hcju/stacksqa huggingface dataset:
The fact that $X/G$ is an algebraic space is immediate from Theorem \ref{theorem-final-bootstrap} and the definitions. Namely, $X/G = X/R$ where $R = G \times_B X$. The morphisms $s, t : G \times_B X \to X$ are flat and locally of finite presentation (clear for $s$ as a base change of $G \to B$ and by symmetry using the inverse it follows for $t$) and the morphism $j : G \times_B X \to X \times_B X$ is a monomorphism by Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-free-action} as the action is free. The assertions about the morphism $X \to X/G$ follow from Lemma \ref{lemma-covering-quotient}.
We may think of $X_i$ as a sheaf on $(\Sch/S_i)_{fppf}$, see Spaces, Section \ref{spaces-section-change-base-scheme}. By Sites, Section \ref{sites-section-glueing-sheaves} the descent datum $(X_i, \varphi_{ij})$ is effective in the sense that there exists a unique sheaf $X$ on $(\Sch/S)_{fppf}$ which recovers the algebraic spaces $X_i$ after restricting back to $(\Sch/S_i)_{fppf}$. Hence we see that $X_i = h_{S_i} \times X$. By Lemma \ref{lemma-locally-algebraic-space} we see that $X$ is an algebraic space, modulo verifying that $\coprod X_i$ is an algebraic space which we do at the end of the proof. By the equivalence of categories in Sites, Lemma \ref{sites-lemma-mapping-property-glue} the action maps $G_i \times_{S_i} X_i \to X_i$ glue to give a map $a : G \times_S X \to X$. Now we have to show that $a$ is an action and that $X$ is a pseudo-torsor, and fppf locally trivial (see Groupoids in Spaces, Definition \ref{spaces-groupoids-definition-principal-homogeneous-space}). These may be checked fppf locally, and hence follow from the corresponding properties of the actions $G_i \times_{S_i} X_i \to X_i$. Hence the lemma is true. \medskip\noindent We suggest the reader skip the rest of the proof, which is purely set theoretical. Pick coverings $\{S_{ij} \to S_j\}_{j \in J_i}$ of $(\Sch/S)_{fppf}$ which trivialize the $G_i$ torsors $X_i$ (possible by assumption, and Topologies, Lemma \ref{topologies-lemma-fppf-induced} part (1)). Then $\{S_{ij} \to S\}_{i \in I, j \in J_i}$ is a covering of $(\Sch/S)_{fppf}$ and hence we may assume that each $X_i$ is the trivial torsor! Of course we may also refine the covering further, hence we may assume that each $S_i$ is affine and that the index set $I$ has cardinality bounded by the cardinality of the set of points of $S$. Choose $U \in \Ob((\Sch/S)_{fppf})$ and a surjective \'etale morphism $U \to G$. Then we see that $U_i = U \times_S S_i$ comes with an \'etale surjective morphism to $X_i \cong G_i$. By Sets, Lemma \ref{sets-lemma-bound-size-fibre-product} we see $\text{size}(U_i) \leq \max\{\text{size}(U), \text{size}(S_i)\}$. By Sets, Lemma \ref{sets-lemma-bound-finite-type} we have $\text{size}(S_i) \leq \text{size}(S)$. Hence we see that $\text{size}(U_i) \leq \max\{\text{size}(U), \text{size}(S)\}$ for all $i \in I$. Together with the bound on $|I|$ we found above we conclude from Sets, Lemma \ref{sets-lemma-bound-size} that $\text{size}(\coprod U_i) \leq \max\{\text{size}(U), \text{size}(S)\}$. Hence Spaces, Lemma \ref{spaces-lemma-coproduct-algebraic-spaces} applies to show that $\coprod X_i$ is an algebraic space which is what we had to prove.
