hcju/stacksqa huggingface dataset:
Let $\mathfrak p \subset A$ be a closed point of $\Spec(A) \setminus V(J)$. We want to show that $I \subset \mathfrak p$. If not, then some $f \in I$ maps to a nonzero element of $A/\mathfrak p$. Note that $V(J) \cap \Spec(A/\mathfrak p)$ is the set of non-generic points. Hence by Lemma \ref{lemma-big-intersection-is-zero} applied to the collection of ideals $J^nA/\mathfrak p$ we conclude that the image of $f$ is zero in $A/\mathfrak p$.
It suffices to show that $M \otimes_A I/I_2 \to IM/I_2M$ is injective, see Algebra, Lemma \ref{algebra-lemma-what-does-it-mean-again}. This is true over $U$ by assumption (2). Thus it suffices to show that $M \otimes_A I/I_2$ injects into its sections over $U$. We have $M \otimes_A I/I_2 = M/IM \otimes_A I/I_2$ and $M/IM$ is a filtered colimit of finite free $A/I$-modules (Algebra, Theorem \ref{algebra-theorem-lazard}). Hence it suffices to show that $I/I_2$ injects into its sections over $U$, which follows from the construction of $I_2$.
Let $T \to S$ be any morphism. Then $h_T : X'_T \to X_T$ is \'etale as a base change of the \'etale morphism $g$. For $t \in T$ denote $Z \subset X_t$ the set of points where $\mathcal{F}_T$ is not flat over $T$, and similarly denote $Z' \subset X'_t$ the set of points where $\mathcal{F}'_T$ is not flat over $T$. As $\mathcal{F}'_T = h_T^*\mathcal{F}_T$ we see that $Z' = h_t^{-1}(Z)$, see Morphisms, Lemma \ref{morphisms-lemma-flat-permanence}. Hence $Z' \to Z$ is an \'etale morphism, so $\dim(Z') \leq \dim(Z)$ (for example by Descent, Lemma \ref{descent-lemma-dimension-at-point-local} or just because an \'etale morphism is smooth of relative dimension $0$). This implies that $F_n \subset F_n'$. \medskip\noindent Finally, suppose that $h(X') \supset \text{Ass}_{X/S}(\mathcal{F})$ and that $T \to S$ is a morphism such that $F_n'(T)$ is nonempty, i.e., such that $\mathcal{F}'_T$ is flat in dimensions $\geq n$ over $T$. Pick a point $t \in T$ and let $Z \subset X_t$ and $Z' \subset X'_t$ be as above. To get a contradiction assume that $\dim(Z) \geq n$. Pick a generic point $\xi \in Z$ corresponding to a component of dimension $\geq n$. Let $x \in \text{Ass}_{X_t}(\mathcal{F}_t)$ be a generalization of $\xi$. Then $x$ maps to a point of $\text{Ass}_{X/S}(\mathcal{F})$ by Divisors, Lemma \ref{divisors-lemma-base-change-relative-assassin} and Remark \ref{divisors-remark-base-change-relative-assassin}. Thus we see that $x$ is in the image of $h_T$, say $x = h_T(x')$ for some $x' \in X'_T$. But $x' \not \in Z'$ as $x \leadsto \xi$ and $\dim(Z') < n$. Hence $\mathcal{F}'_T$ is flat over $T$ at $x'$ which implies that $\mathcal{F}_T$ is flat at $x$ over $T$ (by Morphisms, Lemma \ref{morphisms-lemma-flat-permanence}). Since this holds for every such $x$ we conclude that $\mathcal{F}_T$ is flat over $T$ at $\xi$ by Theorem \ref{theorem-check-flatness-at-associated-points} which is the desired contradiction.
As $X$ is affine and $\mathcal{F}$ is quasi-coherent of finite presentation we know that $\mathcal{F}$ can be generated by $c \geq 0$ elements. Then $\dim_{\kappa(x)}(\mathcal{F}_x \otimes \kappa(x))$ in any point $x \in X$ never exceeds $c$. In particular $H_p = \emptyset$ for $p > c$. Moreover, note that there certainly is an inclusion $\coprod H_p \to F_d$. Having said this the content of the lemma is that, if a base change $\mathcal{F}_T$ is flat in dimensions $\geq d$ over $T$ and if $t \in T$, then $\mathcal{F}_T$ is free of some rank $r$ in an open neighbourhood $U \subset X_T$ of the unique generic point $\xi$ of $X_t$. Namely, then $H_r$ contains the image of $U$ which is an open neighbourhood of $t$. The existence of $U$ follows from More on Morphisms, Lemma \ref{more-morphisms-lemma-flat-and-free-at-point-fibre}.
Proof of (2). Write $E_n = E|_{X_n}$ and $N_n = R\Hom_{X_n}(E_n, \mathcal{F}_n)$. Recall that $R\Hom_{X_n}(-, -)$ is equal to $R\Gamma(X_n, R\SheafHom(-, -))$, see Cohomology, Section \ref{cohomology-section-global-RHom}. Hence by Derived Categories of Schemes, Lemma \ref{perfect-lemma-base-change-RHom-perfect} we see that $N_n$ is a perfect object of $D(A_n)$ whose formation commutes with base change. Thus the maps $N_n \otimes_{A_n}^\mathbf{L} A_{n - 1} \to N_{n - 1}$ coming from $\varphi_n$ are isomorphisms. By More on Algebra, Lemma \ref{more-algebra-lemma-Rlim-perfect-gives-perfect} we find that $R\lim N_n$ is perfect and that its base change back to $A_n$ recovers $N_n$. On the other hand, the exact functor $R\Hom_X(E, -) : D_\QCoh(\mathcal{O}_X) \to D(A)$ of triangulated categories commutes with products and hence with derived limits, whence $$ R\Hom_X(E, K) = R\lim R\Hom_X(E, \mathcal{F}_n) = R\lim R\Hom_X(E_n, \mathcal{F}_n) = R\lim N_n $$ This proves (2). To see that (1) holds, translate it into (2) using Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}.
