hcju/stacksqa huggingface dataset:
Let $x \in X$ be a point with image $y = f(x)$ in $Y$. Choose a separable algebraic closure $\kappa^{sep}$ of $\kappa(x)$. Let $\mathcal{O}_{X, x}^{sh}$ be the strict henselization corresponding to $\kappa^{sep}$ and $\mathcal{O}_{Y, y}^{sh}$ the strict henselization relative to the separable algebraic closure of $\kappa(y)$ in $\kappa^{sep}$. Consider the commutative diagram $$ \xymatrix{ \mathcal{O}_{X, x} \ar[r] & \mathcal{O}_{X, x}^{sh} \\ \mathcal{O}_{Y, y} \ar[u] \ar[r] & \mathcal{O}_{Y, y}^{sh} \ar[u] } $$ local homomorphisms of local rings, see Algebra, Lemma \ref{algebra-lemma-strictly-henselian-functorial}. Since the strict henselization is a filtered colimit of \'etale ring maps, More on Algebra, Lemma \ref{more-algebra-lemma-when-weakly-etale} shows the horizontal maps are weakly \'etale. Moreover, the horizontal maps are faithfully flat by More on Algebra, Lemma \ref{more-algebra-lemma-dumb-properties-henselization}. \medskip\noindent Assume $f$ weakly \'etale. By Lemma \ref{lemma-check-weakly-etale-stalks} the left vertical arrow is weakly \'etale. By More on Algebra, Lemmas \ref{more-algebra-lemma-composition-weakly-etale} and \ref{more-algebra-lemma-weakly-etale-permanence} the right vertical arrow is weakly \'etale. By More on Algebra, Theorem \ref{more-algebra-theorem-olivier} we conclude the right vertical map is an isomorphism. \medskip\noindent Assume $\mathcal{O}_{Y, y}^{sh} \to \mathcal{O}_{X, x}^{sh}$ is an isomorphism. Then $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}^{sh}$ is weakly \'etale. Since $\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^{sh}$ is faithfully flat we conclude that $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is weakly \'etale by More on Algebra, Lemma \ref{more-algebra-lemma-go-down}. Thus (2) implies (1) by Lemma \ref{lemma-check-weakly-etale-stalks}.
We will use Morphisms, Lemmas \ref{morphisms-lemma-base-change-flat} and \ref{morphisms-lemma-composition-flat} without further mention. Write $X \to Y$ as the composition $X \to X \times_S Y \to Y$. The second morphism is flat as the base change of the flat morphism $X \to S$. The first is the base change of the flat morphism $Y \to Y \times_S Y$ by the morphism $X \times_S Y \to Y \times_S Y$, hence flat. Thus $X \to Y$ is flat. The morphism $X \times_Y X \to X \times_S X$ is an immersion. Thus Lemma \ref{lemma-key} implies, that since $X$ is flat over $X \times_S X$ it follows that $X$ is flat over $X \times_Y X$.
Since $f$ is a universal homeomorphism, the diagonal $\Delta : X \to X \times_Y X$ is a surjective closed immersion by Morphisms, Lemmas \ref{morphisms-lemma-homeomorphism-affine} and \ref{morphisms-lemma-universally-injective}. Since $\Delta$ is also flat, we see that $\Delta$ must be an isomorphism by Morphisms, Lemma \ref{morphisms-lemma-characterize-flat-closed-immersions}. In other words, $f$ is a monomorphism (Schemes, Lemma \ref{schemes-lemma-monomorphism}). Since $f$ is a universal homeomorphism it is certainly quasi-compact. Hence by Descent, Lemma \ref{descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism} we find that $f$ is an isomorphism.
The morphism $F_{U/X}$ is a universal homeomorphism by Varieties, Lemma \ref{varieties-lemma-relative-frobenius}. The morphism $F_{U/X}$ is weakly \'etale as a morphism between schemes weakly \'etale over $X$ by Lemma \ref{lemma-weakly-etale-permanence}. Hence $F_{U/X}$ is an isomorphism by Lemma \ref{lemma-weakly-etale-universal-homeomorphism}.
