V, *}\mathcal{I}^\bullet|_{U \times_X V} \to 0 $$ Since $H^1(X, \mathcal{I}^n) = 0$, we see that taking global sections gives an exact sequence of complexes $$ 0 \to \Gamma(X, \mathcal{I}^\bullet) \to \Gamma(U, \mathcal{I}^\bullet) \oplus \Gamma(V, \mathcal{I}^\bullet) \to \Gamma(U \times_X V, \mathcal{I}^\bullet) \to 0 $$ Since these complexes represent $R\Gamma(X, E)$, $R\Gamma(U, E)$, $R\Gamma(V, E)$, and $R\Gamma(U \times_X V, E)$ we  get a distinguished triangle by Derived Categories, Section \ref{derived-section-canonical-delta-functor} and especially Lemma \ref{derived-lemma-derived-canonical-delta-functor}.
This is true because $(j_!\mathcal{F})|_V = j_{W!}(\mathcal{F}|_W)$ for an $\mathcal{O}_X$-module $\mathcal{F}$ as follows immediately  from the construction of the functors $j_!$ and $j_{W!}$, see Modules on Sites, Lemma \ref{sites-modules-lemma-extension-by-zero}.
Let $E$ be an object of $D(\mathcal{O}_X)$ whose cohomology sheaves are supported on $T$. Then we see that $E|_U = 0$ and $E|_{U \times_X V} = 0$ as $T$ doesn't meet $U$ and $j^{-1}T$ doesn't meet $U \times_X V$. Thus (1)(a) follows from Lemma \ref{lemma-exact-sequence-j-star}. In exactly the same way (1)(b) follows from Lemma \ref{lemma-exact-sequence-lower-shriek}. \medskip\noindent Let $F$ be an object of $D(\mathcal{O}_V)$ whose cohomology sheaves are supported on $j^{-1}T$. By Lemma \ref{lemma-restrict-direct-image-open} we have $(Rj_*F)|_U = Rj_{W, *}(F|_W) = 0$ because $F|_W = 0$ by our assumption. Similarly $(j_!F)|_U = j_{W!}(F|_W) = 0$ by Lemma \ref{lemma-restrict-lower-shriek}. Thus $j_!F$ and $Rj_*F$ are supported on $T$ and $(j_!F)|_V$ and $(Rj_*F)|_V$ are supported on $j^{-1}(T)$. To check that the maps (2)(a), (b), (c) are isomorphisms in the derived category, it suffices to check that these map induce isomorphisms on stalks of cohomology sheaves at geometric points of $T$ and $j^{-1}(T)$ by Properties of Spaces, Theorem \ref{spaces-properties-theorem-exactness-stalks}. This we may do after replacing $X$ by $V$, $U$ by $U \times_X V$, $V$ by $V \times_X V$ and $F$ by $F|_{V \times_X V}$ (restriction via first projection), see Lemmas \ref{lemma-restrict-direct-image-open}, \ref{lemma-restrict-lower-shriek}, and \ref{lemma-make-more-elementary-distinguished-squares}. Since $V \times_X V \to V$ has a section this reduces (2) to the case that $j : V \to X$ has a section. \medskip\noindent Assume $j$ has a section $\sigma : X \to V$. Set $V' = \sigma(X)$. This is an open subspace of $V$. Set $U' = j^{-1}(U)$. This is another open subspace of $V$. Then $(U' \subset V, V' \to V)$ is an elementary distinguished square. Observe that $F|_{U'} = 0$ and $F|_{V' \cap U'} = 0$ because $F$ is supported on $j^{-1}(T)$. Denote $j' : V' \to V$ the open immersion and $j_{V'} : V' \to X$ the composition $V' \to V \to X$ which is the inverse of $\sigma$. Set $F' = \sigma^*F$. The distinguished triangles of Lemmas \ref{lemma-exact-sequence-lower-shriek} and \ref{lemma-exact-sequence-j-star} show that $F = j'_!(F|_{V'})$ and $F = Rj'_*(F|_{V'})$. It follows that $j_!F = j_!j'_!(F|_{V'}) = j_{V'!}F = F'$ because $j_{V'} : V' \to X$ is an isomorphism and the inverse of $\sigma$. Similarly, $Rj_*F = Rj_*Rj'_*F = Rj_{V', *}F = F'$. This proves (2)(c). To prove (2)(a) and (2)(b) it suffices to show that $F = F'|_V$. This is clear because both $F$ and $F'|_V$ restrict to zero on $U'$ and $U' \cap V'$ and the same object on $V'$.
