hcju/stacksqa huggingface dataset:
Part (1) follows from Morphisms, Lemma \ref{morphisms-lemma-base-change-projective}. \medskip\noindent Part (2) follows from the fifth characterization of Lemma \ref{lemma-projective} and the fact that compositions of proper morphisms are proper (Morphisms, Lemma \ref{morphisms-lemma-composition-proper}). \medskip\noindent If $X \to S$ and $Y \to S$ are projective, then $X \times_S Y \to Y$ is projective by Morphisms, Lemma \ref{morphisms-lemma-base-change-projective}. Hence (3) follows from (2). \medskip\noindent If $X = Y \amalg Z$ is a disjoint union of schemes and $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module such that $\mathcal{L}|_Y$ and $\mathcal{L}|_Z$ are ample, then $\mathcal{L}$ is ample (details omitted). Thus part (4) follows from the fifth characterization of Lemma \ref{lemma-projective}. \medskip\noindent Part (5) follows from Morphisms, Lemma \ref{morphisms-lemma-finite-projective}.
We may assume $Y$ is affine. Then we find a directed set $I$ and an inverse system of morphisms $X_i \to Y_i$ of schemes with $Y_i$ of finite type over $\mathbf{Z}$, with affine transition morphisms $X_i \to X_{i'}$ and $Y_i \to Y_{i'}$, with $X_i \to Y_i$ proper, such that $X \to Y = \lim (X_i \to Y_i)$. See Limits, Lemma \ref{limits-lemma-proper-limit-of-proper-finite-presentation-noetherian}. After shrinking $I$ we can assume we have a compatible system of invertible $\mathcal{O}_{X_i}$-modules $\mathcal{L}_i$ pulling back to $\mathcal{L}$, see Limits, Lemma \ref{limits-lemma-descend-invertible-modules}. Let $y_i \in Y_i$ be the image of $y$. Then $\kappa(y) = \colim \kappa(y_i)$. Hence for some $i$ we have $\mathcal{L}_{i, y_i}$ is ample on $X_{i, y_i}$ by Limits, Lemma \ref{limits-lemma-limit-ample}. By Cohomology of Schemes, Lemma \ref{coherent-lemma-ample-in-neighbourhood} we find an open neigbourhood $V_i \subset Y_i$ of $y_i$ such that $\mathcal{L}_i$ restricted to $f_i^{-1}(V_i)$ is ample relative to $V_i$. Letting $V \subset Y$ be the inverse image of $V_i$ finishes the proof (hints: use Morphisms, Lemma \ref{morphisms-lemma-ample-base-change} and the fact that $X \to Y \times_{Y_i} X_i$ is affine and the fact that the pullback of an ample invertible sheaf by an affine morphism is ample by Morphisms, Lemma \ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}).
We have $P = \text{Proj}(A)$ by Morphisms, Lemma \ref{morphisms-lemma-proper-ample-is-proj}. Moreover, by Properties, Lemma \ref{properties-lemma-ample-gcd-is-one} via this identification we have $\mathcal{O}_P(m) = \mathcal{L}^{\otimes m}$ for all $m \in \mathbf{Z}$.