Choose a finite affine open covering $S = \bigcup \Spec(A_i)$. Then $K$ is equal to the fraction field of $A_i$ for all $i$. Let $X_i = X \times_S \Spec(A_i)$. Choose $L_i/K$ as in Theorem \ref{theorem-normalized-base-change-with-reduced-fibre} for the morphism $X_i \to \Spec(A_i)$. Let $B_i \subset L_i$ be the integral closure of $A_i$ and let $Y_i$ be the normalized base change of $X$ to $B_i$. Let $L/K$ be a finite extension dominating each $L_i$. Let $T_i \subset T$ be the inverse image of $\Spec(A_i)$. For each $i$ we get a commutative diagram $$ \xymatrix{ g^{-1}(T_i) \ar[r] \ar[d] & Y_i \ar[r] \ar[d] & X \times_S \Spec(A_i) \ar[d] \\ T_i \ar[r] & \Spec(B_i) \ar[r] & \Spec(A_i) } $$ and in fact the left hand square is a normalized base change as discussed at the beginning of the section. In the proof of Theorem \ref{theorem-normalized-base-change-with-reduced-fibre} we have seen that the smooth locus of $Y \to T$ contains the inverse image in $g^{-1}(T_i)$ of the set of points where $Y_i$ is smooth over $B_i$. This proves the lemma.
The equivalence (1) $\Leftrightarrow$ (2) follows from the definitions and Morphisms, Lemma \ref{morphisms-lemma-characterize-quasi-affine}. For (2) $\Rightarrow$ (4), let $U$ and $V$ be as in (4). By Schemes, Lemma \ref{schemes-lemma-quasi-compact-permanence}, the induced morphism $U \to V$ is quasi-compact. Thus, for every affine open $V' \subset V$, the fiber product $V' \times_V U$ is quasi-compact, so, by (2), the induced map $V' \times_V U \to V'$ is quasi-affine. Thus, $U \to V$ is also quasi-affine by Morphisms, Lemma \ref{morphisms-lemma-characterize-quasi-affine}. This argument also gives (3) $\Rightarrow$ (4): indeed, keeping the same notation, those affine opens $V' \subset V$ that lie in one of the $V_j$ cover $V$, so one needs to argue that the quasi-compact map $V' \times_V U \to V'$ is quasi-affine. However, by (3), the composition $V' \times_V U \to V' \to V_j$ is quasi-affine and, by Schemes, Lemma \ref{schemes-lemma-compose-after-separated}, the map $V' \to V_j$ is quasi-separated. Thus, $V' \times_V U \to V'$ is quasi-affine by Morphisms, Lemma \ref{morphisms-lemma-quasi-affine-permanence}. The final implications (4) $\Rightarrow$ (2) and (4) $\Rightarrow$ (3) are evident.
Let $f : X \to Y$ and $g : Y \to Z$ be ind-quasi-affine morphisms. Let $V \subset Z$ and $U \subset f^{-1}(g^{-1}(V))$ be quasi-compact opens such that $V$ is also quasi-separated. The image $f(U)$ is a quasi-compact subset of $g^{-1}(V)$, so it is contained in some quasi-compact open $W \subset g^{-1}(V)$ (a union of finitely many affines). We obtain a factorization $U \to W \to V$. The map $W \to V$ is quasi-affine by Lemma \ref{lemma-ind-quasi-affine-alternative-definition}, so, in particular, $W$ is quasi-separated. Then, by Lemma \ref{lemma-ind-quasi-affine-alternative-definition} again, $U \to W$ is quasi-affine as well. Consequently, by Morphisms, Lemma \ref{morphisms-lemma-composition-quasi-affine}, the composition $U \to V$ is also quasi-affine, and it remains to apply Lemma \ref{lemma-ind-quasi-affine-alternative-definition} once more.
By Lemma \ref{lemma-ind-quasi-affine-alternative-definition}, we may work Zariski locally on $Z$ and then on $Y$, so we lose no generality by assuming that $Z$, and then also $Y$, is affine. Then any quasi-compact open of $X$ is quasi-affine, so Lemma \ref{lemma-ind-quasi-affine-alternative-definition} gives the claim.