Note that properties (2) and (3) of the lemma and the corresponding properties (2) and (3) of Algebraic Spaces, Definition \ref{spaces-definition-algebraic-space} are independent of the topology. This is true because these properties involve only the notion of a fibre product of presheaves, maps of presheaves, the notion of a representable transformation of functors, and what it means for such a transformation to be surjective and \'etale. Thus all we have to prove is that an \'etale sheaf $F$ with properties (2) and (3) is also an fppf sheaf. \medskip\noindent To do this, let $R = U \times_F U$. By (2) the presheaf $R$ is representable by a scheme and by (3) the projections $R \to U$ are \'etale. Thus $j : R \to U \times_S U$ is an \'etale equivalence relation. Moreover $U \to F$ identifies $F$ as the quotient of $U$ by $R$ for the \'etale topology: (a) if $T \to F$ is a morphism, then $\{T \times_F U \to T\}$ is an \'etale covering, hence $U \to F$ is a surjection of sheaves for the \'etale topology, (b) if $a, b : T \to U$ map to the same section of $F$, then $(a, b) : T \to R$ hence $a$ and $b$ have the same image in the quotient of $U$ by $R$ for the \'etale topology. Next, let $U/R$ denote the quotient sheaf in the fppf topology which is an algebraic space by Spaces, Theorem \ref{spaces-theorem-presentation}. Thus we have morphisms (transformations of functors) $$ U \to F \to U/R. $$ By the aforementioned Spaces, Theorem \ref{spaces-theorem-presentation} the composition is representable, surjective, and \'etale. Hence for any scheme $T$ and morphism $T \to U/R$ the fibre product $V = T \times_{U/R} U$ is a scheme surjective and \'etale over $T$. In other words, $\{V \to U\}$ is an \'etale covering. This proves that $U \to U/R$ is surjective as a map of sheaves in the \'etale topology. It follows that $F \to U/R$ is surjective as a map of sheaves in the \'etale topology. On the other hand, the map $F \to U/R$ is injective (as a map of presheaves) since $R = U \times_{U/R} U$ again by Spaces, Theorem \ref{spaces-theorem-presentation}. It follows that $F \to U/R$ is an isomorphism of \'etale sheaves, see Sites, Lemma \ref{sites-lemma-mono-epi-sheaves} which concludes the proof.
Set $R = U \times_F U$. This is an algebraic space as $U \to F$ is assumed representable by algebraic spaces. The projections $s, t : R \to U$ are \'etale morphisms of algebraic spaces as $U \to F$ is assumed \'etale. The map $j = (t, s) : R \to U \times_S U$ is a monomorphism and an equivalence relation as $R = U \times_F U$. By Theorem \ref{theorem-final-bootstrap} the fppf quotient sheaf $F' = U/R$ is an algebraic space. The morphism $U \to F'$ is surjective, flat, and locally of finite presentation by Lemma \ref{lemma-covering-quotient}. The map $R \to U \times_{F'} U$ is surjective as a map of fppf sheaves by Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-quotient-pre-equivalence} and since $j$ is a monomorphism it is an isomorphism. Hence the base change of $U \to F'$ by $U \to F'$ is \'etale, and we conclude that $U \to F'$ is \'etale by Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-etale}. Thus $U \to F'$ is surjective as a map of \'etale sheaves. This means that $F'$ is equal to the quotient sheaf $U/R$ in the \'etale topology (small check omitted). Hence we obtain a canonical factorization $U \to F' \to F$ and $F' \to F$ is an injective map of sheaves. On the other hand, $U \to F$ is surjective as a map of \'etale sheaves and hence so is $F' \to F$. This means that $F' = F$ and the proof is complete.
The proof mirrors the proof of Lemma \ref{lemma-spaces-etale}. Let $R = U \times_F U$. By (2) the presheaf $R$ is an algebraic space and by (3) the projections $R \to U$ are smooth and surjective. Denote $(U, R, s, t, c)$ the groupoid associated to the equivalence relation $j : R \to U \times_S U$ (see Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-equivalence-groupoid}). By Theorem \ref{theorem-final-bootstrap} we see that $X = U/R$ (quotient in the fppf-topology) is an algebraic space. Using that the smooth topology and the \'etale topology have the same sheaves (by More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-dominates-smooth}) we see the map $U \to F$ identifies $F$ as the quotient of $U$ by $R$ for the smooth topology (details omitted). Thus we have morphisms (transformations of functors) $$ U \to F \to X. $$ By Lemma \ref{lemma-covering-quotient} we see that $U \to X$ is surjective, flat and locally of finite presentation. By Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-quotient-pre-equivalence} (and the fact that $j$ is a monomorphism) we have $R = U \times_X U$. By Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-smooth} we conclude that $U \to X$ is smooth and surjective (as the projections $R \to U$ are smooth and surjective and $\{U \to X\}$ is an fppf covering). Hence for any scheme $T$ and morphism $T \to X$ the fibre product $T \times_X U$ is an algebraic space surjective and smooth over $T$. Choose a scheme $V$ and a surjective \'etale morphism $V \to T \times_X U$. Then $\{V \to T\}$ is a smooth covering such that $V \to T \to X$ lifts to a morphism $V \to U$. This proves that $U \to X$ is surjective as a map of sheaves in the smooth topology. It follows that $F \to X$ is surjective as a map of sheaves in the smooth topology. On the other hand, the map $F \to X$ is injective (as a map of presheaves) since $R = U \times_X U$. It follows that $F \to X$ is an isomorphism of smooth ($=$ \'etale) sheaves, see Sites, Lemma \ref{sites-lemma-mono-epi-sheaves} which concludes the proof.
