hcju/stacksqa huggingface dataset:
By Proposition \ref{proposition-separated-closed-in-finite-presentation} we can find a closed immersion $X \to Y$ with $Y$ separated and of finite presentation over $S$. By Lemma \ref{lemma-chow-finite-type} we can find a diagram $$ \xymatrix{ Y \ar[rd] & Y' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_S \ar[dl] \\ & S & } $$ where $Y' \to \mathbf{P}^n_S$ is an immersion, and $\pi : Y' \to Y$ is proper and surjective. By Lemma \ref{lemma-closed-is-limit-closed-and-finite-presentation} we can write $X = \lim X_i$ with $X_i \to Y$ a closed immersion of finite presentation. Denote $X'_i \subset Y'$, resp.\ $X' \subset Y'$ the scheme theoretic inverse image of $X_i \subset Y$, resp.\ $X \subset Y$. Then $\lim X'_i = X'$. Since $X' \to S$ is proper (Morphisms, Lemmas \ref{morphisms-lemma-composition-proper}), we see that $X' \to \mathbf{P}^n_S$ is a closed immersion (Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}). Hence for $i$ large enough we find that $X'_i \to \mathbf{P}^n_S$ is a closed immersion by Lemma \ref{lemma-eventually-closed-immersion}. Thus $X'_i$ is proper over $S$. For such $i$ the morphism $X_i \to S$ is proper by Morphisms, Lemma \ref{morphisms-lemma-image-proper-is-proper}.
By Lemma \ref{lemma-proper-limit-of-proper-finite-presentation} we can write $X = \lim_{k \in K} X_k$ with $X_k \to S$ proper and of finite presentation. Next, by absolute Noetherian approximation (Proposition \ref{proposition-approximate}) we can write $S = \lim_{j \in J} S_j$ with $S_j$ of finite type over $\mathbf{Z}$. For each $k$ there exists a $j$ and a morphism $X_{k, j} \to S_j$ of finite presentation with $X_k \cong S \times_{S_j} X_{k, j}$ as schemes over $S$, see Lemma \ref{lemma-descend-finite-presentation}. After increasing $j$ we may assume $X_{k, j} \to S_j$ is proper, see Lemma \ref{lemma-eventually-proper}. The set $I$ will be consist of these pairs $(k, j)$ and the corresponding morphism is $X_{k, j} \to S_j$. For every $k' \geq k$ we can find a $j' \geq j$ and a morphism $X_{j', k'} \to X_{j, k}$ over $S_{j'} \to S_j$ whose base change to $S$ gives the morphism $X_{k'} \to X_k$ (follows again from Lemma \ref{lemma-descend-finite-presentation}). These morphisms form the transition morphisms of the system. Some details omitted.
Choose a closed immersion $Y \to Y'$ with $Y'$ proper and of finite presentation over $X$, see Lemma \ref{lemma-proper-limit-of-proper-finite-presentation}. Then choose an $i$ and a proper morphism $Y'_i \to X_i$ such that $Y' = X \times_{X_i} Y'_i$. This is possible by Lemmas \ref{lemma-descend-finite-presentation} and \ref{lemma-eventually-proper}. Then after replacing $i$ by a larger index we have that $Y \to Y'_i$ is a closed immersion, see Lemma \ref{lemma-finite-type-eventually-closed}.
We may replace $X_0$ by the scheme theoretic support of $\mathcal{F}_0$. By Morphisms, Lemma \ref{morphisms-lemma-support-finite-type} this guarantees that $X_i$ is the support of $\mathcal{F}_i$ and $X$ is the support of $\mathcal{F}$. Then, if $Z \subset X$ denotes the scheme theoretic support of $\mathcal{F}$, we see that $Z \to X$ is a universal homeomorphism. We conclude that $X \to Y$ is proper as this is true for $Z \to Y$ by assumption, see Morphisms, Lemma \ref{morphisms-lemma-image-proper-is-proper}. By Lemma \ref{lemma-eventually-proper} we see that $X_i \to Y$ is proper for some $i$. Then it follows that the scheme theoretic support $Z_i$ of $\mathcal{F}_i$ is proper over $Y$ by Morphisms, Lemmas \ref{morphisms-lemma-closed-immersion-proper} and \ref{morphisms-lemma-composition-proper}.