Let $f : X \to Y$ be an ind-quasi-affine morphism. For checking that every base change of $f$ is ind-quasi-affine, by Lemma \ref{lemma-ind-quasi-affine-alternative-definition}, we may work Zariski locally on $Y$, so we assume that $Y$ is affine. Furthermore, we may also assume that in the base change morphism $Z \to Y$ the scheme $Z$ is affine, too. The base change $X \times_Y Z \to X$ is an affine morphism, so, by Lemmas \ref{lemma-ind-quasi-affine-composition} and \ref{lemma-ind-quasi-affine-examples}, the map $X \times_Y Z \to Y$ is ind-quasi-affine. Then, by Lemma \ref{lemma-ind-quasi-affine-permanence}, the base change $X \times_Y Z \to Z$ is ind-quasi-affine, as desired.
The stability of ind-quasi-affineness under base change supplied by Lemma \ref{lemma-base-change-ind-quasi-affine} gives one direction. For the other, let $f : X \to Y$ be a morphism of schemes and let $\{g_i : Y_i \to Y\}$ be an fpqc covering such that the base change $f_i : X_i \to Y_i$ is ind-quasi-affine for all $i$. We need to show $f$ is ind-quasi-affine. \medskip\noindent By Lemma \ref{lemma-ind-quasi-affine-alternative-definition}, we may work Zariski locally on $Y$, so we assume that $Y$ is affine. Then we use stability under base change ensured by Lemma \ref{lemma-base-change-ind-quasi-affine} to refine the cover and assume that it is given by a single affine, faithfully flat morphism $g : Y' \to Y$. For any quasi-compact open $U \subset X$, its $Y'$-base change $U \times_Y Y' \subset X \times_Y Y'$ is also quasi-compact. It remains to observe that, by Descent, Lemma \ref{descent-lemma-descending-property-quasi-affine}, the map $U \to Y$ is quasi-affine if and only if so is $U \times_Y Y' \to Y'$.
Let $f : X \to Y$ be a separated locally quasi-finite morphism of schemes. Let $V \subset Y$ be affine and $U \subset f^{-1}(V)$ quasi-compact open. We have to show $U$ is quasi-affine. Since $U \to V$ is a separated quasi-finite morphism of schemes, this follows from Zariski's Main Theorem. See Lemma \ref{lemma-quasi-finite-separated-quasi-affine}.
Let $y \in Y$. Choose an affine open $U \subset X$ such that $j^{-1}(\{y\}) \subset i^{-1}(U)$ (possible by assumption). Choose an affine open $V \subset Y$ neighbourhood of $y$ such that $j^{-1}(V) \subset i^{-1}(U)$. This is possible because $j : Z \to Y$ is a closed morphism (Morphisms, Lemma \ref{morphisms-lemma-integral-universally-closed}) and $i^{-1}(U)$ contains the fibre over $y$. Since $j$ is integral, the scheme theoretic fibre $Z_y$ is the spectrum of an algebra integral over a field. By Limits, Lemma \ref{limits-lemma-ample-profinite-set-in-principal-affine} we can find an $\overline{f} \in \Gamma(i^{-1}(U), \mathcal{O}_{i^{-1}(U)})$ such that $Z_y \subset D(\overline{f}) \subset j^{-1}(V)$. Since $i|_{i^{-1}(U)} : i^{-1}(U) \to U$ is a closed immersion of affines, we can choose an $f \in \Gamma(U, \mathcal{O}_U)$ whose restriction to $i^{-1}(U)$ is $\overline{f}$. After replacing $U$ by the principal open $D(f) \subset U$ we find affine opens $y \in V \subset Y$ and $U \subset X$ with $$ j^{-1}(\{y\}) \subset i^{-1}(U) \subset j^{-1}(V) $$ Now we (in some sense) repeat the argument. Namely, we choose $g \in \Gamma(V, \mathcal{O}_V)$ such that $y \in D(g)$ and $j^{-1}(D(g)) \subset i^{-1}(U)$ (possible by the same argument as above). Then we can pick $f \in \Gamma(U, \mathcal{O}_U)$ whose restriction to $i^{-1}(U)$ is the pullback of $g$ by $i^{-1}(U) \to V$ (again possible by the same reason as above). Then we finally have affine opens $y \in V' = D(g) \subset V \subset Y$ and $U' = D(f) \subset U \subset X$ with $j^{-1}(V') = i^{-1}(V')$.