The proof is identical to the proof of Lemma \ref{lemma-spaces-etale-locally-representable}. Set $R = U \times_F U$. This is an algebraic space as $U \to F$ is assumed representable by algebraic spaces. The projections $s, t : R \to U$ are smooth morphisms of algebraic spaces as $U \to F$ is assumed smooth. The map $j = (t, s) : R \to U \times_S U$ is a monomorphism and an equivalence relation as $R = U \times_F U$. By Theorem \ref{theorem-final-bootstrap} the fppf quotient sheaf $F' = U/R$ is an algebraic space. The morphism $U \to F'$ is surjective, flat, and locally of finite presentation by Lemma \ref{lemma-covering-quotient}. The map $R \to U \times_{F'} U$ is surjective as a map of fppf sheaves by Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-quotient-pre-equivalence} and since $j$ is a monomorphism it is an isomorphism. Hence the base change of $U \to F'$ by $U \to F'$ is smooth, and we conclude that $U \to F'$ is smooth by Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-smooth}. Thus $U \to F'$ is surjective as a map of \'etale sheaves (as the smooth topology is equal to the \'etale topology by More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-dominates-smooth}). This means that $F'$ is equal to the quotient sheaf $U/R$ in the \'etale topology (small check omitted). Hence we obtain a canonical factorization $U \to F' \to F$ and $F' \to F$ is an injective map of sheaves. On the other hand, $U \to F$ is surjective as a map of \'etale sheaves (as the smooth topology is the same as the \'etale topology) and hence so is $F' \to F$. This means that $F' = F$ and the proof is complete.
We will prove this using the induction principle of Lemma \ref{lemma-induction-principle}. \medskip\noindent If $X$ is affine, then $\mathcal{O}_X$ is a perfect generator. This follows from Lemma \ref{lemma-derived-quasi-coherent-small-etale-site} and Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-compare-bounded}. \medskip\noindent Assume that $(U \subset X, f : V \to X)$ is an elementary distinguished square with $U$ quasi-compact such that the theorem holds for $U$ and $V$ is an affine scheme. Let $P$ be a perfect object of $D(\mathcal{O}_U)$ which is a generator for $D_\QCoh(\mathcal{O}_U)$. Using Lemma \ref{lemma-direct-summand-of-a-restriction} we may choose a perfect object $Q$ of $D(\mathcal{O}_X)$ whose restriction to $U$ is a direct sum one of whose summands is $P$. Say $V = \Spec(A)$. Let $Z \subset V$ be the reduced closed subscheme which is the inverse image of $X \setminus U$ and maps isomorphically to it (see Definition \ref{definition-elementary-distinguished-square}). This is a retrocompact closed subset of $V$. Choose $f_1, \ldots, f_r \in A$ such that $Z = V(f_1, \ldots, f_r)$. Let $K \in D(\mathcal{O}_V)$ be the perfect object corresponding to the Koszul complex on $f_1, \ldots, f_r$ over $A$. Note that since $K$ is supported on $Z$, the pushforward $K' = Rf_*K$ is a perfect object of $D(\mathcal{O}_X)$ whose restriction to $V$ is $K$ (see Lemmas \ref{lemma-pushforward-perfect} and \ref{lemma-pushforward-with-support-in-open}). We claim that $Q \oplus K'$ is a generator for $D_\QCoh(\mathcal{O}_X)$. \medskip\noindent Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$ such that there are no nontrivial maps from any shift of $Q \oplus K'$ into $E$. By Lemma \ref{lemma-pushforward-with-support-in-open} we have $K' =  f_! K$ and hence $$ \Hom_{D(\mathcal{O}_X)}(K'[n], E) = \Hom_{D(\mathcal{O}_V)}(K[n], E|_V) $$ Thus by Derived Categories of Schemes, Lemma \ref{perfect-lemma-orthogonal-koszul-complex} (using also Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}) the vanishing of these groups implies that $E|_V$ is isomorphic to $R(U \times_X V \to V)_*E|_{U \times_X V}$. This implies that $E = R(U \to X)_*E|_U$ (small detail omitted). If this is the case then $$ \Hom_{D(\mathcal{O}_X)}(Q[n], E) = \Hom_{D(\mathcal{O}_U)}(Q|_U[n], E|_U) $$ which contains $\Hom_{D(\mathcal{O}_U)}(P[n], E|_U)$ as a direct summand. Thus by our choice of $P$ the vanishing of these groups implies that $E|_U$ is zero. Whence $E$ is zero.