It is clear that (1) implies (2). Let us prove that (2) implies (1). Suppose that the base change $X_T \to T$ is not closed for some scheme $T$ over $S$. By Schemes, Lemma \ref{schemes-lemma-quasi-compact-closed} this means that there exists some specialization $t_1 \leadsto t$ in $T$ and a point $\xi \in X_T$ mapping to $t_1$ such that $\xi$ does not specialize to a point in the fibre over $t$. Set $Z = \overline{\{\xi\}} \subset X_T$. Then $Z \cap X_t = \emptyset$. Apply Lemma \ref{lemma-separate}. We find an open neighbourhood $V \subset T$ of $t$, a commutative diagram $$ \xymatrix{ V \ar[d] \ar[r]_a & T' \ar[d]^b \\ T \ar[r]^g & S, } $$ and a closed subscheme $Z' \subset X_{T'}$ such that \begin{enumerate} \item the morphism $b : T' \to S$ is locally of finite presentation, \item with $t' = a(t)$ we have $Z' \cap X_{t'} = \emptyset$, and \item $Z \cap X_V$ maps into $Z'$ via the morphism $X_V \to X_{T'}$. \end{enumerate} Clearly this means that $X_{T'} \to T'$ maps the closed subset $Z'$ to a subset of $T'$ which contains $a(t_1)$ but not $t' = a(t)$. Since $a(t_1) \leadsto a(t) = t'$ we conclude that $X_{T'} \to T'$ is not closed. Hence we have shown that $X \to S$ not universally closed implies that $X_{T'} \to T'$ is not closed for some $T' \to S$ which is locally of finite presentation. In order words (2) implies (1). \medskip\noindent Assume that $\mathbf{A}^n \times X \to \mathbf{A}^n \times S$ is closed for every integer $n$. We want to prove that $X_T \to T$ is closed for every scheme $T$ which is locally of finite presentation over $S$. We may of course assume that $T$ is affine and maps into an affine open $V$ of $S$ (since $X_T \to T$ being a closed is local on $T$). In this case there exists a closed immersion $T \to \mathbf{A}^n \times V$ because $\mathcal{O}_T(T)$ is a finitely presented $\mathcal{O}_S(V)$-algebra, see Morphisms, Lemma \ref{morphisms-lemma-locally-finite-presentation-characterize}. Then $T \to \mathbf{A}^n \times S$ is a locally closed immersion. Hence we get a cartesian diagram $$ \xymatrix{ X_T \ar[d]_{f_T} \ar[r] & \mathbf{A}^n \times X \ar[d]^{f_n} \\ T \ar[r] & \mathbf{A}^n \times S } $$ of schemes where the horizontal arrows are locally closed immersions. Hence any closed subset $Z \subset X_T$ can be written as $X_T \cap Z'$ for some closed subset $Z' \subset \mathbf{A}^n \times X$. Then $f_T(Z) = T \cap f_n(Z')$ and we see that if $f_n$ is closed, then also $f_T$ is closed.
