hcju/stacksqa huggingface dataset:
Recall that $Rf_*$ is computed on an object $K \in D_\QCoh(\mathcal{O}_X)$ by choosing a K-injective complex $\mathcal{I}^\bullet$ of $\mathcal{O}_X$-modules representing $K$ and taking $f_*\mathcal{I}^\bullet$. Thus we let $\Phi(K)$ be the complex $f_*\mathcal{I}^\bullet$ viewed as a complex of $f_*\mathcal{O}_X$-modules. Denote $g : (X, \mathcal{O}_X) \to (Y, f_*\mathcal{O}_X)$ the obvious morphism of ringed spaces. Then $g$ is a flat morphism of ringed spaces (see below for a description of the stalks) and $\Phi$ is the restriction of $Rg_*$ to $D_\QCoh(\mathcal{O}_X)$. We claim that $Lg^*$ is a quasi-inverse. First, observe that $Lg^*$ sends $D_\QCoh(f_*\mathcal{O}_X)$ into $D_\QCoh(\mathcal{O}_X)$ because $g^*$ transforms quasi-coherent modules into quasi-coherent modules (Modules, Lemma \ref{modules-lemma-pullback-quasi-coherent}). To finish the proof it suffices to show that the adjunction mappings $$ Lg^*\Phi(K) = Lg^*Rg_*K \to K \quad\text{and}\quad M \to Rg_*Lg^*M = \Phi(Lg^*M) $$ are isomorphisms for $K \in D_\QCoh(\mathcal{O}_X)$ and $M \in D_\QCoh(f_*\mathcal{O}_X)$. This is a local question, hence we may assume $Y$ and therefore $X$ are affine. \medskip\noindent Assume $Y = \Spec(B)$ and $X = \Spec(A)$. Let $\mathfrak p = x \in \Spec(A) = X$ be a point mapping to $\mathfrak q = y \in \Spec(B) = Y$. Then $(f_*\mathcal{O}_X)_y = A_\mathfrak q$ and $\mathcal{O}_{X, x} = A_\mathfrak p$ hence $g$ is flat. Hence $g^*$ is exact and $H^i(Lg^*M) = g^*H^i(M)$ for any $M$ in $D(f_*\mathcal{O}_X)$. For $K \in D_\QCoh(\mathcal{O}_X)$ we see that $$ H^i(\Phi(K)) = H^i(Rf_*K) = f_*H^i(K) $$ by the vanishing of higher direct images (Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing}) and Lemma \ref{lemma-application-nice-K-injective} (small detail omitted). Thus it suffice to show that $$ g^*g_*\mathcal{F} \to \mathcal{F} \quad\text{and}\quad \mathcal{G} \to g_*g^*\mathcal{F} $$ are isomorphisms where $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module and $\mathcal{G}$ is a quasi-coherent $f_*\mathcal{O}_X$-module. This follows from Morphisms, Lemma \ref{morphisms-lemma-affine-equivalence-modules}.
Set $U = X \setminus T$ and denote $j : U \to X$ the inclusion. By Cohomology, Lemma \ref{cohomology-lemma-triangle-sections-with-support-sheaves} there is a distinguished triangle $$ i_*R\mathcal{H}_T(E) \to E \to Rj_*(E|_U) \to i_*R\mathcal{H}_Z(E)[1] $$ in $D(\mathcal{O}_X)$. By Lemma \ref{lemma-quasi-coherence-direct-image} the complex $Rj_*(E|_U)$ has quasi-coherent cohomology sheaves (this is where we use that $U$ is retrocompact in $X$). Thus we see that (1) is true. Part (2) follows from this and the adjointness of functors in Cohomology, Lemma \ref{cohomology-lemma-complexes-with-support-on-closed}.
Set $U = X \setminus T$ and denote $j : U \to X$ the inclusion. By Cohomology, Lemma \ref{cohomology-lemma-triangle-sections-with-support-sheaves} there is a distinguished triangle $$ i_*R\mathcal{H}_T(E) \to E \to Rj_*(E|_U) \to i_*R\mathcal{H}_Z(E)[1] $$ in $D(\mathcal{O}_X)$ for any $E$ in $D(\mathcal{O}_X)$. The functor $E \mapsto Rj_*(E|_U)$ commutes with direct sums on $D_\QCoh(\mathcal{O}_X)$ by Lemma \ref{lemma-quasi-coherence-pushforward-direct-sums}. It follows that the same is true for the functor $i_* \circ R\mathcal{H}_T$ (details omitted). Since $i_* : D(i^{-1}\mathcal{O}_X) \to D_T(\mathcal{O}_X)$ is an equivalence (Cohomology, Lemma \ref{cohomology-lemma-complexes-with-support-on-closed}) we conclude.