The pushout $Y \amalg_Z X$ exists by Proposition \ref{proposition-pushout-along-closed-immersion-and-integral}. To see that the pushout $Y' \amalg_{Z'} X'$ exists, we check condition (3) of Situation \ref{situation-pushout-along-closed-immersion-and-integral} holds for $(X', Y', Z', i', j')$. Namely, let $y' \in Y'$ and denote $y \in Y$ the image. Choose $U \subset X$ affine open with $i(j^{-1}(y)) \subset U$. Choose a quasi-compact open $U' \subset X'$ contained in $f^{-1}(U)$ containing the quasi-compact subset $i'((j')^{-1}(\{y'\}))$. By Lemma \ref{lemma-etale-separated-ind-quasi-affine} we see that $U'$ is quasi-affine. Since $Z'_{y'}$ is the spectrum of an algebra integral over a field, we can apply Limits, Lemma \ref{limits-lemma-ample-profinite-set-in-principal-affine} and we find there exists an affine open subscheme of $U'$ containing $i'((j')^{-1}(\{y'\}))$ as desired. \medskip\noindent Having verified existence we check the other assertions. Affine locally we are exactly in the situation of More on Algebra, Lemma \ref{more-algebra-lemma-properties-algebras-over-fibre-product} with $B \to D$ and $A' \to C'$ locally quasi-finite\footnote{To be precise $X, Y, Z, Y \amalg_Z X, X', Y', Z', Y' \amalg_{Z'} X'$ correspond to $A', B, A, B', C', D, C, D'$.}. In particular, the morphism $Y' \amalg_{Z'} X' \to Y \amalg_Z X$ is locally of finite type. The squares in of the diagram are cartesian by More on Algebra, Lemma \ref{more-algebra-lemma-module-over-fibre-product}. Since being locally quasi-finite can be checked on fibres (Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-at-point-characterize}) we conclude that $Y' \amalg_{Z'} X' \to Y \amalg_Z X$ is locally quasi-finite. \medskip\noindent We still have to check $Y' \amalg_{Z'} X' \to Y \amalg_Z X$ is separated. Observe that $Y' \amalg X' \to Y' \amalg_{Z'} X'$ is universally closed and surjective by  Proposition \ref{proposition-pushout-along-closed-immersion-and-integral}. Since also the morphism $Y' \amalg X' \to Y \amalg_Z X$ is separated (as it factors as $Y' \amalg X' \to Y \amalg X \to Y \amalg_Z X$) we conclude by Morphisms, Lemma \ref{morphisms-lemma-image-universally-closed-separated}.
If we start with $(X', Y', Z', i', j', f, g, h)$ as in Lemma \ref{lemma-pushout-functor} with $f, g, h$ flat or \'etale, then $Y' \amalg_{Z'} X' \to Y \amalg_Z X$ is flat or \'etale by More on Algebra, Lemma \ref{more-algebra-lemma-properties-algebras-over-fibre-product}. \medskip\noindent For the converse, let $W \to Y \amalg_Z X$ be a separated and locally quasi-finite morphism. Set $X' = W \times_{Y \amalg_Z X} X$, $Y' = W \times_{Y \amalg_Z X} Y$, and $Z' = W \times_{Y \amalg_Z X} Z$ with obvious morphisms $i', j', f, g, h$. Form the pushout $Y' \amalg_{Z'} X'$. We obtain a morphism $$ Y' \amalg_{Z'} X' \longrightarrow W $$ of schemes over $Y \amalg_X Z$ by the universal property of the pushout. If we do not assume that $W \to Y \amalg_Z X$ is flat, then in general this morphism won't be an isomorphism. (In fact, More on Algebra, Lemma \ref{more-algebra-lemma-module-over-fibre-product-bis} shows the displayed arrow is a closed immersion but not an isomorphism in general.) However, if $W \to Y \times_Z X$ is flat, then it is an isomorphism by More on Algebra, Lemma \ref{more-algebra-lemma-properties-algebras-over-fibre-product}.