By the non-Noetherian version of this lemma (Morphisms, Lemma \ref{morphisms-lemma-reach-points-scheme-theoretic-image}) there exists a point $x \in X$ such that $f(x)$ specializes to $y$. We may replace $x$ by any point specializing to $x$, hence we may assume that $x$ is a generic point of an irreducible component of $X$. This produces a ring map $\mathcal{O}_{Y, y} \to \kappa(x)$ (see Schemes, Section \ref{schemes-section-points}). Let $R \subset \kappa(x)$ be the image. Then $R$ is Noetherian as a quotient of the Noetherian local ring $\mathcal{O}_{Y, y}$. On the other hand, the extension $\kappa(x)$ is a finitely generated extension of the fraction field of $R$ as $f$ is of finite type. Thus there exists a discrete valuation ring $A \subset \kappa(x)$ with fraction field $\kappa(x)$ dominating $R$ by Algebra, Lemma \ref{algebra-lemma-exists-dvr}. Then $$ \xymatrix{ \Spec(\kappa(x)) \ar[d] \ar[rrr] & & & X \ar[d] \\ \Spec(A) \ar[r] & \Spec(R) \ar[r] & \Spec(\mathcal{O}_{Y, y}) \ar[r] & Y } $$ gives the desired diagram.
(1) implies (2) implies (3) implies (4).  We will now show (4) implies (1).  As in the proof of Lemma \ref{lemma-Noetherian-dvr-valuative-separation}, we can reduce to the case $S$ is affine, since properness is local on the base, and if $X \to S$ satisfies (4), then $X_\alpha \to S_\alpha$ does as well for open $S_\alpha \subset S$ and $X_\alpha = f^{-1}(S_\alpha)$. \medskip\noindent Now $S$ is a Noetherian scheme, and so $X$ is as well, since $X \to S$ is of finite type.  Now we may use Chow's lemma (Cohomology of Schemes, Lemma \ref{coherent-lemma-chow-Noetherian}) to get a surjective, proper, birational $X' \to X$ and an immersion $X' \to \mathbf{P}^n_S$.  We wish to show $X \to S$ is universally closed.  As in the proof of Lemma \ref{lemma-limited-base-change}, it is enough to check that $X' \to \mathbf{P}^n_S$ is a closed immersion. For the sake of contradiction, assume that $X' \to \mathbf{P}^n_S$ is not a closed immersion.  Then there is some $y \in \mathbf{P}^n_S$ that is in the closure of the image of $X'$, but is not in the image.  So $y$ is in the closure of the image of an irreducible component $X_0'$ of $X'$, but not in the image. Let $\bar X_0' \subset \mathbf{P}^n_S$ be the closure of the image of $X_0'$. As $X' \to \mathbf{P}^n_S$ is an immersion of Noetherian schemes, the morphism $X'_0 \to \bar X_0'$ is open and dense. By Algebra, Lemma \ref{algebra-lemma-exists-dvr} or Properties, Lemma \ref{properties-lemma-locally-Noetherian-specialization-dvr} we can find a discrete valuation ring $A$ dominating $\mathcal{O}_{\bar X_0', y}$ and with identical field of fractions $K$. It is clear that $K$ is the residue field at the generic point of $X_0'$. Thus the solid commutative diagram \begin{equation} \label{equation-solid} \xymatrix{ \Spec K \ar[r] \ar[d] & X' \ar [r] \ar[d] & \mathbf{P}^n_S \ar[d] \\ \Spec A \ar@{-->}[r] \ar@{-->}[ru] \ar[urr] & X \ar[r] & S\\ } \end{equation} Note that the closed point of $A$ maps to $y \in \mathbf{P}^n_S$.  By construction, there does not exist a set theoretic lift to $X'$. As $X' \to X$ is birational, the image of $X'_0$ in $X$ is an irreducible component $X_0$ of $X$ and $K$ is also identified with the function field of $X_0$. Hence, as $X \to S$ is assumed to satisfy (4), the dotted arrow $\Spec(A) \to X$ exists. Since $X' \to X$ is proper, the dotted arrow lifts to the dotted arrow $\Spec(A) \to X'$ (use Schemes, Proposition \ref{schemes-proposition-characterize-universally-closed}). We can compose this with the immersion $X' \to \mathbf{P}^n_S$ to obtain another morphism (not depicted in the diagram) from $\Spec(A) \to \mathbf{P}^n_S$.  Since $\mathbf{P}^n_S$ is proper over $S$, it satisfies (2), and so these two morphisms agree.  This is a contradiction, for we have constructed the forbidden lift of our original map $\Spec(A) \to \mathbf{P}^n_S$ to $X'$.