Let $W \subset X \setminus Z$ be an open subset. Evaluating the complex of sheaves (\ref{equation-extended-alternating}) on $W$ we obtain the complex $$ \mathcal{F}(W) \to \bigoplus\nolimits_{i_0} \mathcal{F}(U_{i_0} \cap W) \to \bigoplus\nolimits_{i_0 < i_1} \mathcal{F}(U_{i_0i_1} \cap W) \to \ldots $$ In other words, we obtain the extended ordered {\v C}ech complex for the covering $W = \bigcup U_i \cap W$ and the standard ordering on $\{1, \ldots, c\}$, see Cohomology, Section \ref{cohomology-section-alternating-cech}. By Cohomology, Lemma \ref{cohomology-lemma-alternating-cech-trivial} this complex is homotopic to zero as soon as $W$ is contained in $V(f_i)$ for some $1 \leq i \leq c$. This finishes the proof.
Let us denote $\mathcal{F}^\bullet$ the complex (\ref{equation-extended-alternating}). The statement of the lemma means that the map $\mathcal{F}^\bullet \to i_*R\mathcal{H}_Z(\mathcal{F})$ of Remark \ref{remark-extended-alternating-map-to-support} is an isomorphism. Since $\mathcal{F}^\bullet$ is in $D_Z(\mathcal{O}_X)$ (see remark cited), we see that $i_*R\mathcal{H}_Z(\mathcal{F}^\bullet) = \mathcal{F}^\bullet$ by Cohomology, Lemma \ref{cohomology-lemma-complexes-with-support-on-closed}. The morphism $U_{i_0 \ldots i_p} \to X$ is affine as it is given over affine opens of $X$ by inverting the function $f_{i_0} \ldots f_{i_p}$. Thus we see that $$ \mathcal{F}_{i_0 \ldots i_p} = (U_{i_0 \ldots i_p} \to X)_*\mathcal{F}|_{U_{i_0 \ldots i_p}} = R(U_{i_0 \ldots i_p} \to X)_*\mathcal{F}|_{U_{i_0 \ldots i_p}} $$ by Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing} and the assumption that $\mathcal{F}$ is quasi-coherent. We conclude that $R\mathcal{H}_Z(\mathcal{F}_{i_0 \ldots i_p}) = 0$ by Cohomology, Lemma \ref{cohomology-lemma-sections-support-in-closed-disjoint-open}. Thus $i_*R\mathcal{H}_Z(\mathcal{F}^p) = 0$ for $p > 0$. Putting everything together we obtain $$ \mathcal{F}^\bullet = i_*R\mathcal{H}_Z(\mathcal{F}^\bullet) = i_*R\mathcal{H}_Z(\mathcal{F}) $$ as desired.
By the description of $R\mathcal{H}_Z(\mathcal{F})$ given in Lemma \ref{lemma-extended-alternating-represented} this boils down to the following algebra statement: given a ring $R$, a Koszul regular sequence $f_1, \ldots, f_c \in R$, and a flat $R$-module $M$, the extended alternating {\v C}ech complex $M \to \bigoplus\nolimits_{i_0} M_{f_{i_0}} \to \bigoplus\nolimits_{i_0 < i_1} M_{f_{i_0}f_{i_1}} \to \ldots \to M_{f_1 \ldots f_c}$ from More on Algebra, Section \ref{more-algebra-section-alternating-cech} only has cohomology in degree $c$. By More on Algebra, Lemma \ref{more-algebra-lemma-vanishing-extended-alternating-koszul} we obtain the desired vanishing for the extended alternating {\v C}ech complex of $R$. Since the complex for $M$ is obtained by tensoring this with the flat $R$-module $M$ (More on Algebra, Lemma \ref{more-algebra-lemma-extended-alternating-form-module}) we conclude.
The functor $f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)$ is exact, see Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing}. 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}. The diagram commutes by Lemma \ref{lemma-pushforward-affine-morphism}.
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 Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. Recall that $\QCoh(\mathcal{O}_X)$ is a Grothendieck abelian category (Properties, Proposition \ref{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, Lemma \ref{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.