Choose an affine open covering $S = \bigcup U_i$ such that $\Gamma(Z \times_S U_i, \mathcal{O}_{Z \times_S U_i})$ is finite free over $\mathcal{O}_S(U_i)$. Let $F_i \subset \mathit{Mor}_S(Z, X)$ be the subfunctor which assigns to $T/S$ the empty set if $T \to S$ does not factor through $U_i$ and $\mathit{Mor}_S(Z, X)(T)$ otherwise. Then the collection of these subfunctors satisfy the conditions (2)(a), (2)(b), (2)(c) of Schemes, Lemma \ref{schemes-lemma-glue-functors} which proves the lemma. Condition (2)(a) follows from Lemma \ref{lemma-hom-from-finite-free-into-affine} and the other two follow from straightforward arguments.
Consider the set $I$ of pairs $(U, V)$ where $U \subset X$ and $V \subset S$ are affine open and $U \to S$ factors through $V$. For $i \in I$ denote $(U_i, V_i)$ the corresponding pair. Set $F_i = \mathit{Mor}_{V_i}(Z_{V_i}, U_i)$. It is immediate that $F_i$ is a subfunctor of $\mathit{Mor}_S(Z, X)$. Then we claim that conditions (2)(a), (2)(b), (2)(c) of Schemes, Lemma \ref{schemes-lemma-glue-functors} which proves the lemma. \medskip\noindent Condition (2)(a) follows from Lemma \ref{lemma-hom-from-finite-locally-free-into-affine}. \medskip\noindent To check condition (2)(b) consider $T/S$ and $b \in \mathit{Mor}_S(Z, X)$. Thinking of $b$ as a morphism $T \times_S Z \to X$ we find an open $b^{-1}(U_i) \subset T \times_S Z$. Clearly, $b \in F_i(T)$ if and only if $b^{-1}(U_i) = T \times_S Z$. Since the projection $p : T \times_S Z \to T$ is finite hence closed, the set $U_{i, b} \subset T$ of points $t \in T$ with $p^{-1}(\{t\}) \subset b^{-1}(U_i)$ is open. Then $f : T' \to T$ factors through $U_{i, b}$ if and only if $b \circ f \in F_i(T')$ and we are done checking (2)(b). \medskip\noindent Finally, we check condition (2)(c) and this is where our condition on $X \to S$ is used. Namely, consider $T/S$ and $b \in \mathit{Mor}_S(Z, X)$. It suffices to prove that every $t \in T$ is contained in one of the opens $U_{i, b}$ defined in the previous paragraph. This is equivalent to the condition that $b(p^{-1}(\{t\})) \subset U_i$ for some $i$ where $p : T \times_S Z \to T$ is the projection and $b : T \times_S Z \to X$ is the given morphism. Since $p$ is finite, the set $b(p^{-1}(\{t\})) \subset X$ is finite and contained in the fibre of $X \to S$ over the image $s$ of $t$ in $S$. Thus our condition on $X \to S$ exactly shows a suitable pair exists.
The question is local, hence we may assume all schemes are affine (some details omitted). Observe that $$ \xymatrix{ X \times_S Y \ar[r] \ar[d] & X' \times_{S'} Y' \ar[d] \\ X \ar[r] & X' } $$ is cartesian with flat vertical arrows. Write $X = \Spec(A)$, $X' = \Spec(A')$, $X' \times_{S'} Y' = \Spec(B')$. Then $X \times_S Y = \Spec(A \otimes_{A'} B')$. Write $Z' = \Spec(C')$. We have to show $$ \text{Tor}_p^{B'}(A \otimes_{A'} B', C') = 0, \quad\text{for } p > 0 $$ Since $A' \to B'$ is flat we have $A \otimes_{A'} B' = A \otimes_{A'}^\mathbf{L} B'$. Hence $$ (A \otimes_{A'} B') \otimes_{B'}^\mathbf{L} C' = (A \otimes_{A'}^\mathbf{L} B') \otimes_{B'}^\mathbf{L} C' = A \otimes_{A'}^\mathbf{L} C' = A \otimes_{A'} C' $$ The second equality by More on Algebra, Lemma \ref{more-algebra-lemma-double-base-change}. The last equality because $A' \to C'$ is flat. This proves the lemma.