The equivalence of (1) and (2) is a special case of Lemma \ref{lemma-test-universally-closed}. The equivalence of (1) and (3) is a special case of Schemes, Proposition \ref{schemes-proposition-characterize-universally-closed}. Trivially (3) implies (4). Thus all we have to do is prove that (4) implies (2). We will prove that $\mathbf{A}^n \times X \to \mathbf{A}^n \times S$ is closed by the criterion of Schemes, Lemma \ref{schemes-lemma-quasi-compact-closed}. Pick $n$ and a specialization $z \leadsto z'$ of points in $\mathbf{A}^n \times S$ and a point $y \in \mathbf{A}^n \times X$ lying over $z$. Note that $\kappa(y)$ is a finitely generated field extension of $\kappa(z)$ as $\mathbf{A}^n \times X \to \mathbf{A}^n \times S$ is of finite type. Hence by Properties, Lemma \ref{properties-lemma-locally-Noetherian-specialization-dvr} or Algebra, Lemma \ref{algebra-lemma-exists-dvr} implies that there exists a discrete valuation ring $A \subset \kappa(y)$ with fraction field $\kappa(z)$ dominating the image of $\mathcal{O}_{\mathbf{A}^n \times S, z'}$ in $\kappa(z)$. This gives a commutative diagram $$ \xymatrix{ \Spec(\kappa(y)) \ar[r] \ar[d] & \mathbf{A}^n \times X \ar[d] \ar[r] & X \ar[d] \\ \Spec(A) \ar[r] & \mathbf{A}^n \times S \ar[r] & S } $$ Now property (4) implies that there exists a morphism $\Spec(A) \to X$ which fits into this diagram. Since we already have the morphism $\Spec(A) \to \mathbf{A}^n$ from the left lower horizontal arrow we also get a morphism $\Spec(A) \to \mathbf{A}^n \times X$ fitting into the left square. Thus the image $y' \in \mathbf{A}^n \times X$ of the closed point is a specialization of $y$ lying over $z'$. This proves that specializations lift along $\mathbf{A}^n \times X \to \mathbf{A}^n \times S$ and we win.
There is an immediate reduction to the case where $S$ is affine. Then $U$ is quasi-compact. Let $U = U_1 \cup \ldots \cup U_n$ be an affine open covering. We may replace $U$ by $U_1 \amalg \ldots \amalg U_n$ without changing the assumptions, hence we may assume $U$ is affine. Thus we can find an open immersion $U \to Y$ over $X$ with $Y$ proper over $X$. (First put $U$ inside $\mathbf{A}^n_X$ using Morphisms, Lemma \ref{morphisms-lemma-quasi-affine-finite-type-over-S} and then take the closure inside $\mathbf{P}^n_X$, or you can directly use Morphisms, Lemma \ref{morphisms-lemma-quasi-projective-open-projective}.) We can assume $U$ is dense in $Y$ (replace $Y$ by the scheme theoretic closure of $U$ if necessary, see Morphisms, Section \ref{morphisms-section-scheme-theoretic-closure}). Note that $g : Y \to X$ is surjective as the image is closed and contains the dense subset $h(U)$. We will show that $Y \to S$ is proper. This will imply that $X \to S$ is proper by Morphisms, Lemma \ref{morphisms-lemma-image-proper-is-proper} thereby finishing the proof. To show that $Y \to S$ is proper we will use part (4) of Lemma \ref{lemma-Noetherian-dvr-valuative-proper}. To do this consider a diagram $$ \xymatrix{ \Spec(K) \ar[r]_y \ar[d] & Y \ar[d]^{f \circ g} \\ \Spec(A) \ar[r] \ar@{..