Denote $j_U$, $j_V$, $j_{U \times_X V}$ the corresponding morphisms towards $X$. Choose a distinguished triangle $$ F \to Rj_{U, *}A \oplus Rj_{V, *}B \to Rj_{U \times_X V, *}(B|_{U \times_X V}) \to F[1] $$ Here the map $Rj_{V, *}B \to Rj_{U \times_X V, *}(B|_{U \times_X V})$ is the obvious one. The map $Rj_{U, *}A \to Rj_{U \times_X V, *}(B|_{U \times_X V})$ is the composition of $Rj_{U, *}A \to Rj_{U \times_X V, *}(A|_{U \times_X V})$ with $Rj_{U \times_X V, *}c$. Restricting to $U$ we obtain $$ F|_U \to A \oplus (Rj_{V, *}B)|_U \to (Rj_{U \times_X V, *}(B|_{U \times_X V}))|_U \to F|_U[1] $$ Denote $j : U \times_X V \to U$. Compatibility of restriction and total direct image (Lemma \ref{lemma-restrict-direct-image-open}) shows that both $(Rj_{V, *}B)|_U$ and $(Rj_{U \times_X V, *}(B|_{U \times_X V}))|_U$ are canonically isomorphic to $Rj_*(B|_{U \times_X V})$. Hence the second arrow of the last displayed equation has a section, and we conclude that the morphism $F|_U \to A$ is an isomorphism. \medskip\noindent To see that the morphism $F|_V \to B$ is an isomorphism we will use a trick. Namely, choose a distinguished triangle $$ F|_V \to B \to B' \to F[1]|_V $$ in $D(\mathcal{O}_V)$. Since $F|_U \to A$ is an isomorphism, and since we have the isomorphism $c : A|_{U \times_X V} \to B|_{U \times_X V}$ the restriction of $F|_V \to B$ is an isomorphism over $U \times_X V$. Thus $B'$ is supported on $j_V^{-1}(T)$ where $T = |X| \setminus |U|$. On the other hand, there is a morphism of distinguished triangles $$ \xymatrix{ F \ar[r] \ar[d] & Rj_{U, *}F|_U \oplus Rj_{V, *}F|_V \ar[r] \ar[d] & Rj_{U \times_X V, *}F|_{U \times_X V} \ar[r] \ar[d] & F[1] \ar[d] \\ F \ar[r] & Rj_{U, *}A \oplus Rj_{V, *}B \ar[r] & Rj_{U \times_X V, *}(B|_{U \times_X V}) \ar[r] & F[1] } $$ The all of the vertical maps in this diagram are isomorphisms, except for the map $Rj_{V, *}F|_V \to Rj_{V, *}B$, hence that is an isomorphism too (Derived Categories, Lemma \ref{derived-lemma-third-isomorphism-triangle}). This implies that $Rj_{V, *}B' = 0$. Hence $B' = 0$ by Lemma \ref{lemma-pushforward-with-support-in-open}. \medskip\noindent The existence of the morphism $F \to E$ follows from the Mayer-Vietoris sequence for $\Hom$, see Lemma \ref{lemma-mayer-vietoris-hom}.
The functor $f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)$ is exact, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-affine-vanishing-higher-direct-images}. Hence $f_*$ defines a derived functor $f_* : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))$ by simply applying $f_*$ to any representative complex, see Derived Categories, Lemma \ref{derived-lemma-right-derived-exact-functor}. For any complex of $\mathcal{O}_X$-modules $\mathcal{F}^\bullet$ there is a canonical map $f_*\mathcal{F}^\bullet \to Rf_*\mathcal{F}^\bullet$. To finish the proof we show this is a quasi-isomorphism when $\mathcal{F}^\bullet$ is a complex with each $\mathcal{F}^n$ quasi-coherent. The statement is \'etale local on $Y$ hence we may assume $Y$ affine. As an affine morphism is representable we reduce to the case of schemes by the compatibility of Remark \ref{remark-match-total-direct-images}. The case of schemes is Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-pushforward}.
