hcju/stacksqa huggingface dataset:
Observe that $h$ is representable by algebraic spaces by Morphisms of Stacks, Lemmas \ref{stacks-morphisms-lemma-aut-iso-unramified} and \ref{stacks-morphisms-lemma-stabilizer-preserving}. Choose an affine scheme $U$ and a surjective, flat, finite, and finitely presented morphism $U \to \mathcal{X}$. Then $U' = \mathcal{X}' \times_\mathcal{X} U$ is an algebraic space with a finite (in particular affine) morphism $U' \to \mathcal{X}'$. By Lemma \ref{lemma-affine-over-well-nigh-affine} we conclude that $U'$ is affine. Setting $R = U \times_\mathcal{X} U$ and $R' = U' \times_{\mathcal{X}'} U'$ we obtain groupoids $(U, R, s, t, c)$ and $(U', R', s', t', c')$ such that $\mathcal{X} = [U/R]$ and $\mathcal{X}' = [U'/R']$, see proof of Lemma \ref{lemma-well-nigh-affine}. we see that the diagrams $$ \xymatrix{ R' \ar[d]_{s'} \ar[r]_f & R \ar[d]^s \\ U' \ar[r]^f & U } \quad \quad \xymatrix{ R' \ar[d]_{t'} \ar[r]_f & R \ar[d]^t \\ U' \ar[r]^f & U } \quad \quad \xymatrix{ G' \ar[d] \ar[r]_f & G \ar[d] \\ U' \ar[r]^f & U } $$ are cartesian where $G$ and $G'$ are the stabilizer group schemes. This follows for the first two by transitivity of fibre products and for the last one this follows because it is the pullback of the isomorphism $\mathcal{I}_{\mathcal{X}'} \to \mathcal{X}' \times_\mathcal{X} \mathcal{I}_\mathcal{X}$ (by the already used Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-aut-iso-unramified}). Recall that $M$, resp.\ $M'$ was constructed in Lemma \ref{lemma-well-nigh-affine-moduli-space} as the spectrum of the ring of $R$-invariant functions on $U$, resp.\ the ring of $R'$-invariant functions on $U'$. Thus $M' \to M$ is \'etale and $U' = M' \times_M U$ by Groupoids, Lemma \ref{groupoids-lemma-etale}. It follows that $R' = M' \times_M U$, in other words the groupoid $(U', R', s', t', c')$ is the base change of $(U, R, s, t, c)$ by $M' \to M$. This implies that the diagram in the lemma is cartesian by Quotients of Groupoids, Lemma \ref{groupoids-quotients-lemma-base-change-quotient-stack}.
For any $x \in |\mathcal{X}|$ we can choose $g : \mathcal{U} \to \mathcal{X}$, $\mathcal{U} = [U/R]$, and $u$ as in Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-etale-local-quasi-DM-at-x}. Then by Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-stabilizer-preserving-unramified} we see that there exists an open substack $\mathcal{U}' \subset \mathcal{U}$ containing $u$ such that $\mathcal{I}_{\mathcal{U}'} \to \mathcal{U}' \times_\mathcal{X} \mathcal{I}_\mathcal{X}$ is an isomorphism. Let $U' \subset U$ be the $R$-invariant open corresponding to the open substack $\mathcal{U}'$. Let $u' \in U'$ be a point of $U'$ mapping to $u$. Observe that $t(s^{-1}(\{u'\}))$ is finite as $s : R \to U$ is finite. By Properties, Lemma \ref{properties-lemma-ample-finite-set-in-affine} and Groupoids, Lemma \ref{groupoids-lemma-find-invariant-affine} we can find an $R$-invariant affine open $U'' \subset U'$ containing $u'$. Let $R''$ be the restriction of $R$ to $U''$. Then $\mathcal{U}'' = [U''/R'']$ is an open substack of $\mathcal{U}'$ containing $u$, is well-nigh affine, $\mathcal{I}_{\mathcal{U}''} \to \mathcal{U}'' \times_\mathcal{X} \mathcal{I}_\mathcal{X}$ is an isomorphism, and $\mathcal{U}'' \to \mathcal{X}$ and is representable by algebraic spaces and \'etale. Finally, $\mathcal{U}'' \to \mathcal{X}$ is separated as $\mathcal{U}''$ is separated (Lemma \ref{lemma-well-nigh-affine}) the diagonal of $\mathcal{X}$ is separated (Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-diagonal-diagonal}) and separatedness follows from Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-compose-after-separated}. Since the point $x \in |\mathcal{X}|$ is arbitrary the proof is complete.
By Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-aut-iso-unramified} we see that $\mathcal{I}_{\mathcal{X}'} \to \mathcal{X}' \times_\mathcal{X} \mathcal{I}_\mathcal{X}$ is an isomorphism. Hence $\mathcal{I}_{\mathcal{X}'} \to \mathcal{X}'$ is finite as a base change of $\mathcal{I}_\mathcal{X} \to \mathcal{X}$. Let $f' : \mathcal{X}' \to M'$ and $f : \mathcal{X} \to M$ be as in Theorem \ref{theorem-keel-mori}. We obtain a commutative diagram as in the lemma because $f'$ is categorical moduli space. Choose $I$ and $g'_i : \mathcal{X}'_i \to \mathcal{X}'$ as in Lemma \ref{lemma-etale-local-finite-inertia}. Observe that $g_i = h \circ g'_i$ is \'etale, separated, and induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark \ref{stacks-morphisms-remark-identify-automorphism-groups}). Let $f'_i : \mathcal{X}'_i \to M'_i$ be as in Lemma \ref{lemma-well-nigh-affine-moduli-space}. In the proof of Theorem \ref{theorem-keel-mori} we have seen that the diagrams $$ \xymatrix{ \mathcal{X}'_i \ar[d]_{f'_i} \ar[r]_{g'_i} & \mathcal{X}' \ar[d]^{f'} \\ M'_i \ar[r] & M' } \quad\text{and}\quad \xymatrix{ \mathcal{X}'_i \ar[d]_{f'_i} \ar[r]_{g_i} & \mathcal{X} \ar[d]^f \\ M'_i \ar[r] & M } $$ are cartesian and that $M'_i \to M'$ and $M'_i \to M$ are \'etale (this also follows directly from the properties of the morphisms $g'_i, g_i, f', f'_i, f$ listed sofar by arguing in exactly the same way). This first implies that $M' \to M$ is \'etale and second that the diagram in the lemma is cartesian. We still need to show that $M' \to M$ is separated. To do this we contemplate the diagram $$ \xymatrix{ \mathcal{X}' \ar[r] \ar[d] & \mathcal{X}' \times_\mathcal{X} \mathcal{X}' \ar[d] \\ M' \ar[r] & M' \times_M M' } $$ The top horizontal arrow is universally closed as $\mathcal{X}' \to \mathcal{X}$ is separated. The vertical arrows are as in Theorem \ref{theorem-keel-mori} (as flat base changes of $\mathcal{X} \to M$) hence universal homeomorphisms. Thus the lower horizontal arrow is universally closed. This (combined with it being an \'etale monomorphism of algebraic spaces) proves it is a closed immersion as desired.
Since $f$ is a uniform categorical moduli space we obtain the morphism $M \to Y$. It suffices to check that $M \to Y$ is locally of finite type \'etale locally on $M$ and $Y$. Since $f$ is a uniform categorical moduli space, we may first replace $Y$ by an affine scheme \'etale over $Y$. Next, we may choose $I$ and $g_i : \mathcal{X}_i \to \mathcal{X}$ as in Lemma \ref{lemma-etale-local-finite-inertia}. Then by Lemma \ref{lemma-etale-separated-over-keel-mori} we reduce to the case $\mathcal{X} = \mathcal{X}_i$. In other words, we may assume $\mathcal{X}$ is well-nigh affine. In this case we have $Y = \Spec(D)$, we have $\mathcal{X} = [U/R]$ with $U = \Spec(A)$ and $M = \Spec(C)$ where $C \subset A$ is the set of $R$-invariant functions on $U$. See Lemmas \ref{lemma-well-nigh-affine} and \ref{lemma-well-nigh-affine-moduli-space}. Then $A_0$ is Noetherian and $A_0 \to A$ is of finite type. Moreover $A$ is integral over $C$ by Groupoids, Lemma \ref{groupoids-lemma-integral-over-invariants}, hence finite over $C$ (being of finite type over $A_0$). Thus we may finally apply Algebra, Lemma \ref{algebra-lemma-Artin-Tate} to conclude.
To prove this consider the following diagram $$ \xymatrix{ \mathcal{X} \ar[d]_f \ar[r]_{\Delta_\mathcal{X}} & \mathcal{X} \times \mathcal{X} \ar[d]^{f \times f} \\ M \ar[r]^{\Delta_M} & M \times M } $$ Since $f$ is a universal homeomorphism, we see that $f \times f$ is a universal homeomorphism. \medskip\noindent If $\mathcal{X}$ is separated, then $\Delta_\mathcal{X}$ is proper, hence $\Delta_\mathcal{X}$ is universally closed, hence $\Delta_M$ is universally closed, hence $M$ is separated by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-separated-diagonal-proper}. \medskip\noindent If $\mathcal{X}$ is quasi-separated, then $\Delta_\mathcal{X}$ is quasi-compact, hence $\Delta_M$ is quasi-compact, hence $M$ is quasi-separated.