Since $b^{-1}(U) = c^{-1}(V)$ and since $c$ is a closed immersion over $V$, we see that $c \circ (b|_U)^{-1}$ is a closed immersion. It is of finite presentation because $U$ and $V$ are of finite presentation over $A$, see Morphisms, Lemma \ref{morphisms-lemma-finite-presentation-permanence}. First we have $$ Rq_*(Lp^*K \otimes^\mathbf{L} E)|_V = Rq'_*\left((Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V}\right) $$ where $q' : X \times_A V \to V$ is the projection because formation of total direct image commutes with localization. Set $W = b^{-1}(U) = c^{-1}(V)$ and denote $i : W \to X \times_A V$ the closed immersion $i = (b, c)|_W$. Then $$ Rq'_*\left((Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V}\right) = Rq'_*(Lp^*K|_{X \times_A V} \otimes^\mathbf{L} i_*\mathcal{O}_W) $$ by property (5). Since $i$ is a closed immersion we have $i_*\mathcal{O}_W = Ri_*\mathcal{O}_W$. Using Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change} we can rewrite this as $$ Rq'_* Ri_* Li^* Lp^*K|_{X \times_A V} = R(q' \circ i)_* Lb^*K|_W = R(U \to V)_* K|_U $$ which is what we want.
Assume $K \in D_\QCoh(\mathcal{O}_X)$ and $R\Gamma(X, E \otimes^\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_X)$. Let $x \in X$. We will show that $K$ is pseudo-coherent relative to $A$ in a neighbourhood of $x$ and this will prove the lemma. \medskip\noindent Choose $U, n, V, Z, z, E$ as in Lemma \ref{lemma-derived-chow}. Denote $p : X \times \mathbf{P}^n \to X$ and $q : X \times \mathbf{P}^n \to \mathbf{P}^n_A$ the projections. Then for any $i \in \mathbf{Z}$ we have \begin{align*} & R\Gamma(\mathbf{P}^n_A, Rq_*(Lp^*K \otimes^\mathbf{L} E) \otimes^\mathbf{L} \mathcal{O}_{\mathbf{P}^n_A}(i)) \\ & = R\Gamma(X \times \mathbf{P}^n, Lp^*K \otimes^\mathbf{L} E \otimes^\mathbf{L} Lq^*\mathcal{O}_{\mathbf{P}^n_A}(i)) \\ & = R\Gamma(X, K \otimes^\mathbf{L} Rq_*(E \otimes^\mathbf{L} Lq^*\mathcal{O}_{\mathbf{P}^n_A}(i))) \end{align*} by  Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change}. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-pseudo-coherent-direct-image-general} the complex $Rq_*(E \otimes^\mathbf{L} Lq^*\mathcal{O}_{\mathbf{P}^n_A}(i))$ is pseudo-coherent on $X$. Hence the assumption tells us the expression in the displayed formula is a pseudo-coherent object of $D(A)$. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-pseudo-coherent-on-projective-space} we conclude that $Rq_*(Lp^*K \otimes^\mathbf{L} E)$ is pseudo-coherent on $\mathbf{P}^n_A$. By Lemma \ref{lemma-compute-Fourier-Mukai-for-derived-chow} we have $$ Rq_*(Lp^*K \otimes^\mathbf{L} E)|_{X \times_A V} = R(U \to V)_*K|_U $$ Since $U \to V$ is a closed immersion into an open subscheme of $\mathbf{P}^n_A$ this means $K|_U$ is pseudo-coherent relative to $A$ by Lemma \ref{lemma-check-relative-pseudo-coherence-on-charts}.
In view of Lemma \ref{lemma-characterize-pseudo-coherent}, it suffices to show $R \Gamma (X, E \otimes ^{\mathbf{L}} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent  $E \in D(\mathcal{O}_X)$. By Derived Categories of Schemes, Proposition \ref{perfect-proposition-detecting-bounded-above} it follows that $K \in D^-_\QCoh (\mathcal{O}_X)$. Now the result follows by Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-enough}.