Let $K^\bullet$ be a K-injective complex of $\mathcal{O}$-modules which is perfect and generates $D_\QCoh(\mathcal{O}_X)$. Such a thing exists by Theorem \ref{theorem-bondal-van-den-Bergh} and the existence of K-injective resolutions. We will show the theorem holds with $$ (E, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet) $$ where $\text{Comp}^{dg}(\mathcal{O}_X)$ is the differential graded category of complexes of $\mathcal{O}$-modules. Please see Differential Graded Algebra, Section \ref{dga-section-variant-base-change}. Since $K^\bullet$ is K-injective we have \begin{equation} \label{equation-E-is-OK} H^n(E) = \Ext^n_{D(\mathcal{O}_X)}(K^\bullet, K^\bullet) \end{equation} for all $n \in \mathbf{Z}$. Only a finite number of these Exts are nonzero by Lemma \ref{lemma-ext-from-perfect-into-bounded-QCoh}. Consider the functor $$ - \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \longrightarrow D(\mathcal{O}_X) $$ of Differential Graded Algebra, Lemma \ref{dga-lemma-tensor-with-complex-derived}. Since $K^\bullet$ is perfect, it defines a compact object of $D(\mathcal{O}_X)$, see Proposition \ref{proposition-compact-is-perfect}. Combined with (\ref{equation-E-is-OK}) the functor above is fully faithful as follows from Differential Graded Algebra, Lemmas \ref{dga-lemma-fully-faithful-in-compact-case}. It has a right adjoint $$ R\Hom(K^\bullet, - ) : D(\mathcal{O}_X) \longrightarrow D(E, \text{d}) $$ by Differential Graded Algebra, Lemmas \ref{dga-lemma-tensor-with-complex-hom-adjoint} which is a left quasi-inverse functor by generalities on adjoint functors. On the other hand, it follows from Lemma \ref{lemma-tensor-with-QCoh-complex} that we obtain $$ - \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \longrightarrow D_\QCoh(\mathcal{O}_X) $$ and by our choice of $K^\bullet$ as a generator of $D_\QCoh(\mathcal{O}_X)$ the kernel of the adjoint restricted to $D_\QCoh(\mathcal{O}_X)$ is zero. A formal argument shows that we obtain the desired equivalence, see Derived Categories, Lemma \ref{derived-lemma-fully-faithful-adjoint-kernel-zero}.
Represent $E$ by a K-injective complex $\mathcal{I}^\bullet$ of $\mathcal{O}_X$-modules. Then $Rf_*(E) = f_*\mathcal{I}^\bullet$ and $Rg_*(E|_U) = g_*(\mathcal{I}^\bullet|_U)$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-restrict-K-injective-to-open}. Hence the result follows from Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules}.
The second assertion follows from the first by Modules on Sites, Lemma \ref{sites-modules-lemma-flat-pullback-exact}. To prove the first assertion we have to show that $\mathcal{O}_\etale$ is a flat $\epsilon^{-1}\mathcal{O}_X$-module. To do this it suffices to check $\mathcal{O}_{X, x} \to \mathcal{O}_{\etale, \overline{x}}$ is flat for any geometric point $\overline{x}$ of $X$, see Modules on Sites, Lemma \ref{sites-modules-lemma-check-flat-stalks}, Sites, Lemma \ref{sites-lemma-point-morphism-sites}, and \'Etale Cohomology, Remarks \ref{etale-cohomology-remarks-enough-points}. By \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-etale-local-ring} we see that $\mathcal{O}_{\etale, \overline{x}}$ is the strict henselization of $\mathcal{O}_{X, x}$. Thus $\mathcal{O}_{X, x} \to \mathcal{O}_{\etale, \overline{x}}$ is faithfully flat by More on Algebra, Lemma \ref{more-algebra-lemma-dumb-properties-henselization}. The final statement follows also: if $\epsilon^*\mathcal{F} = 0$, then $$ 0 = \epsilon^*\mathcal{F}_{\overline{x}} = \mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \mathcal{O}_\etale $$ (Modules on Sites, Lemma \ref{sites-modules-lemma-pullback-stalk}) for all geometric points $\overline{x}$. By faithful flatness of $\mathcal{O}_{X, x} \to \mathcal{O}_{\etale, \overline{x}}$ we conclude $\mathcal{F}_x = 0$ for all $x \in X$.