The functor $Q_X$ is the functor $$ \mathcal{F} \mapsto \widetilde{\Gamma(X, \mathcal{F})} $$ by Schemes, Lemma \ref{schemes-lemma-compare-constructions}. This immediately implies (1) and (2). The third assertion follows from (the proof of) Lemma \ref{lemma-affine-compare-bounded}.
Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$ and let $A$ be an object of $D(\QCoh(\mathcal{O}_X))$. We have to show that the adjunction maps $$ RQ_X(i_X(A)) \to A \quad\text{and}\quad E \to i_X(RQ_X(E)) $$ are isomorphisms. Consider the hypothesis $H_n$: the adjunction maps above are isomorphisms whenever $E$ and $i_X(A)$ are supported (Definition \ref{definition-supported-on}) on a closed subset of $X$ which is contained in the union of $n$ affine opens of $X$. We will prove $H_n$ by induction on $n$. \medskip\noindent Base case: $n = 0$. In this case $E = 0$, hence the map $E \to i_X(RQ_X(E))$ is an isomorphism. Similarly $i_X(A) = 0$. Thus the cohomology sheaves of $i_X(A)$ are zero. Since the inclusion functor $\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$ is fully faithful and exact, we conclude that the cohomology objects of $A$ are zero, i.e., $A = 0$ and $RQ_X(i_X(A)) \to A$ is an isomorphism as well. \medskip\noindent Induction step. Suppose that $E$ and $i_X(A)$ are supported on a closed subset $T$ of $X$ contained in $U_1 \cup \ldots \cup U_n$ with $U_i \subset X$ affine open. Set $U = U_n$. Consider the distinguished triangles $$ A \to \Phi(A|_U) \to A' \to A[1] \quad\text{and}\quad E \to Rj_*(E|_U) \to E' \to E[1] $$ where $\Phi$ is as in the statement of the lemma. Note that $E \to Rj_*(E|_U)$ is a quasi-isomorphism over $U = U_n$. Since $i_X \circ \Phi = Rj_* \circ i_U$ by assumption and since $i_X(A)|_U = i_U(A|_U)$ we see that $i_X(A) \to i_X(\Phi(A|_U))$ is a quasi-isomorphism over $U$. Hence $i_X(A')$ and $E'$ are supported on the closed subset $T \setminus U$ of $X$ which is contained in $U_1 \cup \ldots \cup U_{n - 1}$. By induction hypothesis the statement is true for $A'$ and $E'$. By Derived Categories, Lemma \ref{derived-lemma-third-isomorphism-triangle} it suffices to prove the maps $$ RQ_X(i_X(\Phi(A|_U))) \to \Phi(A|_U) \quad\text{and}\quad Rj_*(E|_U) \to i_X(RQ_X(Rj_*E|_U)) $$ are isomorphisms. By assumption and by Lemma \ref{lemma-flat-pushforward-coherator} (the inclusion morphism $j : U \to X$ is flat, quasi-compact, and quasi-separated) we have $$ RQ_X(i_X(\Phi(A|_U))) = RQ_X(Rj_*(i_U(A|_U))) = \Phi(RQ_U(i_U(A|_U))) $$ and $$ i_X(RQ_X(Rj_*(E|_U))) = i_X(\Phi(RQ_U(E|_U))) = Rj_*(i_U(RQ_U(E|_U))) $$ Finally, the maps $$ RQ_U(i_U(A|_U)) \to A|_U \quad\text{and}\quad E|_U \to i_U(RQ_U(E|_U)) $$ are isomorphisms by Lemma \ref{lemma-affine-coherator}. The result follows.