Note that the two assertions are equivalent by Algebra, Lemma \ref{algebra-lemma-image-dense-generic-points}. To start the proof let $C$ be an $A$-algebra of finite type and $\mathfrak q$ a prime of $C$ such that $B = C_{\mathfrak q}$. Of course we may assume that $C$ is a domain and that $g \in C$. After replacing $C$ by a localization we see that $\dim(C/\mathfrak m_AC) = \dim(B/\mathfrak m_AB) + \text{trdeg}_{\kappa(\mathfrak m_A)}(\kappa(\mathfrak m_B))$, see Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-at-a-point}. Setting $K$ equal to the fraction field of $A$ we see by the same reference that $\dim(C \otimes_A K) = \text{trdeg}_A(B)$. Hence assumption (4) means that the generic and closed fibres of the morphism $\Spec(C) \to \Spec(A)$ have the same dimension. \medskip\noindent Suppose that the lemma is false. Then $(B/gB) \otimes_A K = 0$. This means that $g \otimes 1$ is invertible in $B \otimes_A K = C_{\mathfrak q} \otimes_A K$. As $C_{\mathfrak q}$ is a limit of principal localizations we conclude that $g \otimes 1$ is invertible in $C_h \otimes_A K$ for some $h \in C$, $h \not \in \mathfrak q$. Thus after replacing $C$ by $C_h$ we may assume that $(C/gC) \otimes_A K = 0$. We do one more replacement of $C$ to make sure that the minimal primes of $C/\mathfrak m_AC$ correspond one-to-one with the minimal primes of $B/\mathfrak m_AB$. At this point we apply Lemma \ref{lemma-locally-principal-vertical} to $X = \Spec(C) \to \Spec(A) = S$ and the locally closed subscheme $Z = \Spec(C/gC)$. Since $Z_K = \emptyset$ we see that $Z \otimes \kappa(\mathfrak m_A)$ has to contain an irreducible component of $X \otimes \kappa(\mathfrak m_A) = \Spec(C/\mathfrak m_AC)$. But this contradicts the assumption that $g$ is not contained in any prime minimal over $\mathfrak m_AB$. The lemma follows.
Let $C$ be an $A$-algebra of finite type and $\mathfrak q$ a prime of $C$ such that $B = C_{\mathfrak q}$. We may assume $C$ is a domain. We have $\dim_{\mathfrak q}(C/\mathfrak m_AC) = \dim(B/\mathfrak m_AB) + \text{trdeg}_{\kappa(\mathfrak m_A)}(\kappa(\mathfrak m_B))$, see Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-at-a-point}. Setting $K$ equal to the fraction field of $A$ we see by the same reference that $\dim(C \otimes_A K) = \text{trdeg}_A(B)$. Thus we are really trying to prove that $\dim_{\mathfrak q}(C/\mathfrak m_AC) = \dim(C \otimes_A K)$. Choose a valuation ring $A'$ in $K$ dominating $A$, see Algebra, Lemma \ref{algebra-lemma-dominate}. Set $C' = C \otimes_A A'$. Choose a prime $\mathfrak q'$ of $C'$ lying over $\mathfrak q$; such a prime exists because $$ C'/\mathfrak m_{A'}C' = C/\mathfrak m_AC \otimes_{\kappa(\mathfrak m_A)} \kappa(\mathfrak m_{A'}) $$ which proves that $C/\mathfrak m_AC \to C'/\mathfrak m_{A'}C'$ is faithfully flat. This also proves that $\dim_{\mathfrak q}(C/\mathfrak m_AC) = \dim_{\mathfrak q'}(C'/\mathfrak m_{A'}C')$, see Algebra, Lemma \ref{algebra-lemma-dimension-at-a-point-preserved-field-extension}. Note that $B' = C'_{\mathfrak q'}$ is a localization of $B \otimes_A A'$. Hence $B'$ is flat over $A'$. The generic fibre $B' \otimes_{A'} K$ is a localization of $B \otimes_A K$. Hence $B'$ is a domain. If we prove the lemma for $A' \subset B'$, then we get the equality $\dim_{\mathfrak q'}(C'/\mathfrak m_{A'}C') = \dim(C' \otimes_{A'} K)$ which implies the desired equality $\dim_{\mathfrak q}(C/\mathfrak m_AC) = \dim(C \otimes_A K)$ by what was said above. This reduces the lemma to the case where $A$ is a valuation ring. \medskip\noindent Let $A \subset B$ be as in the lemma with $A$ a valuation ring. As before write $B = C_{\mathfrak q}$ for some domain $C$ of finite type over $A$. By Algebra, Lemma \ref{algebra-lemma-finite-type-domain-over-valuation-ring-dim-fibres} we obtain $\dim(C/\mathfrak m_AC) = \dim(C \otimes_A K)$ and we win.