>}[ru] & S } $$ where $A$ is a discrete valuation ring with fraction field $K$ and where $y : \Spec(K) \to Y$ is the inclusion of a generic point. We have to show there exists a unique dotted arrow. Uniqueness holds by the converse to the valuative criterion for separatedness (Schemes, Lemma \ref{schemes-lemma-separated-implies-valuative}) since $Y \to S$ is separated as the composition of the separated morphisms $Y \to X$ and $X \to S$ (Schemes, Lemma \ref{schemes-lemma-separated-permanence}). Existence can be seen as follows. As $y$ is a generic point of $Y$, it is contained in $U$. By assumption of the lemma there exists a morphism $a : \Spec(A) \to X$ such that $$ \xymatrix{ \Spec(K) \ar[r]_y \ar[d] & U \ar[r] & X \ar[d]^f \\ \Spec(A) \ar[rr] \ar[rru]^a & & S } $$ is commutative. Then since $Y \to X$ is proper, we can apply the valuative criterion for properness (Morphisms, Lemma \ref{morphisms-lemma-characterize-proper}) to find a morphism $b : \Spec(A) \to Y$ such that $$ \xymatrix{ \Spec(K) \ar[r]_y \ar[d] & Y \ar[d]^g \\ \Spec(A) \ar[r]^a \ar[ru]^b & X } $$ is commutative. This finishes the proof since $b$ can serve as the dotted arrow above.
We will apply Lemma \ref{lemma-refined-valuative-criterion-proper} to the morphisms $U \to X$ and $\Delta : X \to X \times_S X$. We check the conditions. Observe that $\Delta$ is quasi-compact by Properties, Lemma \ref{properties-lemma-locally-Noetherian-quasi-separated} (and Schemes, Lemma \ref{schemes-lemma-compose-after-separated}). Of course $\Delta$ is locally of finite type and separated (true for any diagonal morphism). Finally, suppose given a commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r]^h & X \ar[d]^\Delta \\ \Spec(A) \ar[rr]^{(a, b)} \ar@{-->}[rru] & & X \times_S X } $$ where $A$ is a discrete valuation ring with field of fractions $K$. Then $a$ and $b$ give two dotted arrows in the diagram of the lemma and have to be equal. Hence as dotted arrow we can use $a = b$ which gives existence. This finishes the proof.
For each $i$ let $U_i = \{x \in X_i \mid \dim_x((X_i)_{f_i(x)}) \leq d\}$. This is an open subset of $X_i$, see Morphisms, Lemma \ref{morphisms-lemma-openness-bounded-dimension-fibres}. Set $Z_i = X_i \setminus U_i$ (with reduced induced scheme structure). We have to show that $Z_i = \emptyset$ for some $i$. If not, then $Z = \lim Z_i \not = \emptyset$, see Lemma \ref{lemma-limit-nonempty}. Say $z \in Z$ is a point. Note that $Z \subset X$ is a closed subscheme. Set $s = f(z)$. For each $i$ let $s_i \in S_i$ be the image of $s$. We remark that $Z_s$ is the limit of the schemes $(Z_i)_{s_i}$ and $Z_s$ is also the limit of the schemes $(Z_i)_{s_i}$ base changed to $\kappa(s)$. Moreover, all the morphisms $$ Z_s \longrightarrow (Z_{i'})_{s_{i'}} \times_{\Spec(\kappa(s_{i'}))} \Spec(\kappa(s)) \longrightarrow (Z_i)_{s_i} \times_{\Spec(\kappa(s_i))} \Spec(\kappa(s)) \longrightarrow X_s $$ are closed immersions by assumption (4). Hence $Z_s$ is the scheme theoretic intersection of the closed subschemes $(Z_i)_{s_i} \times_{\Spec(\kappa(s_i))} \Spec(\kappa(s))$ in $X_s$. Since all the irreducible components of the schemes $(Z_i)_{s_i} \times_{\Spec(\kappa(s_i))} \Spec(\kappa(s))$ have dimension $> d$ and contain $z$ we conclude that $Z_s$ contains an irreducible component of dimension $> d$ passing through $z$ which contradicts the fact that $Z_s \subset X_s$ and $\dim(X_s) \leq d$.