We will prove this by showing that $RQ_Y \circ Rf_*$ and $\Phi \circ RQ_X$ are right adjoint to the same functor $D(\QCoh(\mathcal{O}_Y)) \to D(\mathcal{O}_X)$. \medskip\noindent Since $f$ is quasi-compact and quasi-separated, we see that $f_*$ preserves quasi-coherence, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}. Recall that $\QCoh(\mathcal{O}_X)$ is a Grothendieck abelian category (Properties of Spaces, Proposition \ref{spaces-properties-proposition-coherator}). Hence any $K$ in $D(\QCoh(\mathcal{O}_X))$ can be represented by a K-injective complex $\mathcal{I}^\bullet$ of $\QCoh(\mathcal{O}_X)$, see Injectives, Theorem \ref{injectives-theorem-K-injective-embedding-grothendieck}. Then we can define $\Phi(K) = f_*\mathcal{I}^\bullet$. \medskip\noindent Since $f$ is flat, the functor $f^*$ is exact. Hence $f^*$ defines $f^* : D(\mathcal{O}_Y) \to D(\mathcal{O}_X)$ and also $f^* : D(\QCoh(\mathcal{O}_Y)) \to D(\QCoh(\mathcal{O}_X))$. The functor $f^* = Lf^* : D(\mathcal{O}_Y) \to D(\mathcal{O}_X)$ is left adjoint to $Rf_* : D(\mathcal{O}_X) \to D(\mathcal{O}_Y)$, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-adjoint}. Similarly, the functor $f^* : D(\QCoh(\mathcal{O}_Y)) \to D(\QCoh(\mathcal{O}_X))$ is left adjoint to $\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))$ by Derived Categories, Lemma \ref{derived-lemma-derived-adjoint-functors}. \medskip\noindent Let $A$ be an object of $D(\QCoh(\mathcal{O}_Y))$ and $E$ an object of $D(\mathcal{O}_X)$. Then \begin{align*} \Hom_{D(\QCoh(\mathcal{O}_Y))}(A, RQ_Y(Rf_*E)) & = \Hom_{D(\mathcal{O}_Y)}(A, Rf_*E) \\ & = \Hom_{D(\mathcal{O}_X)}(f^*A, E) \\ & = \Hom_{D(\QCoh(\mathcal{O}_X))}(f^*A, RQ_X(E)) \\ & = \Hom_{D(\QCoh(\mathcal{O}_Y))}(A, \Phi(RQ_X(E))) \end{align*} This implies what we want.
Let $X_0 = \Spec(A)$ be the affine scheme representing $X$. Recall that there is a morphism of ringed sites $\epsilon : X_\etale \to X_{0, Zar}$ which induces equivalences $$ \xymatrix{ \QCoh(\mathcal{O}_X) \ar@<1ex>[r]^{{\epsilon_*}} & \QCoh(\mathcal{O}_{X_0}) \ar@<1ex>[l]^{{\epsilon^*}} } $$ see Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}. Hence we see that $Q_X = \epsilon^* \circ Q_{X_0} \circ \epsilon_*$ by uniqueness of adjoint functors. Hence (1) follows from the description of $Q_{X_0}$ in Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-coherator} and the fact that $\Gamma(X_0, \epsilon_*\mathcal{F}) = \Gamma(X, \mathcal{F})$. Part (2) follows from (1) and the fact that the functor from $A$-modules to quasi-coherent $\mathcal{O}_X$-modules is exact. The third assertion now follows from the result for schemes (Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-coherator}) and Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}.
Let $S = \Spec(A)$. Let $\mathcal{J}$ be an injective object of $\QCoh(\mathcal{O}_S)$. Since $\QCoh(\mathcal{O}_S)$ is equivalent to the category of $A$-modules we see that $\mathcal{J}$ is equal to $\widetilde{J}$ for some injective $A$-module $J$. By Dualizing Complexes, Proposition \ref{dualizing-proposition-structure-injectives-noetherian} we can write $J = \bigoplus E_\alpha$ with $E_\alpha$ indecomposable and therefore isomorphic to the injective hull of a reside field at a point. Thus (because finite disjoint unions of Artinian schemes are Artinian) we may assume that $J$ is the injective hull of $\kappa(\mathfrak p)$ for some prime $\mathfrak p$ of $A$. Then $J = \bigcup J[\mathfrak p^n]$ where $J[\mathfrak p^n]$ is the injective hull of $\kappa(\mathfrak p)$ over $A_\mathfrak/\mathfrak p^nA_\mathfrak p$, see Dualizing Complexes, Lemma \ref{dualizing-lemma-union-artinian}. Thus $\widetilde{J}$ is the colimit of the sheaves $(Z_n \to X)_*\mathcal{G}_n$ where $Z_n = \Spec(A_\mathfrak p/\mathfrak p^nA_\mathfrak p)$ and $\mathfrak G_n$ the coherent sheaf associated to the finite $A_\mathfrak/\mathfrak p^nA_\mathfrak p$-module $J[\mathfrak p^n]$. Finiteness follows from Dualizing Complexes, Lemma \ref{dualizing-lemma-finite}.