Since $A$ is Noetherian and $A \to B$ is essentially of finite type, we know that $B$ is Noetherian too. By Properties, Lemma \ref{properties-lemma-complement-closed-point-Jacobson} the topological space $\Spec(B) \setminus \{\mathfrak m_B\}$ is Jacobson. Hence we can choose a closed point $\mathfrak q$ which is contained in the nonempty open $$ V \setminus \{\mathfrak q \subset B \mid \mathfrak m_A = \mathfrak q \cap A\}. $$ (Nonempty by assumption, open because $\{\mathfrak m_A\}$ is a closed subset of $\Spec(A)$.) Then $\Spec(B/\mathfrak q)$ has two points, namely $\mathfrak m_B$ and $\mathfrak q$ and $\mathfrak q$ does not lie over $\mathfrak m_A$. Write $B/\mathfrak q = C_{\mathfrak m}$ for some finite type $A$-algebra $C$ and prime ideal $\mathfrak m$. Then $A \to C$ is quasi-finite at $\mathfrak m$ by Algebra, Lemma \ref{algebra-lemma-isolated-point-fibre} (2). Hence by Algebra, Lemma \ref{algebra-lemma-quasi-finite-open} we see that after replacing $C$ by a principal localization the ring map $A \to C$ is quasi-finite.
Namely, we can reduce this to the cited lemma as follows: First we replace $X$ and $S$ by affine neighbourhoods of $x$ and $s$. Then $X$ is Noetherian, in particular $U$ is quasi-compact (see Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian} and Topology, Lemmas \ref{topology-lemma-Noetherian} and \ref{topology-lemma-Noetherian-quasi-compact}). Hence there exists a specialization $x' \leadsto x$ with $x' \in U$ (see Morphisms, Lemma \ref{morphisms-lemma-reach-points-scheme-theoretic-image}). Note that $f(x') \not = s$. Thus we see all hypotheses of the lemma are satisfied and we win.
By Morphisms, Lemma \ref{morphisms-lemma-universally-closed-quasi-compact} the morphism $f$ is quasi-compact. Hence the normalization $S'$ of $S$ in $X$ is defined (Morphisms, Definition \ref{morphisms-definition-normalization-X-in-Y}) and we have the factorization $X \to S' \to S$. By Morphisms, Lemma \ref{morphisms-lemma-normalization-in-universally-closed} we have (2), (4), and (5). The morphism $f'$ is universally closed by Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}. It is quasi-compact by Schemes, Lemma \ref{schemes-lemma-quasi-compact-permanence} and quasi-separated by Schemes, Lemma \ref{schemes-lemma-compose-after-separated}. \medskip\noindent To show the remaining statements we may assume the base scheme $S$ is affine, say $S = \Spec(R)$. Then $S' = \Spec(A)$ with $A = \Gamma(X, \mathcal{O}_X)$ an integral $R$-algebra. Thus it is clear that $f'_*\mathcal{O}_X$ is $\mathcal{O}_{S'}$ (because $f'_*\mathcal{O}_X$ is quasi-coherent, by Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}, and hence equal to $\widetilde{A}$). This proves (3). \medskip\noindent Let us show that $f'$ is surjective. As $f'$ is universally closed (see above) the image of $f'$ is a closed subset $V(I) \subset S' = \Spec(A)$. Pick $h \in I$. Then $h|_X = f^\sharp(h)$ is a global section of the structure sheaf of $X$ which vanishes at every point. As $X$ is quasi-compact this means that $h|_X$ is a nilpotent section, i.e., $h^n|X = 0$ for some $n > 0$. But $A = \Gamma(X, \mathcal{O}_X)$, hence $h^n = 0$. In other words $I$ is contained in the Jacobson radical ideal of $A$ and we conclude that $V(I) = S'$ as desired.
Recall that there are no specializations among the points of $\pi^{-1}(\{s\})$, see Algebra, Lemma \ref{algebra-lemma-integral-no-inclusion}. As $f'$ is surjective, we find that $|X_s| \to \pi^{-1}(\{s\})$ is surjective. Observe that $X_s$ is a quasi-separated scheme of finite type over a field (quasi-compactness was shown in the proof of the referenced lemma). Thus $X_s$ is Noetherian (Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}). A topological argument (omitted) now shows that $\pi^{-1}(\{s\})$ is finite. For each $i$ we can pick a finite type point $x_i \in X_s$ mapping to $s_i$ (Morphisms, Lemma \ref{morphisms-lemma-enough-finite-type-points}). We conclude that $\kappa(s_i)/\kappa(s)$ is finite: $x_i$ can be represented by a morphism $\Spec(k_i) \to X_s$ of finite type (by our definition of finite type points) and hence $\Spec(k_i) \to s = \Spec(\kappa(s))$ is of finite type (as a composition of finite type morphisms), hence $k_i/\kappa(s)$ is finite (Morphisms, Lemma \ref{morphisms-lemma-point-finite-type}).