By Lemma \ref{lemma-limit-dimension} we may assume all fibres of $f_0$ have dimension $\leq d$. By Morphisms, Lemma \ref{morphisms-lemma-openness-bounded-dimension-fibres-finite-presentation} the set $U_0 \subset X_0$ of points $x \in X_0$ such that the dimension of the fibre of $X_0 \to Y_0$ at $x$ is $\leq d - 1$ is open and retrocompact in $X_0$. Hence the complement $E = X_0 \setminus U_0$ is constructible. Moreover the image of $X \to X_0$ is contained in $E$ by Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-after-base-change}. Thus for $i \gg 0$ we have that the image of $X_i \to X_0$ is contained in $E$ (Lemma \ref{lemma-limit-contained-in-constructible}). Then all fibres of $X_i \to Y_i$ have dimension $d$ by the aforementioned Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-after-base-change}.
By Proposition \ref{proposition-approximate} we can write $S = \lim S_i$ as the limit of a directed inverse system of Noetherian schemes with affine transition maps. By Lemma \ref{lemma-descend-finite-presentation} we may assume that there exist a system of morphisms $f_i : X_i \to S_i$ of finite presentation such that $X_{i'} = X_i \times_{S_i} S_{i'}$ for all $i' \geq i$ and such that $X = X_i \times_{S_i} S$. Let $Z_i \subset X_i$ be the scheme theoretic image of $Z \to X \to X_i$. Then for $i' \geq i$ the morphism $X_{i'} \to X_i$ maps $Z_{i'}$ into $Z_i$ and the induced morphism $Z_{i'} \to Z_i \times_{S_i} S_{i'}$ is a closed immersion. By Lemma \ref{lemma-limit-dimension} we see that the dimension of the fibres of $Z_i \to S_i$ all have dimension $\leq d$ for a suitable $i \in I$. Fix such an $i$ and set $Z' = Z_i \times_{S_i} S \subset X$. Since $S_i$ is Noetherian, we see that $X_i$ is Noetherian, and hence the morphism $Z_i \to X_i$ is of finite presentation. Therefore also the base change $Z' \to X$ is of finite presentation. Moreover, the fibres of $Z' \to S$ are base changes of the fibres of $Z_i \to S_i$ and hence have dimension $\leq d$.
The module $R^df_*\mathcal{F}$ is quasi-coherent by Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images}. The question is local on $Y$ hence we may assume $Y$ is affine. Say $Y = \Spec(R)$. Then it suffices to prove that $H^d(X, \mathcal{F})$ is a finite $R$-module. \medskip\noindent By Lemma \ref{lemma-proper-limit-of-proper-finite-presentation} we can write $X = \lim X_i$ as a cofiltered limit with $X_i \to Y$ proper and of finite presentation and such that both $X \to X_i$ and transition morphisms are closed immersions. For some $i$ we have that $X_i \to Y$ has fibres of dimension $\leq d$, see Lemma \ref{lemma-limit-dimension}. We have $R^pf_*\mathcal{F} = R^pf_{i, *}(X \to X_i)_*\mathcal{F}$ by Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing} and Leray (Cohomology, Lemma \ref{cohomology-lemma-relative-Leray}). Thus we may replace $X$ by $X_i$ and reduce to the case discussed in the next paragraph. \medskip\noindent Assume $Y$ is affine and $f : X \to Y$ is proper and of finite presentation and all fibres have dimension $\leq d$. We can write $\mathcal{F}$ as a quotient of a finitely presented $\mathcal{O}_X$-module $\mathcal{F}'$, see Properties, Lemma \ref{properties-lemma-finite-directed-colimit-surjective-maps}. The map $H^d(X, \mathcal{F}') \to H^d(X, \mathcal{F})$ is surjective, as we have $H^{d + 1}(X, \Ker(\mathcal{F}' \to \mathcal{F})) = 0$ by the vanishing of higher cohomology seen in Lemma \ref{lemma-higher-direct-images-zero-above-dimension-fibre} (or its proof). Thus we reduce to the case discussed in the next paragraph. \medskip\noindent Assume $Y = \Spec(R)$ is affine and $f : X \to Y$ is proper and of finite presentation and all fibres have dimension $\leq d$ and $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation. Write $Y = \lim_{i \in I} Y_i$ as a cofiltered limit of affine schemes with $Y_i = \Spec(R_i)$ the spectrum of a Noetherian ring (for example a finite type $\mathbf{Z}$-algebra). We can choose an element $0 \in I$ and a finite type morphism $X_0 \to Y_0$ such that $X \cong Y \times_{Y_0} X_0$, see Lemma \ref{lemma-descend-finite-presentation}. After increasing $0$ we may assume $X_0 \to Y_0$ is proper (Lemma \ref{lemma-eventually-proper}) and that the fibres of $X_0 \to Y_0$ have dimension $\leq d$ (Lemma \ref{lemma-limit-dimension}). After increasing $0$ we can assume there is a coherent $\mathcal{O}_{X_0}$-module $\mathcal{F}_0$ which pulls back to $\mathcal{F}$, see Lemma \ref{lemma-descend-modules-finite-presentation}. By Lemma \ref{lemma-top-cohomology-functor} we have $$ H^d(X, \mathcal{F}) = H^d(X_0, \mathcal{F}_0) \otimes_{R_0} R $$ This finishes the proof because the cohomology module $H^d(X_0, \mathcal{F}_0)$ is finite by Cohomology of Schemes, Lemma \ref{coherent-lemma-proper-over-affine-cohomology-finite}.
Let $W \subset S$ be an open neighbourhood of $s$. By glueing of relative schemes, see Constructions, Section \ref{constructions-section-relative-glueing}, the functor $$ \left\{ \begin{matrix} X \to S\text{ of finite presentation} \end{matrix} \right\} \longrightarrow \left\{ \vcenter{ \xymatrix{ X' \ar[d] & Y' \ar[d] \ar[l] \ar[r] & Y \ar[d] \\ U & W \setminus \{s\} \ar[l] \ar[r] & W } } \right\} $$ is an equivalence of categories. We have $\mathcal{O}_{S, s} = \colim \mathcal{O}_W(W)$ where $W$ runs over the affine open neighbourhoods of $s$. Hence $\Spec(\mathcal{O}_{S, s}) = \lim W$ where $W$ runs over the affine open neighbourhoods of $s$. Thus the category of schemes of finite presentation over $\Spec(\mathcal{O}_{S, s})$ is the limit of the category of schemes of finite presentation over $W$ where $W$ runs over the affine open neighbourhoods of $s$, see Lemma \ref{lemma-descend-finite-presentation}. For every affine open $s \in W$ we see that $U \cap W$ is quasi-compact as $U \to S$ is quasi-compact. Hence $V = \lim W \cap U = \lim W \setminus \{s\}$ is a limit of quasi-compact and quasi-separated schemes (see Lemma \ref{lemma-directed-inverse-system-has-limit}). Thus also the category of schemes of finite presentation over $V$ is the limit of the categories of schemes of finite presentation over $W \cap U$ where $W$ runs over the affine open neighbourhoods of $s$. The lemma follows formally from a combination of these results.
You can either prove this by redoing the proof of Lemma \ref{lemma-glueing-near-closed-point} using Lemma \ref{lemma-descend-modules-finite-presentation} or you can deduce it from Lemma \ref{lemma-glueing-near-closed-point} using the equivalence between quasi-coherent modules and ``vector bundles'' from Constructions, Section \ref{constructions-section-vector-bundle}. We omit the details.