By Lemma \ref{lemma-keel-mori-finite-type} we see that $M \to Y$ is locally of finite type. By Lemma \ref{lemma-keel-mori-diagonal} we see that $M \to Y$ is separated. Of course $M \to Y$ is quasi-compact and universally closed as these are topological properties and $\mathcal{X} \to Y$ has these properties and $\mathcal{X} \to M$ is a universal homeomorphism.
Consider the category $p : \mathcal{C} \to W_{spaces, \etale}$ introduced in Remark \ref{remark-gerbe-of-lifts}. The proposition states that there exists an object of $\mathcal{C}$ lying over $W$. Namely, if we have such an object $(W, W', a, i, y', \alpha)$ then $W = \mathcal{X} \times_{\mathcal{X}'} W'$. Hence $W \to W'$ is a thickening of algebraic spaces so $W'$ is affine by More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-thickening-scheme} and More on Morphisms, Lemma \ref{more-morphisms-lemma-thickening-affine-scheme}. \medskip\noindent Lemma \ref{lemma-gerbe-of-lifts} tells us $\mathcal{C}$ is a gerbe over $W_{spaces, \etale}$. This means we can \'etale locally find a solution and these local solutions are \'etale locally isomorphic; this part does not require the assumption that the thickening is first order. By Lemma \ref{lemma-gerbe-of-lifts-first-order} the automorphism sheaves of objects of our gerbe are abelian and fit together to form a quasi-coherent module $\mathcal{G}$ on $W_{spaces, \etale}$. We will verify conditions (1) and (2) of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-existence} to conclude the existence of an object of $\mathcal{C}$ lying over $W$. Condition (1) is true: the \'etale coverings $\{W_i \to W\}$ with each $W_i$ affine are cofinal in the collection of all coverings. For such a covering $W_i$ and $W_i \times_W W_j$ are affine and $H^1(W_i, \mathcal{G})$ and $H^1(W_i \times_W W_j, \mathcal{G})$ are zero: the cohomology of a quasi-coherent module over an affine algebraic space is zero for example by Cohomology of Spaces, Proposition \ref{spaces-cohomology-proposition-vanishing}. Finally, condition (2) is that $H^2(W, \mathcal{G}) = 0$ for our quasi-coherent sheaf $\mathcal{G}$ which again follows from Cohomology of Spaces, Proposition \ref{spaces-cohomology-proposition-vanishing}. This finishes the proof.
We will prove this using the induction principle of Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}. \medskip\noindent If $X$ is affine, then $\mathcal{O}_X$ is a perfect generator. This follows from Lemma \ref{lemma-affine-compare-bounded}. \medskip\noindent Assume that $X = U \cup V$ is an open covering with $U$ quasi-compact such that the theorem holds for $U$ and $V$ is an affine open. 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 $Z = X \setminus U$. This is a closed subset of $V$ with $V \setminus Z$ quasi-compact. 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 $Z \subset V$ closed, the pushforward $K' = R(V \to X)_*K$ is a perfect object of $D(\mathcal{O}_X)$ whose restriction to $V$ is $K$ (see Cohomology, Lemma \ref{cohomology-lemma-pushforward-perfect}). 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 Cohomology, Lemma \ref{cohomology-lemma-pushforward-restriction} we have $K' =  R(V \to X)_! K$ and hence $$ \Hom_{D(\mathcal{O}_X)}(K'[n], E) = \Hom_{D(\mathcal{O}_V)}(K[n], E|_V) $$ Thus by Lemma \ref{lemma-orthogonal-koszul-complex} the vanishing of these groups implies that $E|_V$ is isomorphic to $R(U \cap V \to V)_*E|_{U \cap V}$. 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.
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 Schemes, Section \ref{schemes-section-quasi-coherent}.
Let $\mathcal{B}$ be the set of affine opens of $X$. Since $H^q(K_n)$ is quasi-coherent we have $H^p(U, H^q(K_n)) = 0$ for $U \in \mathcal{B}$ by 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 Schemes, Lemma \ref{schemes-lemma-equivalence-quasi-coherent}. Part (1) follows from Cohomology, Lemma \ref{cohomology-lemma-derived-limit-suitable-system} whose conditions we have just verified. Parts (2) and (3) follow from Cohomology, Lemma \ref{cohomology-lemma-inverse-limit-is-derived-limit}.