If $X_s$ is geometrically connected, then any base change of it is connected. On the other hand, suppose that $X_s$ is not geometrically connected. Then by Varieties, Lemma \ref{varieties-lemma-characterize-geometrically-disconnected} we see that $X_s \times_{\Spec(\kappa(s)} \Spec(k)$ is disconnected for some finite separable field extension $\kappa(s) \subset k$. By Lemma \ref{lemma-realize-prescribed-residue-field-extension-etale} there exists an affine \'etale neighbourhood $(U, u) \to (S, s)$ such that $\kappa(s) \subset \kappa(u)$ is identified with $\kappa(s) \subset k$. In this case $X_u$ is disconnected.
Apply Theorem \ref{theorem-stein-factorization-general} to get a factorization $X \to S' \to S$. It is enough to show that $S' = S$. This will follow from Morphisms, Lemma \ref{morphisms-lemma-finite-birational-over-normal}. Namely, $S'$ is reduced because $X$ is reduced (Morphisms, Lemma \ref{morphisms-lemma-normalization-in-reduced}). The morphism $S' \to S$ is integral by the theorem cited above. Every generic point of $S'$ lies over $\xi$ by Morphisms, Lemma \ref{morphisms-lemma-normalization-generic} and assumption (5). On the other hand, since $S'$ is the relative spectrum of $f_*\mathcal{O}_X$ we see that the scheme theoretic fibre $S'_\xi$ is the spectrum of $H^0(X_\xi, \mathcal{O})$ which is equal to $\kappa(\xi)$ by assumption. Hence $S'$ is an integral scheme with function field equal to the function field of $S$. This finishes the proof.
Let $s \in S$. Set $n = n_{X/S}(s)$. Note that $n < \infty$ as the geometric fibre of $X \to S$ at $s$ is a proper scheme over a field, hence Noetherian, hence has a finite number of connected components. We have to find an open neighbourhood $V$ of $s$ such that $n_{X/S}|_V \geq n$. Let $X \to S' \to S$ be the Stein factorization as in Theorem \ref{theorem-stein-factorization-general}. By Lemma \ref{lemma-stein-universally-closed-residue-fields} there are finitely many points $s'_1, \ldots, s'_m \in S'$ lying over $s$ and the extensions $\kappa(s'_i)/\kappa(s)$ are finite. Then Lemma \ref{lemma-etale-makes-integral-split} tells us that after replacing $S$ by an \'etale neighbourhood of $s$ we may assume $S' = V_1 \amalg \ldots \amalg V_m$ as a scheme with $s'_i \in V_i$ and $\kappa(s'_i)/\kappa(s)$ purely inseparable. Then the schemes $X_{s_i'}$ are geometrically connected over $\kappa(s)$, hence $m = n$. The schemes $X_i = (f')^{-1}(V_i)$, $i = 1, \ldots, n$ are flat and of finite presentation over $S$. Hence the image of $X_i \to S$ is open (Morphisms, Lemma \ref{morphisms-lemma-fppf-open}). Thus in a neighbourhood of $s$ we see that $n_{X/S}$ is at least $n$.
By Lemma \ref{lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous} the function $n_{X/S}$ is lower semincontinuous. For $s \in S$ consider the $\kappa(s)$-algebra $$ A = H^0(X_s, \mathcal{O}_{X_s}) $$ By Varieties, Lemma \ref{varieties-lemma-proper-geometrically-reduced-global-sections} and the fact that $X_s$ is geometrically reduced $A$ is finite product of finite separable extensions of $\kappa(s)$. Hence $A \otimes_{\kappa(s)} \kappa(\overline{s})$ is a product of $\beta_0(s) = \dim_{\kappa(s)} H^0(E \otimes^\mathbf{L} \kappa(s))$ copies of $\kappa(\overline{s})$. Thus $X_{\overline{s}}$ has $\beta_0(s) = \dim_{\kappa(s)} A$ connected components. In other words, we have $n_{X/S} = \beta_0$ as functions on $S$. Thus $n_{X/S}$ is upper semi-continuous by Derived Categories of Schemes, Lemma \ref{perfect-lemma-jump-loci-geometric}. This finishes the proof.