Choose an affine scheme $U$ and a surjective \'etale morphism $j : U \to X$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Then $U$ is a Noetherian affine scheme. Choose an injective object $\mathcal{J}'$ of $\QCoh(\mathcal{O}_U)$ such that there exists an injection $\mathcal{J}|_U \to \mathcal{J}'$. Then $$ \mathcal{J} \to j_*\mathcal{J}' $$ is an injective morphism in $\QCoh(\mathcal{O}_X)$, hence identifies $\mathcal{J}$ as a direct summand of $j_*\mathcal{J}'$. Thus the result follows from the corresponding result for $\mathcal{J}'$ proved in Lemma \ref{lemma-affine-injective-colimit-direct-sum-pushforwards-artin}.
Since $f$ is quasi-compact and quasi-separated, the functor $f_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}). The functor $f^*$ is a left adjoint to $f_*$ which transforms injections into injections. Hence the result follows from Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}
Proof of (1). Write $\mathcal{J}$ as a direct summand of $\colim \mathcal{F}_i$ with $\mathcal{F}_i = (Z_i \to X)_*\mathcal{G}_i$ as in Lemma \ref{lemma-injective-colimit-direct-sum-pushforwards-artin}. It is clear that it suffices to prove the vanishing for $\colim \mathcal{F}_i$. Since pullback commutes with colimits and since $U$ is quasi-compact and quasi-separated, it suffices to prove $H^p(U, \mathcal{F}_i|_U) = 0$ for $p > 0$, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-colimits}. Observe that $Z_i \to X$ is an affine morphism, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-Artinian-affine}. Thus $$ \mathcal{F}_i|_U = (Z_i \times_X U \to U)_*\mathcal{G}'_i = R(Z_i \times_X U \to U)_*\mathcal{G}'_i $$ where $\mathcal{G}'_i$ is the pullback of $\mathcal{G}_i$ to $Z_i \times_X U$, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-affine-base-change}. Since $Z_i \times_X U$ is affine we conclude that $\mathcal{G}'_i$ has no higher cohomology on $Z_i \times_X U$. By the Leray spectral sequence we conclude the same thing is true for $\mathcal{F}_i|_U$ (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray}). \medskip\noindent Proof of (2). Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $V \to Y$ be an \'etale morphism with $V$ affine. Then $V \times_Y X \to X$ is an \'etale morphism and $V \times_Y X$ is a quasi-compact and quasi-separated algebraic space \'etale over $X$ (details omitted). Hence $H^p(V \times_Y X, \mathcal{J})$ is zero by part (1). Since $R^pf_*\mathcal{J}$ is the sheaf associated to the presheaf $V \mapsto H^p(V \times_Y X, \mathcal{J})$ the result is proved.
Since $X$ and $Y$ are Noetherian the morphism is quasi-compact and quasi-separated (see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-quasi-separated-permanence}). Thus $f_*$ preserve quasi-coherence, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}. Next, let $K$ be an object of $D(\QCoh(\mathcal{O}_X))$. Since $\QCoh(\mathcal{O}_X)$ is a Grothendieck abelian category (Properties of Spaces, Proposition \ref{spaces-properties-proposition-coherator}), we can represent $K$ by a K-injective complex $\mathcal{I}^\bullet$ such that each $\mathcal{I}^n$ is an injective object of $\QCoh(\mathcal{O}_X)$, see Injectives, Theorem \ref{injectives-theorem-K-injective-embedding-grothendieck}. Thus we see that the functor $\Phi$ is defined by setting $$ \Phi(K) = f_*\mathcal{I}^\bullet $$ where the right hand side is viewed as an object of $D(\QCoh(\mathcal{O}_Y))$. To finish the proof of the lemma it suffices to show that the canonical map $$ f_*\mathcal{I}^\bullet \longrightarrow Rf_*\mathcal{I}^\bullet $$ is an isomorphism in $D(\mathcal{O}_Y)$. To see this it suffices to prove the map induces an isomorphism on cohomology sheaves. Pick any $m \in \mathbf{Z}$. Let $N = N(X, Y, f)$ be as in Lemma \ref{lemma-quasi-coherence-direct-image}. Consider the short exact sequence $$ 0 \to \sigma_{\geq m - N - 1}\mathcal{I}^\bullet \to \mathcal{I}^\bullet \to \sigma_{\leq m - N - 2}\mathcal{I}^\bullet \to 0 $$ of complexes of quasi-coherent sheaves on $X$. By Lemma \ref{lemma-quasi-coherence-direct-image} we see that the cohomology sheaves of $Rf_*\sigma_{\leq m - N - 2}\mathcal{I}^\bullet$ are zero in degrees $\geq m - 1$. Thus we see that $R^mf_*\mathcal{I}^\bullet$ is isomorphic to $R^mf_*\sigma_{\geq m - N - 1}\mathcal{I}^\bullet$. In other words, we may assume that $\mathcal{I}^\bullet$ is a bounded below complex of injective objects of $\QCoh(\mathcal{O}_X)$. This case follows from Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) with required vanishing because of Lemma \ref{lemma-injective-pushforward}.