Observe that the horizontal arrows in the diagram are equivalences of categories by Proposition \ref{proposition-quasi-compact-affine-diagonal}. Hence we can identify these categories (and similarly for other quasi-compact schemes with affine diagonal). The statement of the lemma is that the canonical map $\Phi(K) \to Rf_*(K)$ is an isomorphism for all $K$ in $D(\QCoh(\mathcal{O}_X))$. Note that if $K_1 \to K_2 \to K_3 \to K_1[1]$ is a distinguished triangle in $D(\QCoh(\mathcal{O}_X))$ and the statement is true for two-out-of-three, then it is true for the third. \medskip\noindent Let $U \subset X$ be an affine open. Since the diagonal of $X$ is affine, the inclusion morphism $j : U \to X$ is affine (Morphisms, Lemma \ref{morphisms-lemma-affine-permanence}). Similarly, the composition $g = f \circ j : U \to Y$ is affine. Let $\mathcal{I}^\bullet$ be a K-injective complex in $\QCoh(\mathcal{O}_U)$. Since $j_* : \QCoh(\mathcal{O}_U) \to \QCoh(\mathcal{O}_X)$ has an exact left adjoint $j^* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_U)$ we see that $j_*\mathcal{I}^\bullet$ is a K-injective complex in $\QCoh(\mathcal{O}_X)$, see Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}. It follows that $$ \Phi(j_*\mathcal{I}^\bullet) = f_*j_*\mathcal{I}^\bullet = g_*\mathcal{I}^\bullet $$ By Lemma \ref{lemma-affine-pushforward} we see that $j_*\mathcal{I}^\bullet$ represents $Rj_*\mathcal{I}^\bullet$ and $g_*\mathcal{I}^\bullet$ represents $Rg_*\mathcal{I}^\bullet$. On the other hand, we have $Rf_* \circ Rj_* = Rg_*$. Hence $f_*j_*\mathcal{I}^\bullet$ represents $Rf_*(j_*\mathcal{I}^\bullet)$. We conclude that the lemma is true for any complex of the form $j_*\mathcal{G}^\bullet$ with $\mathcal{G}^\bullet$ a complex of quasi-coherent modules on $U$. (Note that if $\mathcal{G}^\bullet \to \mathcal{I}^\bullet$ is a quasi-isomorphism, then $j_*\mathcal{G}^\bullet \to j_*\mathcal{I}^\bullet$ is a quasi-isomorphism as well since $j_*$ is an exact functor on quasi-coherent modules.) \medskip\noindent Let $\mathcal{F}^\bullet$ be a complex of quasi-coherent $\mathcal{O}_X$-modules. Let $T \subset X$ be a closed subset such that the support of $\mathcal{F}^p$ is contained in $T$ for all $p$. We will use induction on the minimal number $n$ of affine opens $U_1, \ldots, U_n$ such that $T \subset U_1 \cup \ldots \cup U_n$. The base case $n = 0$ is trivial. If $n \geq 1$, then set $U = U_1$ and denote $j : U \to X$ the open immersion as above. We consider the map of complexes $c : \mathcal{F}^\bullet \to j_*j^*\mathcal{F}^\bullet$. We obtain two short exact sequences of complexes: $$ 0 \to \Ker(c) \to \mathcal{F}^\bullet \to \Im(c) \to 0 $$ and $$ 0 \to \Im(c) \to j_*j^*\mathcal{F}^\bullet \to \Coker(c) \to 0 $$ The complexes $\Ker(c)$ and $\Coker(c)$ are supported on $T \setminus U \subset U_2 \cup \ldots \cup U_n$ and the result holds for them by induction. The result holds for $j_*j^*\mathcal{F}^\bullet$ by the discussion in the preceding paragraph. We conclude by looking at the distinguished triangles associated to the short exact sequences and using the initial remark of the proof.
Let $U \subset X$ be an open subset and let $s \in \mathcal{J}(U)$ be a section. Let $\mathcal{I} \subset X$ be the quasi-coherent sheaf of ideals defining the reduced induced scheme structure on $X \setminus U$ (see Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}). By Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open} the section $s$ corresponds to a map $\sigma : \mathcal{I}^n \to \mathcal{J}$ for some $n$. As $\mathcal{J}$ is an injective object of $\QCoh(\mathcal{O}_X)$ we can extend $\sigma$ to a map $\tilde s : \mathcal{O}_X \to \mathcal{J}$. Then $\tilde s$ corresponds to a global section of $\mathcal{J}$ restricting to $s$.
Since $X$ and $Y$ are Noetherian schemes the morphism is quasi-compact and quasi-separated (see Properties, Lemma \ref{properties-lemma-locally-Noetherian-quasi-separated} and Schemes, Remark \ref{schemes-remark-quasi-compact-and-quasi-separated}). Thus $f_*$ preserve quasi-coherence, see Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. Next, let $K$ be an object of $D(\QCoh(\mathcal{O}_X))$. Since $\QCoh(\mathcal{O}_X)$ is a Grothendieck abelian category (Properties, Proposition \ref{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 by Lemma \ref{lemma-acyclicity-lemma} it suffices to show that $\mathcal{I}^n$ is right $f_*$-acyclic for all $n \in \mathbf{Z}$. This is true because $f_*\mathcal{I}^n$ is flasque by Lemma \ref{lemma-injective-quasi-coherent-sheaf-Noetherian} and flasque modules are right $f_*$-acyclic by Cohomology, Lemma \ref{cohomology-lemma-flasque-acyclic-pushforward}.