We will prove this by showing the functor $R\epsilon_* : D(\mathcal{O}_\etale) \to D(\mathcal{O}_X)$ induces a quasi-inverse. We will use freely that $\epsilon_*$ is given by restriction to $X_{Zar} \subset X_\etale$ and the description of $\epsilon^* = \text{id}_{small, \etale, Zar}^*$ in Descent, Lemma \ref{descent-lemma-compare-sites}. \medskip\noindent For a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the adjunction map $\mathcal{F} \to \epsilon_*\epsilon^*\mathcal{F}$ is an isomorphism by the fact that $\mathcal{F}^a$ (Descent, Definition \ref{descent-definition-structure-sheaf}) is a sheaf as proved in Descent, Lemma \ref{descent-lemma-sheaf-condition-holds}. Conversely, every quasi-coherent $\mathcal{O}_\etale$-module $\mathcal{H}$ is of the form $\epsilon^*\mathcal{F}$ for some quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$, see Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}. Then $\mathcal{F} = \epsilon_*\mathcal{H}$ by what we just said and we conclude that the adjunction map $\epsilon^*\epsilon_*\mathcal{H} \to \mathcal{H}$ is an isomorphism for all quasi-coherent $\mathcal{O}_\etale$-modules $\mathcal{H}$. \medskip\noindent Let $E$ be an object of $D_\QCoh(\mathcal{O}_\etale)$ and denote $\mathcal{H}^q = H^q(E)$ its $q$th cohomology sheaf. Let $\mathcal{B}$ be the set of affine objects of $X_\etale$. Then $H^p(U, \mathcal{H}^q) = 0$ for all $p > 0$, all $q \in \mathbf{Z}$, and all $U \in \mathcal{B}$, see Descent, Proposition \ref{descent-proposition-same-cohomology-quasi-coherent} and Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}. By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-over-U-trivial} this means that $$ H^q(U, E) = H^0(U, \mathcal{H}^q) $$ for all $U \in \mathcal{B}$. In particular, we find that this holds for affine opens $U \subset X$. It follows that the $q$th cohomology of $R\epsilon_*E$ over $U$ is the value of the sheaf $\epsilon_*\mathcal{H}^q$ over $U$. Applying sheafification we obtain $$ H^q(R\epsilon_*E) = \epsilon_*\mathcal{H}^q $$ which in particular shows that $R\epsilon_*$ induces a functor $D_\QCoh(\mathcal{O}_\etale) \to D_\QCoh(\mathcal{O}_X)$. Since $\epsilon^*$ is exact we then obtain $H^q(\epsilon^*R\epsilon_*E) = \epsilon^*\epsilon_*\mathcal{H}^q = \mathcal{H}^q$ (by discussion above). Thus the adjunction map $\epsilon^*R\epsilon_*E \to E$ is an isomorphism. \medskip\noindent Conversely, for $F \in D_\QCoh(\mathcal{O}_X)$ the adjunction map $F \to R\epsilon_*\epsilon^*F$ is an isomorphism for the same reason, i.e., because the cohomology sheaves of $R\epsilon_*\epsilon^*F$ are isomorphic to $\epsilon_*H^m(\epsilon^*F) = \epsilon_*\epsilon^*H^m(F) = H^m(F)$.
By Injectives, Lemma \ref{injectives-lemma-derived-products} the derived category $D(\mathcal{O}_X)$ has direct sums and they are computed by taking termwise direct sums of any representatives. Thus it is clear that the cohomology sheaf of a direct sum is the direct sum of the cohomology sheaves as taking direct sums is an exact functor (in any Grothendieck abelian category). The lemma follows as the direct sum of quasi-coherent sheaves is quasi-coherent, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-properties-quasi-coherent}.