Being separated and being locally quasi-finite are properties of morphisms of schemes which are preserved under any base change, see Schemes, Lemma \ref{schemes-lemma-separated-permanence} and Morphisms, Lemma \ref{morphisms-lemma-base-change-quasi-finite}. Hence Descent, Lemma \ref{descent-lemma-descending-types-morphisms} applies and it suffices to prove the statement of the lemma in case the fppf-covering is given by a single $\{X \to S\}$ flat surjective morphism of finite presentation of affines. Say $X = \Spec(A)$ and $S = \Spec(R)$ so that $R \to A$ is a faithfully flat ring map. Let $(V, \varphi)$ be a descent datum relative to $X$ over $S$ and assume that $\pi : V \to X$ is separated and locally quasi-finite. \medskip\noindent Let $W^1 \subset V$ be any affine open. Consider $W = \text{pr}_1(\varphi(W^1 \times_S X)) \subset V$. Here is a picture $$ \xymatrix{ W^1 \times_S X \ar[rrrrr] \ar[ddd] \ar[rd] & & & & & \varphi(W^1 \times_S X) \ar[ddd] \ar[ld] \\ & V \times_S X \ar[rrr]^\varphi \ar[rd] \ar[dd] & & & X \times_S V \ar[ld] \ar[dd] & \\ & & X \times_S X \ar[r]^1 \ar[d]_{\text{pr}_0} & X \times_S X \ar[d]^{\text{pr}_1} & & \\ W^1 \ar[r] & V \ar[r] & X & X & V \ar[l] & W \ar[l] } $$ Ok, and now since $X \to S$ is flat and of finite presentation it is universally open (Morphisms, Lemma \ref{morphisms-lemma-fppf-open}). Hence we conclude that $W$ is open. Moreover, it is also clearly the case that $W$ is quasi-compact, and $W^1 \subset W$. Moreover, we note that $\varphi(W \times_S X) = X \times_S W$ by the cocycle condition for $\varphi$. Hence we obtain a new descent datum $(W, \varphi')$ by restricting $\varphi$ to $W \times_S X$. Note that the morphism $W \to X$ is quasi-compact, separated and locally quasi-finite. This implies that it is separated and quasi-finite by definition. Hence it is quasi-affine by Lemma \ref{lemma-quasi-finite-separated-quasi-affine}. Thus by Descent, Lemma \ref{descent-lemma-quasi-affine} we see that the descent datum $(W, \varphi')$ is effective. \medskip\noindent In other words, we find that there exists an open covering $V = \bigcup W_i$ by quasi-compact opens $W_i$ which are stable for the descent morphism $\varphi$. Moreover, for each such quasi-compact open $W \subset V$ the corresponding descent data $(W, \varphi')$ is effective. This means the original descent datum is effective by glueing the schemes obtained from descending the opens $W_i$, see Descent, Lemma \ref{descent-lemma-effective-for-fpqc-is-local-upstairs}.