The implication (1) $\Rightarrow$ (2) is a general fact, see Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback}. Assume (2). By assumption there exists an \'etale covering $\{f_i : X_i \to X\}$ such that $\epsilon^*\mathcal{F}|_{(X_i)_\etale}$ is generated by finitely many sections. Let $x \in X$. We will show that $\mathcal{F}$ is generated by finitely many sections in a neighbourhood of $x$. Say $x$ is in the image of $X_i \to X$ and denote $X' = X_i$. Let $s_1, \ldots, s_n \in \Gamma(X', \epsilon^*\mathcal{F}|_{X'_\etale})$ be generating sections. As $\epsilon^*\mathcal{F} = \epsilon^{-1}\mathcal{F} \otimes_{\epsilon^{-1}\mathcal{O}_X} \mathcal{O}_\etale$ we can find an \'etale morphism $X'' \to X'$ such that $x$ is in the image of $X' \to X$ and such that $s_i|_{X''} = \sum s_{ij} \otimes a_{ij}$ for some sections $s_{ij} \in \epsilon^{-1}\mathcal{F}(X'')$ and $a_{ij} \in \mathcal{O}_\etale(X'')$. Denote $U \subset X$ the image of $X'' \to X$. This is an open subscheme as $f'' : X'' \to X$ is \'etale (Morphisms, Lemma \ref{morphisms-lemma-etale-open}). After possibly shrinking $X''$ more we may assume $s_{ij}$ come from elements $t_{ij} \in \mathcal{F}(U)$ as follows from the construction of the inverse image functor $\epsilon^{-1}$. Now we claim that $t_{ij}$ generate $\mathcal{F}|_U$ which finishes the proof of the lemma. Namely, the corresponding map $\mathcal{O}_U^{\oplus N} \to \mathcal{F}|_U$ has the property that its pullback by $f''$ to $X''$ is surjective. Since $f'' : X'' \to U$ is a surjective flat morphism of schemes, this implies that $\mathcal{O}_U^{\oplus N} \to \mathcal{F}|_U$ is surjective by looking at stalks and using that $\mathcal{O}_{U, f''(z)} \to \mathcal{O}_{X'', z}$ is faithfully flat for all $z \in X''$.
The implication (1) $\Rightarrow$ (2) is a general fact, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pseudo-coherent-pullback}. Assume $\epsilon^*E$ is $m$-pseudo-coherent. We will use without further mention that $\epsilon^*$ is an exact functor and that therefore $$ \epsilon^*H^i(E) = H^i(\epsilon^*E). $$ To show that $E$ is $m$-pseudo-coherent we may work locally on $X$, hence we may assume that $X$ is quasi-compact (for example affine). Since $X$ is quasi-compact every \'etale covering $\{U_i \to X\}$ has a finite refinement. Thus we see that $\epsilon^*E$ is an object of $D^{-}(\mathcal{O}_\etale)$, see comments following Cohomology on Sites, Definition \ref{sites-cohomology-definition-pseudo-coherent}. By Lemma \ref{lemma-epsilon-flat} it follows that $E$ is an object of $D^-(\mathcal{O}_X)$. \medskip\noindent Let $n \in \mathbf{Z}$ be the largest integer such that $H^n(E)$ is nonzero; then $n$ is also the largest integer such that $H^n(\epsilon^*E)$ is nonzero. We will prove the lemma by induction on $n - m$. If $n < m$, then the lemma is clearly true. If $n \geq m$, then $H^n(\epsilon^*E)$ is a finite $\mathcal{O}_\etale$-module, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-finite-cohomology}. Hence $H^n(E)$ is a finite $\mathcal{O}_X$-module, see Lemma \ref{lemma-descend-finite-type}. After replacing $X$ by the members of an open covering, we may assume there exists a surjection $\mathcal{O}_X^{\oplus t} \to H^n(E)$. We may locally on $X$ lift this to a map of complexes $\alpha : \mathcal{O}_X^{\oplus t}[-n] \to E$ (details omitted). Choose a distinguished triangle $$ \mathcal{O}_X^{\oplus t}[-n] \to E \to C \to \mathcal{O}_X^{\oplus t}[-n + 1] $$ Then $C$ has vanishing cohomology in degrees $\geq n$. On the other hand, the complex $\epsilon^*C$ is $m$-pseudo-coherent, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cone-pseudo-coherent}. Hence by induction we see that $C$ is $m$-pseudo-coherent. Applying Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cone-pseudo-coherent} once more we conclude.