Let $\mathcal{B} \subset \Ob(X_\etale)$ be the set of affine objects. Since $H^q(K_n)$ is quasi-coherent we have $H^p(U, H^q(K_n)) = 0$ for $U \in \mathcal{B}$ by the discussion in Cohomology of Spaces, Section \ref{spaces-cohomology-section-higher-direct-image} and Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}. Moreover, the maps $H^0(U, H^q(K_{n + 1})) \to H^0(U, H^q(K_n))$ are surjective for $U \in \mathcal{B}$ by similar reasoning. Part (1) follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-derived-limit-suitable-system} whose conditions we have just verified. Parts (2) and (3) follow from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-inverse-limit-is-derived-limit}.
Choose a diagram $$ \xymatrix{ U \ar[d]_a \ar[r]_h & V \ar[d]^b \\ X \ar[r]^f & Y } $$ where $U$ and $V$ are schemes, the vertical arrows are \'etale, and $a$ is surjective. Since $a^* \circ Lf^* = Lh^* \circ b^*$ the result follows from Lemma \ref{lemma-check-quasi-coherence-on-covering} and the case of schemes which is Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-pullback}.
Let $\varphi : U \to X$ be a surjective \'etale morphism from a scheme $U$. Since $\varphi^*(K \otimes_{\mathcal{O}_X}^\mathbf{L} L) = \varphi^*K \otimes_{\mathcal{O}_U}^\mathbf{L} \varphi^*L$ we see from Lemma \ref{lemma-check-quasi-coherence-on-covering} that this follows from the case of schemes which is Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-tensor-product}.
Denote $\mathcal{H}^i = H^i(E)$ the $i$th cohomology sheaf of $E$. Let $\mathcal{B}$ be the set of affine objects of $X_\etale$. Then $H^p(U, \mathcal{H}^i) = 0$ for all $p > 0$, all $i \in \mathbf{Z}$, and all $U \in \mathcal{B}$ as $U$ is an affine scheme. See discussion in Cohomology of Spaces, Section \ref{spaces-cohomology-section-higher-direct-image} and Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}. Thus the lemma follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-is-limit-dimension} with $d = 0$.
Statement (1) is Derived Categories, Lemma \ref{derived-lemma-negative-vanishing}. \medskip\noindent Proof of statement (2). Write $E_n = \tau_{\geq -n}E$. We have $E = R\lim E_n$, see Lemma \ref{lemma-nice-K-injective}. Thus $RF(E) = R\lim RF(E_n)$ in $D(\textit{Ab})$ by Injectives, Lemma \ref{injectives-lemma-RF-commutes-with-Rlim}. Thus for every $i \in \mathbf{Z}$ we have a short exact sequence $$ 0 \to R^1\lim H^{i - 1}(RF(E_n)) \to H^i(RF(E)) \to \lim H^i(RF(E_n)) \to 0 $$ see More on Algebra, Remark \ref{more-algebra-remark-compare-derived-limit}. To prove (2) we will show that the term on the left is zero and that the term on the right equals $H^i(RF(E_{-b + N - 1})$ for any $b$ with $i \geq b$. \medskip\noindent For every $n$ we have a distinguished triangle $$ H^{-n}(E)[n] \to E_n \to E_{n - 1} \to H^{-n}(E)[n + 1] $$ (Derived Categories, Remark \ref{derived-remark-truncation-distinguished-triangle}) in $D(\mathcal{O}_X)$. Since $H^{-n}(E)$ is quasi-coherent we have $$ H^i(RF(H^{-n}(E)[n])) = R^{i + n}F(H^{-n}(E)) = 0 $$ for $i + n \geq N$ and $$ H^i(RF(H^{-n}(E)[n + 1])) = R^{i + n + 1}F(H^{-n}(E)) = 0 $$ for $i + n + 1 \geq N$. We conclude that $$ H^i(RF(E_n)) \to H^i(RF(E_{n - 1})) $$ is an isomorphism for $n \geq N - i$. Thus the systems $H^i(RF(E_n))$ all satisfy the ML condition and the $R^1\lim$ term in our short exact sequence is zero (see discussion in More on Algebra, Section \ref{more-algebra-section-Rlim}). Moreover, the system $H^i(RF(E_n))$ is constant starting with $n = N - i - 1$ as desired. \medskip\noindent Proof of (3). Under the assumption on $E$ we have $\tau_{\leq a - 1}E = 0$ and we get the vanishing of $H^i(RF(E))$ for $i \leq a - 1$ from (1). Similarly, we have $\tau_{\geq b + 1}E = 0$ and hence we get the vanishing of $H^i(RF(E))$ for $i \geq b + n$ from part (2).