The final statement is clear from the equivalence of (1) and (2). It is also clear that (2) implies (1). Assume (1) holds. Let $S = \bigcup V_i$ and $f^{-1}(V_i) = \bigcup U_{ij}$ be affine open coverings as in Definition \ref{definition-relatively-finitely-presented-sheaf}. Let $U \subset X$ and $V \subset S$ be as in (2). By More on Algebra, Lemma \ref{more-algebra-lemma-glue-relative-finite-presentation} it suffices to find a standard open covering $U = \bigcup U_k$ of $U$ such that $\mathcal{F}(U_k)$ is finitely presented relative to $\mathcal{O}_S(V)$. In other words, for every $u \in U$ it suffices to find a standard affine open $u \in U' \subset U$ such that $\mathcal{F}(U')$ is finitely presented relative to $\mathcal{O}_S(V)$. Pick $i$ such that $f(u) \in V_i$ and then pick $j$ such that $u \in U_{ij}$. By Schemes, Lemma \ref{schemes-lemma-standard-open-two-affines} we can find $v \in V' \subset V \cap V_i$ which is standard affine open in $V'$ and $V_i$. Then $f^{-1}V'  \cap U$, resp.\ $f^{-1}V' \cap U_{ij}$ are standard affine opens of $U$, resp.\ $U_{ij}$. Applying the lemma again we can find $u \in U' \subset f^{-1}V' \cap U \cap U_{ij}$ which is standard affine open in both $f^{-1}V'  \cap U$ and $f^{-1}V' \cap U_{ij}$. Thus $U'$ is also a standard affine open of $U$ and $U_{ij}$. By More on Algebra, Lemma \ref{more-algebra-lemma-localize-relative-finite-presentation} the assumption that $\mathcal{F}(U_{ij})$ is finitely presented relative to $\mathcal{O}_S(V_i)$ implies that $\mathcal{F}(U')$ is finitely presented relative to $\mathcal{O}_S(V_i)$. Since $\mathcal{O}_X(U') = \mathcal{O}_X(U') \otimes_{\mathcal{O}_S(V_i)} \mathcal{O}_S(V')$ we see from More on Algebra, Lemma \ref{more-algebra-lemma-base-change-relative-finite-presentation} that $\mathcal{F}(U')$ is finitely presented relative to $\mathcal{O}_S(V')$. Applying More on Algebra, Lemma \ref{more-algebra-lemma-localize-relative-finite-presentation} again we conclude that $\mathcal{F}(U')$ is finitely presented relative to $\mathcal{O}_S(V)$. This finishes the proof.
Say $S = \Spec(R)$ and $X = \Spec(A)$. Let $i$ correspond to the surjection $\alpha : R[x_1, \ldots, x_n] \to A$ and let $X \to \mathbf{A}^m_S$ correspond to $\beta : R[y_1, \ldots, y_m] \to A$. Choose $f_j \in R[x_1, \ldots, x_n]$ with $\alpha(f_j) = \beta(y_j)$ and $g_i \in R[y_1, \ldots, y_m]$ with $\beta(g_i) = \alpha(x_i)$. Then we get a commutative diagram $$ \xymatrix{ R[x_1, \ldots, x_n, y_1, \ldots, y_m] \ar[d]^{x_i \mapsto g_i} \ar[rr]_-{y_j \mapsto f_j} & & R[x_1, \ldots, x_n] \ar[d] \\ R[y_1, \ldots, y_m] \ar[rr] & & A } $$ corresponding to the commutative diagram of closed immersions $$ \xymatrix{ \mathbf{A}^{n + m}_S & \mathbf{A}^n_S \ar[l] \\ \mathbf{A}^m_S \ar[u] & X \ar[u] \ar[l] } $$ Thus it suffices to show that under a closed immersion $$ f : \mathbf{A}^m_S \to \mathbf{A}^{n + m}_S $$ an object $E$ of $D(\mathcal{O}_{\mathbf{A}^m_S})$ is $m$-pseudo-coherent if and only if $Rf_*E$ is $m$-pseudo-coherent. This follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-closed-push-pseudo-coherent} and the fact that $f_*\mathcal{O}_{\mathbf{A}^m_S}$ is a pseudo-coherent $\mathcal{O}_{\mathbf{A}^{n + m}_S}$-module. The pseudo-coherence of $f_*\mathcal{O}_{\mathbf{A}^m_S}$ is straightforward to prove directly, but it also follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-pseudo-coherent-affine} and More on Algebra, Lemma \ref{more-algebra-lemma-relatively-pseudo-coherent}.