Denote $\mathcal{H}^i = H^i(E)$ the $i$th cohomology sheaf of $E$. Let $\mathcal{B}$ be the set of affine open subsets of $X$. Then $H^p(U, \mathcal{H}^i) = 0$ for all $p > 0$, all $i \in \mathbf{Z}$, and all $U \in \mathcal{B}$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}. Thus the lemma follows from Cohomology, Lemma \ref{cohomology-lemma-is-limit-dimension}.
The functor $R\Gamma(X, -)$ gives a functor $D(\mathcal{O}_X) \to D(A)$ and hence by restriction a functor \begin{equation} \label{equation-back} R\Gamma(X, -) : D_\QCoh(\mathcal{O}_X) \longrightarrow D(A). \end{equation} We will show this functor is quasi-inverse to (\ref{equation-compare}) via the equivalence between quasi-coherent modules on $X$ and the category of $A$-modules. \medskip\noindent Elucidation. Denote $(Y, \mathcal{O}_Y)$ the one point space with sheaf of rings given by $A$. Denote $\pi : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ the obvious morphism of ringed spaces. Then $R\Gamma(X, -)$ can be identified with $R\pi_*$ and the functor (\ref{equation-compare}) via the equivalence $\textit{Mod}(\mathcal{O}_Y) = \text{Mod}_A = \QCoh(\mathcal{O}_X)$ can be identified with $L\pi^* = \pi^* = \widetilde{\ }$ (see Modules, Lemma \ref{modules-lemma-construct-quasi-coherent-sheaves} and Schemes, Lemmas \ref{schemes-lemma-compare-constructions} and \ref{schemes-lemma-equivalence-quasi-coherent}). Thus the functors $$ \xymatrix{ D(A) \ar@<1ex>[r] & D(\mathcal{O}_X) \ar@<1ex>[l] } $$ are adjoint (by Cohomology, Lemma \ref{cohomology-lemma-adjoint}). In particular we obtain canonical adjunction mappings $$ a : \widetilde{R\Gamma(X, E)} \longrightarrow E $$ for $E$ in $D(\mathcal{O}_X)$ and $$ b : M^\bullet \longrightarrow R\Gamma(X, \widetilde{M^\bullet}) $$ for $M^\bullet$ a complex of $A$-modules. \medskip\noindent Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. We may apply Lemma \ref{lemma-application-nice-K-injective} to the functor $F(-) = \Gamma(X, -)$ with $N = 1$ by Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}. Hence $$ H^0(R\Gamma(X, E)) = H^0(R\Gamma(X, \tau_{\geq 0}E)) = \Gamma(X, H^0(E)) $$ (the last equality by definition of the canonical truncation). Using this we will show that the adjunction mappings $a$ and $b$ induce isomorphisms $H^0(a)$ and $H^0(b)$. Thus $a$ and $b$ are quasi-isomorphisms (as the statement is invariant under shifts) and the lemma is proved. \medskip\noindent In both cases we use that $\widetilde{\ }$ is an exact functor (Schemes, Lemma \ref{schemes-lemma-spec-sheaves}). Namely, this implies that $$ H^0\left(\widetilde{R\Gamma(X, E)}\right) = \widetilde{H^0(R\Gamma(X, E))} = \widetilde{\Gamma(X, H^0(E))} $$ which is equal to $H^0(E)$ because $H^0(E)$ is quasi-coherent. Thus $H^0(a)$ is an isomorphism. For the other direction we have $$ H^0(R\Gamma(X, \widetilde{M^\bullet})) = \Gamma(X, H^0(\widetilde{M^\bullet})) = \Gamma(X, \widetilde{H^0(M^\bullet)}) = H^0(M^\bullet) $$ which proves that $H^0(b)$ is an isomorphism.
By More on Algebra, Lemma \ref{more-algebra-lemma-base-change-K-flat} we see that $K^\bullet \otimes_A A_\mathfrak p$ is a K-flat complex of $A_\mathfrak p$-modules for every $\mathfrak p \in \Spec(A)$. Hence we conclude from Cohomology, Lemma \ref{cohomology-lemma-check-K-flat-stalks} (and Schemes, Lemma \ref{schemes-lemma-spec-sheaves}) that $\widetilde{K^\bullet}$ is K-flat.