The easy implication follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-tor-amplitude-pullback}. For the converse, assume that $\epsilon^*E$ has tor amplitude in $[a, b]$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. As $\epsilon$ is a flat morphism of ringed sites (Lemma \ref{lemma-epsilon-flat}) we have $$ \epsilon^*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{F}) = \epsilon^*E \otimes^\mathbf{L}_{\mathcal{O}_\etale} \epsilon^*\mathcal{F} $$ Thus the (assumed) vanishing of cohomology sheaves on the right hand side implies the desired vanishing of the cohomology sheaves of $E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{F}$ via Lemma \ref{lemma-epsilon-flat}.
The easy direction in (1) follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-tor-amplitude-pullback}. Let $x \in X$ be a point and let $\overline{x}$ be a geometric point lying over $x$. Let $y = f(x)$ and denote $\overline{y}$ the geometric point of $Y$ coming from $\overline{x}$. Then $(f^{-1}\mathcal{O}_Y)_x = \mathcal{O}_{Y, y}$ (Sheaves, Lemma \ref{sheaves-lemma-stalk-pullback}) and $$ (f_{small}^{-1}\mathcal{O}_{Y_\etale})_{\overline{x}} = \mathcal{O}_{Y_\etale, \overline{y}} = \mathcal{O}_{Y, y}^{sh} $$ is the strict henselization (by \'Etale Cohomology, Lemmas \ref{etale-cohomology-lemma-stalk-pullback} and \ref{etale-cohomology-lemma-describe-etale-local-ring}). Since the stalk of $\mathcal{O}_{X_\etale}$ at $X$ is $\mathcal{O}_{X, x}^{sh}$ we obtain $$ (\epsilon^*E)_{\overline{x}} = E_x \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \mathcal{O}_{X, x}^{sh} $$ by transitivity of pullbacks. If $\epsilon^*E$ has tor amplitude in $[a, b]$ as a complex of $f_{small}^{-1}\mathcal{O}_{Y_\etale}$-modules, then $(\epsilon^*E)_{\overline{x}}$ has tor amplitude in $[a, b]$ as a complex of $\mathcal{O}_{Y, y}^{sh}$-modules (because taking stalks is a pullback and lemma cited above). By More on Flatness, Lemma \ref{flat-lemma-tor-amplitude-up-down-henselization} we find the tor amplitude of $(\epsilon^*E)_{\overline{x}}$ as a complex of $\mathcal{O}_{Y, y}$-modules is in $[a, b]$. Since $\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^{sh}$ is faithfully flat (More on Algebra, Lemma \ref{more-algebra-lemma-dumb-properties-henselization}) and since $(\epsilon^*E)_{\overline{x}} = E_x \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \mathcal{O}_{X, x}^{sh}$ we may apply More on Algebra, Lemma \ref{more-algebra-lemma-no-change-tor-amplitude} to conclude the tor amplitude of $E_x$ as a complex of $\mathcal{O}_{Y, y}$-modules is in $[a, b]$. By Cohomology, Lemma \ref{cohomology-lemma-tor-amplitude-stalk} we conclude that $E$ as an object of $D(f^{-1}\mathcal{O}_Y)$ has tor amplitude in $[a, b]$. This gives the reverse implication in (1). Part (2) follows formally from (1).
The easy implication follows from the general result contained in Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-perfect-pullback}. For the converse, we can use the equivalence of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-perfect} and the corresponding results for pseudo-coherent and complexes of finite tor dimension, namely Lemmas \ref{lemma-descend-pseudo-coherent} and \ref{lemma-descend-tor-amplitude}. Some details omitted.