Choose an affine open covering $S = \bigcup V_i$ and for each $i$ an affine open covering $f^{-1}(V_i) = \bigcup U_{ij}$ such that the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} are satisfied for each of the pairs $(U_{ij} \to V_i, E|_{U_{ij}})$. Since being quasi-coherent is local on $X$, we may assume that there exists an closed immersion $i : X \to \mathbf{A}^n_S$ such that $Ri_*E$ is $m$-pseudo-coherent on $\mathbf{A}^n_S$. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-pseudo-coherent} this means that $H^q(Ri_*E)$ is quasi-coherent for $q > m$. Since $i_*$ is an exact functor, we have $i_*H^q(E) = H^q(Ri_*E)$ is quasi-coherent on $\mathbf{A}^n_S$. By Morphisms, Lemma \ref{morphisms-lemma-i-star-equivalence} this implies that $H^q(E)$ is quasi-coherent as desired (strictly speaking it implies there exists some quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ such that $i_*\mathcal{F} = i_*H^q(E)$ and then Modules, Lemma \ref{modules-lemma-i-star-equivalence} tells us that $\mathcal{F} \cong H^q(E)$ hence the result).
Write $S = \Spec(R)$, $V = D(f)$, $X = \Spec(A)$, and $U = D(g)$. Assume the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} are satisfied for the pair $(X \to V, E)$. \medskip\noindent Choose $R_f[x_1, \ldots, x_n] \to A$ surjective. Write $R_f = R[x_0]/(fx_0 - 1)$. Then $R[x_0, x_1, \ldots, x_n] \to A$ is surjective, and $R_f[x_1, \ldots, x_n]$ is pseudo-coherent as an $R[x_0, \ldots, x_n]$-module. Thus we have $$ X \to \mathbf{A}^n_V \to \mathbf{A}^{n + 1}_S $$ and we can apply Derived Categories of Schemes, Lemma \ref{perfect-lemma-closed-push-pseudo-coherent} to conclude that the pushforward $E'$ of $E$ to $\mathbf{A}^{n + 1}_S$ is $m$-pseudo-coherent. \medskip\noindent Choose an element $g' \in R[x_0, x_1, \ldots, x_n]$ which maps to $g \in A$. Consider the surjection $R[x_0, \ldots, x_{n + 1}] \to R[x_0, \ldots, x_n, 1/g']$. We obtain $$ \xymatrix{ X \ar[d] & U \ar[d] \ar[l] \ar[dr] \\ \mathbf{A}^{n + 1}_S & D(g')\ar[l] \ar[r] & \mathbf{A}^{n + 2}_S } $$ where the lower left arrow is an open immersion and the lower right arrow is a closed immersion. We conclude as before that the pushforward of $E'|_{D(g')}$ to $\mathbf{A}^{n + 2}_S$ is $m$-pseudo-coherent. Since this is also the pushforward of $E|_U$ to $\mathbf{A}^{n + 2}_S$ we conclude the lemma is true.
The implication (2) $\Rightarrow$ (1) is Lemma \ref{lemma-localize-relative-pseudo-coherent}. Assume (1). Say $S = \Spec(R)$ and $X = \Spec(A)$ and $U_i = D(f_i)$. Write $1 = \sum f_ig_i$ in $A$. Consider the surjections $$ R[x_i, y_i, z_i] \to R[x_i, y_i, z_i]/(\sum y_iz_i - 1) \to A. $$ which sends $y_i$ to $f_i$ and $z_i$ to $g_i$. Note that $R[x_i, y_i, z_i]/(\sum y_iz_i - 1)$ is pseudo-coherent as an $R[x_i, y_i, z_i]$-module. Thus it suffices to prove that the pushforward of $E$ to $T = \Spec(R[x_i, y_i, z_i]/(\sum y_iz_i - 1))$ is $m$-pseudo-coherent, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-closed-push-pseudo-coherent}. For each $i_0$ it suffices to prove the restriction of this pushforward to $W_{i_0} = \Spec(R[x_i, y_i, z_i, 1/y_{i_0}]/(\sum y_iz_i - 1))$ is $m$-pseudo-coherent. Note that there is a commutative diagram $$ \xymatrix{ X \ar[d] & U_{i_0} \ar[l] \ar[d] \\ T & W_{i_0} \ar[l] } $$ which implies that the pushforward of $E$ to $T$ restricted to $W_{i_0}$ is the pushforward of $E|_{U_{i_0}}$ to $W_{i_0}$. Since $R[x_i, y_i, z_i, 1/y_{i_0}]/(\sum y_iz_i - 1)$ is isomorphic to a polynomial ring over $R$ this proves what we want.