Recall that the alternating {\v C}ech complex is the subcomplex of the usual {\v C}ech complex given by alternating cochains, see Cohomology, Section \ref{cohomology-section-alternating-cech}. As usual we view a $p$-cochain in $\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})$ as an alternating function $s$ on $\{1, \ldots, r\}^{p + 1}$ whose value $s_{i_0\ldots i_p}$ at $(i_0, \ldots, i_p)$ lies in $M_{f_{i_0}\ldots f_{i_p}} = \mathcal{F}(U_{i_0\ldots i_p})$. On the other hand, a $p$-cochain $t$ in $\Hom_A(I^\bullet(f_1^e, \ldots, f_r^e), M)$ is given by a map $t : \wedge^{p + 1}(A^{\oplus r}) \to M$. Write $[i] \in A^{\oplus r}$ for the $i$th basis element and write $$ [i_0, \ldots, i_p] = [i_0] \wedge \ldots \wedge [i_p] \in \wedge^{p + 1}(A^{\oplus r}) $$ Then we send $t$ as above to $s$ with $$ s_{i_0\ldots i_p} = \frac{t([i_0, \ldots, i_p])}{f_{i_0}^e\ldots f_{i_p}^e} $$ It is clear that $s$ so defined is an alternating cochain. The construction of this map is compatible with the transition maps of the system as the transition map $$ I^\bullet(f_1^e, \ldots, f_r^e) \leftarrow I^\bullet(f_1^{e + 1}, \ldots, f_r^{e + 1}), $$ of the (\ref{equation-system}) sends $[i_0, \ldots, i_p]$ to $f_{i_0}\ldots f_{i_p}[i_0, \ldots, i_p]$. It is clear from the description of the localizations $M_{f_{i_0}\ldots f_{i_p}}$ in Algebra, Lemma \ref{algebra-lemma-localization-colimit} that these maps define an isomorphism of cochain modules in degree $p$ in the limit. To finish the proof we have to show that the map is compatible with differentials. To see this recall that \begin{align*} d(s)_{i_0\ldots i_{p + 1}} & = \sum\nolimits_{j = 0}^{p + 1} (-1)^j s_{i_0\ldots \hat i_j \ldots i_p} \\ & =  \sum\nolimits_{j = 0}^{p + 1} (-1)^j \frac{t([i_0, \ldots, \hat i_j, \ldots i_{p + 1}])} {f_{i_0}^e\ldots \hat f_{i_j}^e \ldots f_{i_{p + 1}}^e} \end{align*} On the other hand, we have \begin{align*} \frac{d(t)([i_0, \ldots, i_{p + 1}])}{f_{i_0}^e\ldots f_{i_{p + 1}}^e} & = \frac{t(d[i_0, \ldots, i_{p + 1}])}{f_{i_0}^e\ldots f_{i_{p + 1}}^e} \\ & = \frac{\sum_j (-1)^j f_{i_j}^e t([i_0, \ldots, \hat i_j, \ldots i_{p + 1}])} {f_{i_0}^e \ldots f_{i_{p + 1}}^e} \end{align*} The two formulas agree by inspection.
Let $\mathcal{B}$ be the set of affine opens of $U$. Since the higher cohomology groups of a quasi-coherent module on an affine scheme are zero (Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}) this is a special case of Cohomology, Lemma \ref{cohomology-lemma-alternating-cech-complex-complex-ss}.
Since $U$ is covered by $r$ affine opens we have $H^j(U, \mathcal{F}) = 0$ for $j \geq r$ and any quasi-coherent module (Cohomology of Schemes, Lemma \ref{coherent-lemma-vanishing-nr-affines}). By Lemma \ref{lemma-application-nice-K-injective} we see that $H^0(U, E)$ is equal to $H^0(U, \tau_{\geq -r + 1}E)$. There is a spectral sequence $$ H^j(U, H^i(\tau_{\geq -r + 1}E)) \Rightarrow H^{i + j}(U, \tau_{\geq -N}E) $$ see Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. Hence $H^0(U, E) = 0$ by our assumed vanishing of cohomology sheaves of $E$. We conclude that $s|_U = 0$. Think of $s$ as a morphism $\mathcal{O}_X \to E$ in $D(\mathcal{O}_X)$. By Proposition \ref{proposition-represent-cohomology-class-on-open} the composition $I_e \to \mathcal{O}_X \to E$ is zero for some $e$. By the distinguished triangle $I_e \to \mathcal{O}_X \to K_e \to I_e[1]$ we obtain a morphism $K_e \to E$ such that $s$ is the composition $\mathcal{O}_X \to K_e \to E$.