By Lemma \ref{lemma-characterize-pseudo-coh-improved}, $K$ is pseudo-coherent relative to $A$. By Lemma \ref{lemma-check-relative-pseudo-coherence-on-charts}, $K$ is pseudo-coherent in $D( \mathcal{O}_X)$. By Derived Categories of Schemes, Proposition \ref{perfect-proposition-detecting-bounded-below} we see that $K$ is in $D^-(\mathcal{O}_X)$. Let $\mathfrak{p}$ be a prime ideal of $A$ and denote $i : Y \to X$ the inclusion of the scheme theoretic fibre over $\mathfrak{p}$, i.e., $Y$ is a scheme over $\kappa(\mathfrak p)$. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-bounded-on-fibres}, we will be done if we can show  $Li^*(K)$ is bounded below.  Let $G \in D_{perf} (\mathcal{O}_X)$ be a perfect complex which generates $D_\QCoh (\mathcal{O}_X)$, see Derived Categories of Schemes, Theorem \ref{perfect-theorem-bondal-van-den-Bergh}. We have \begin{align*} R\Hom _{\mathcal{O}_Y}(Li^*(G), Li^*(K)) & = R\Gamma(Y, Li^*(G ^\vee \otimes ^\mathbf{L} K)) \\ & = R\Gamma(X, G^\vee \otimes ^{\mathbf{L}} K) \otimes^\mathbf{L}_A \kappa(\mathfrak{p}) \end{align*} The first equality uses that $Li^*$ preserves perfect objects and duals and Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}; we omit some details. The second equality follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-compare-base-change} as $X$ is flat over $A$. It follows from our hypothesis that this is a perfect object of $D(\kappa(\mathfrak{p}))$. The object $Li^*(G) \in D_{perf}(\mathcal{O}_Y)$  generates $D_\QCoh(\mathcal{O}_Y)$ by Derived Categories of Schemes, Remark \ref{perfect-remark-pullback-generator}. Hence Derived Categories of Schemes, Proposition \ref{perfect-proposition-detecting-bounded-below} now implies that $Li^*(K)$ is bounded below and we win.
Assume $E$ is $m$-pseudo-coherent relative to $S$. The morphisms $f_i$ are pseudo-coherent by Lemma \ref{lemma-flat-finite-presentation-pseudo-coherent}. Hence $Lf_i^*E$ is $m$-pseudo-coherent relative to $S$ by Lemma \ref{lemma-pull-relative-pseudo-coherent}. \medskip\noindent Conversely, assume that $Lf_i^*E$ is $m$-pseudo-coherent relative to $S$ for each $i$. Pick $S = \bigcup U_j$, $W_j \to U_j$, $W_j = \bigcup W_{j, k}$, $T_{j, k} \to W_{j, k}$, and morphisms $\alpha_{j, k} : T_{j, k} \to X_{i(j, k)}$ over $S$ as in Lemma \ref{lemma-dominate-fppf}. Since the morphism $T_{j, K} \to S$ is flat and of finite presentation, we see that $\alpha_{j, k}$ is pseudo-coherent by Lemma \ref{lemma-permanence-pseudo-coherent}. Hence $$ L\alpha_{j, k}^*Lf_{i(j, k)}^*E = L(T_{i, k} \to S)^*E $$ is $m$-pseudo-coherent relative to $S$ by Lemma \ref{lemma-pull-relative-pseudo-coherent}. Now we want to descend this property through the coverings $\{T_{j, k} \to W_{j, k}\}$, $W_j = \bigcup W_{j, k}$, $\{W_j \to U_j\}$, and $S = \bigcup U_j$. Since for Zariski coverings the result is true (by the definition of $m$-pseudo-coherence relative to $S$), this means we may assume we have a single surjective finite locally free morphism $\pi : Y \to X$ such that $L\pi^*E$ is pseudo-coherent relative to $S$. In this case $R\pi_*L\pi^*E$ is pseudo-coherent relative to $S$ by Lemma \ref{lemma-finite-morphism-relative-pseudo-coherence} (this is the first time we use that $E$ has quasi-coherent cohomology sheaves). We have $R\pi_*L\pi^*E = E \otimes^\mathbf{L}_{\mathcal{O}_X} \pi_*\mathcal{O}_Y$ for example by Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change} and locally on $X$ the map $\mathcal{O}_X \to \pi_*\mathcal{O}_Y$ is the inclusion of a direct summand. Hence we conclude by Lemma \ref{lemma-summands-relative-pseudo-coherent}.