We first prove the diagram $$ \xymatrix{ D(B) \ar[r] & D(\mathcal{O}_Y) \\ D(A) \ar[r] \ar[u]^{- \otimes_A^\mathbf{L} B} & D(\mathcal{O}_X) \ar[u]_{Lf^*} } $$ commutes. This is clear from Lemma \ref{lemma-affine-K-flat} and the constructions of the functors in question. To see (1) let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. To see that $Lf^*E$ has quasi-coherent cohomology sheaves we may work locally on $X$. Note that $Lf^*$ is compatible with restricting to open subschemes. Hence we can assume that $f$ is a morphism of affine schemes as in (2). Then we can apply Lemma \ref{lemma-affine-compare-bounded} to see that $E$ comes from a complex of $A$-modules. By the commutativity of the first diagram of the proof the same holds for $Lf^*E$ and we conclude (1) is true.
The equality of (2) follows immediately from Lemma \ref{lemma-affine-K-flat} and the construction of the derived tensor product. To see (1) let $K, L$ be objects of $D_\QCoh(\mathcal{O}_X)$. To check that $K \otimes^\mathbf{L} L$ is in $D_\QCoh(\mathcal{O}_X)$ we may work locally on $X$, hence we may assume $X = \Spec(A)$ is affine. By Lemma \ref{lemma-affine-compare-bounded} we may represent $K$ and $L$ by complexes of $A$-modules. Then part (2) implies the result.
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. The question is local on $S$, hence we may assume $S$ is quasi-compact. Pick $N = N(X, S, f)$ as in Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images}. Thus $R^pf_*\mathcal{F} = 0$ for all quasi-coherent $\mathcal{O}_X$-modules $\mathcal{F}$ and all $p \geq N$ and the same remains 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 $U \subset S$ be affine open. By Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images-application} and our choice of $N$ we have $H^p(f^{-1}(U), \mathcal{F}) = 0$ for $p \geq N$ and any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$. Hence we may apply Lemma \ref{lemma-application-nice-K-injective} to the functor $\Gamma(f^{-1}(U), -)$ to see that $$ R\Gamma(U, Rf_*E) = R\Gamma(f^{-1}(U), E) $$ has vanishing cohomology in degrees $\geq N$. Since this holds for all $U \subset S$ affine open 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.
There is always a canonical map $f_*\mathcal{F}^\bullet \to Rf_*\mathcal{F}^\bullet$. Our task is to show that this is an isomorphism on cohomology sheaves. As the statement is invariant under shifts it suffices to show that $H^0(f_*(\mathcal{F}^\bullet)) \to R^0f_*\mathcal{F}^\bullet$ is an isomorphism. The statement is local on $S$ hence we may assume $S$ affine. By Lemma \ref{lemma-quasi-coherence-direct-image} we have $R^0f_*\mathcal{F}^\bullet = R^0f_*\tau_{\geq -n}\mathcal{F}^\bullet$ for all sufficiently large $n$. Thus we may assume $\mathcal{F}^\bullet$ bounded below. As each $\mathcal{F}^n$ is right $f_*$-acyclic by assumption we see that $f_*\mathcal{F}^\bullet \to Rf_*\mathcal{F}^\bullet$ is a quasi-isomorphism by Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}).
By the construction of the spectral sequence via Cohomology, Lemma \ref{cohomology-lemma-spectral-sequence-filtered-object} using the filtration given by $\tau_{\leq -p}K$, we see that suffices to show that given $n \in \mathbf{Z}$ we have $$ H^n(X, \tau_{\leq -p}K) = 0 \text{ for } p \gg 0 $$ and $$ H^n(X, K) = H^n(X, \tau_{\leq -p}K) \text{ for } p \ll 0 $$ The first follows from Lemma \ref{lemma-application-nice-K-injective} applied with $F = \Gamma(X, -)$ and the bound in Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images}. The second holds whenever $-p \leq n$ for any ringed space $(X, \mathcal{O}_X)$ and any $K \in D(\mathcal{O}_X)$.
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 Schemes, Lemma \ref{coherent-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 local on $S$. Hence we may assume $S$ and therefore $X$ is affine. In this case the statement is clear from the description of the derived categories $D_\QCoh(\mathcal{O}_X)$ and $D_\QCoh(\mathcal{O}_S)$ given in Lemma \ref{lemma-affine-compare-bounded}. Some details omitted.
Since $f$ is affine the map $f_*\mathcal{O}_X \to Rf_*\mathcal{O}_X$ is an isomorphism (Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing}). 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 $S$. Hence we may assume $S$ and therefore $X$ is affine. In this case the statement is clear from the description of the derived categories $D_\QCoh(\mathcal{O}_X)$ and $D_\QCoh(\mathcal{O}_S)$ and the functor $Lf^*$ given in Lemmas \ref{lemma-affine-compare-bounded} and \ref{lemma-quasi-coherence-pullback}. Some details omitted.