It is immediate from the definitions that if $M^\bullet$ is $m$-pseudo-coherent, so is $E$. To prove the converse, assume $E$ is $m$-pseudo-coherent. As $X = \Spec(A)$ is quasi-compact with a basis for the topology given by standard opens, we can find a standard open covering $X = D(f_1) \cup \ldots \cup D(f_n)$ and strictly perfect complexes $\mathcal{E}_i^\bullet$ on $D(f_i)$ and maps $\alpha_i : \mathcal{E}_i^\bullet \to E|_{U_i}$ inducing isomorphisms on $H^j$ for $j > m$ and surjections on $H^m$. By Cohomology, Lemma \ref{cohomology-lemma-local-actual} after refining the open covering we may assume $\alpha_i$ is given by a map of complexes $\mathcal{E}_i^\bullet \to \widetilde{M^\bullet}|_{U_i}$ for each $i$. By Modules, Lemma \ref{modules-lemma-direct-summand-of-locally-free-is-locally-free} the terms $\mathcal{E}_i^n$ are finite locally free modules. Hence after refining the open covering we may assume each $\mathcal{E}_i^n$ is a finite free $\mathcal{O}_{U_i}$-module. From the definition it follows that $M^\bullet_{f_i}$ is an $m$-pseudo-coherent complex of $A_{f_i}$-modules. We conclude by applying More on Algebra, Lemma \ref{more-algebra-lemma-glue-pseudo-coherent}. \medskip\noindent The case ``pseudo-coherent'' follows from the fact that $E$ is pseudo-coherent if and only if $E$ is $m$-pseudo-coherent for all $m$ (by definition) and the same is true for $M^\bullet$ by More on Algebra, Lemma \ref{more-algebra-lemma-pseudo-coherent}.
As $X$ is quasi-compact we see that in both (1) and (2) the object $E$ is bounded above. Thus the question is local on $X$ and we may assume $X$ is affine. Say $X = \Spec(A)$ for some Noetherian ring $A$. In this case $E$ corresponds to a complex of $A$-modules $M^\bullet$ by Lemma \ref{lemma-affine-compare-bounded}. By Lemma \ref{lemma-pseudo-coherent-affine} we see that $E$ is $m$-pseudo-coherent if and only if $M^\bullet$ is $m$-pseudo-coherent. On the other hand, $H^i(E)$ is coherent if and only if $H^i(M^\bullet)$ is a finite $A$-module (Properties, Lemma \ref{properties-lemma-finite-type-module}). Thus the result follows from More on Algebra, Lemma \ref{more-algebra-lemma-Noetherian-pseudo-coherent}.
Part (2) follows trivially from part (1). In the proof of (1) we will use the equivalence $D(A) = D_\QCoh(X)$ of Lemma \ref{lemma-affine-compare-bounded} without further mention. Assume $M^\bullet$ has tor amplitude in $[a, b]$. Then $K^\bullet$ is isomorphic in $D(A)$ to a complex $K^\bullet$ of flat $A$-modules with $K^i = 0$ for $i \not \in [a, b]$, see More on Algebra, Lemma \ref{more-algebra-lemma-tor-amplitude}. Then $E$ is isomorphic to $\widetilde{K^\bullet}$. Since each $\widetilde{K^i}$ is a flat $\mathcal{O}_X$-module, we see that $E$ has tor amplitude in $[a, b]$ by Cohomology, Lemma \ref{cohomology-lemma-tor-amplitude}. \medskip\noindent Assume that $E$ has tor amplitude in $[a, b]$. Then $E$ is bounded whence $M^\bullet$ is in $K^-(A)$. Thus we may replace $M^\bullet$ by a bounded above complex of $A$-modules. We may even choose a projective resolution and assume that $M^\bullet$ is a bounded above complex of free $A$-modules. Then for any $A$-module $N$ we have $$ E \otimes_{\mathcal{O}_X}^\mathbf{L} \widetilde{N} \cong \widetilde{M^\bullet} \otimes_{\mathcal{O}_X}^\mathbf{L} \widetilde{N} \cong \widetilde{M^\bullet \otimes_A N} $$ in $D(\mathcal{O}_X)$. Thus the vanishing of cohomology sheaves of the left hand side implies $M^\bullet$ has tor amplitude in $[a, b]$.