Combinging Lemma \ref{lemma-compute-against-perfect} and Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-enough} we see that $R\Gamma(X, K \otimes^\mathbf{L} E)$ is pseudo-coherent in $D(A)$ for all pseudo-coherent $E$ in $D(\mathcal{O}_X)$. Thus the lemma follows from More on Morphisms, Lemma \ref{more-morphisms-lemma-characterize-pseudo-coherent}.
Fix $n$. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-computing-sections-as-colim} there exists a system of perfect complexes $E_m$ on $X$ such that $R\Gamma(U, K) = \text{hocolim} R\Gamma(X, K \otimes^\mathbf{L} E_m)$. In fact, this formula holds not just for $K$ but for every object of $D_\QCoh(\mathcal{O}_X)$. Applying this to $\mathcal{F}_n$ we obtain \begin{align*} R\Gamma(U_n, \mathcal{F}_n) & = R\Gamma(U, \mathcal{F}_n) \\ & = \text{hocolim}_m R\Gamma(X, \mathcal{F}_n \otimes^\mathbf{L} E_m) \\ & = \text{hocolim}_m R\Gamma(X_n, \mathcal{F}_n \otimes^\mathbf{L} E_m|_{X_n}) \end{align*} Using Lemma \ref{lemma-compute-against-perfect} and the fact that $- \otimes_A^\mathbf{L} A_n$ commutes with homotopy colimits we obtain the result.
Fix $n \geq 1$. By construction there is a canonical map $K \to \mathcal{F}_n$ in $D_\QCoh(\mathcal{O}_X)$ and hence a canonical map $H^0(K) \to \mathcal{F}_n$ of quasi-coherent sheaves. This explains the meaning of part (3). \medskip\noindent Let $x \in X_0$ be a point. We will find an open neighbourhood $W$ of $x$ such that (1), (2), and (3) are true. Since $X_0$ is quasi-compact this will prove the lemma. Let $U \subset X$ be an affine open neighbourhood of $x$. Say $U = \Spec(B)$. Choose a surjection $P \to B$ with $P$ smooth over $A$. By Lemma \ref{lemma-relative-pseudo-coherence} and the definition of relative pseudo-coherence there exists a bounded above complex $F^\bullet$ of finite free $P$-modules representing $Ri_*K$ where $i : U \to \Spec(P)$ is the closed immersion induced by the presentation. Let $M_n$ be the $B$-module corresponding to $\mathcal{F}_n|_U$. By Lemma \ref{lemma-compute-over-affine} $$ H^i(F^\bullet \otimes_A A_n) = \left\{ \begin{matrix} 0 & \text{if} & i \not = 0 \\ M_n & \text{if} & i = 0 \end{matrix} \right. $$ Let $i$ be the maximal index such that $F^i$ is nonzero. If $i \leq 0$, then (1), (2), and (3) are true. If not, then $i > 0$ and we see that the rank of the map $$ F^{i - 1} \to F^i $$ in the point $x$ is maximal. Hence in an open neighbourhood of $x$ inside $\Spec(P)$ the rank is maximal. Thus after replacing $P$ by a principal localization we may assume that the displayed map is surjective. Since $F^i$ is finite free we may choose a splitting $F^{i - 1} = F' \oplus F^i$. Then we may replace $F^\bullet$ by the complex $$ \ldots \to F^{i - 2} \to F' \to 0 \to \ldots $$ and we win by induction on $i$.
Fix $n \geq 1$. Let $I_n = \Ker(A \to A_n)$. By More on Algebra, Lemma \ref{more-algebra-lemma-limit-henselian} the pair $(A, I_n)$ is henselian. Let $Z \subset W$ be the support of $\mathcal{F}$. This is a closed subset as $\mathcal{F}$ is of finite presentation. By part (3) of Lemma \ref{lemma-finitely-presented} we see that $Z \times_{\Spec(A)} \Spec(A_n)$ is equal to the support of $\mathcal{F}_n$ and hence proper over $\Spec(A/I)$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-split-off-proper-part-henselian} we can write $Z = Z_1 \amalg Z_2$ with $Z_1, Z_2$ open and closed in $Z$, with $Z_1$ proper over $A$, and with $Z_1 \times_{\Spec(A)} \Spec(A/I_n)$ equal to the support of $\mathcal{F}_n$. In other words, $Z_2$ does not meet $X_0$. Hence after replacing $W$ by $W \setminus Z_2$ we obtain the lemma.
Set $S_n = \Spec(A_n)$. Let $T_0 \subset T$ be the common image of the factorizations $S_n \to T$. Then $T_0$ is quasi-compact. Let $T' \subset T$ be a quasi-compact open containing $T_0$. Then $S_n \to T$ factors through $T'$. If we can show that $T' = S$, then $T' = T = S$. Hence we may assume $T$ is quasi-compact. \medskip\noindent Assume $T$ is quasi-compact. In this case $T \to S$ is separated and quasi-finite (Morphisms, Lemma \ref{morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}). Using Zariski's Main Theorem (in the form of More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite}) we choose a factorization $T \to W \to S$ with $W \to S$ finite and $T \to W$ an open immersion. Write $W = \Spec(B)$. The (unique) factorizations $S_n \to T$ may be viewed as morphisms into $W$ and we obtain $$ A \longrightarrow B \longrightarrow \lim A_n = A $$ Consider the morphism $h : S = \Spec(A) \to \Spec(B) = W$ coming from the arrow on the right. Then $$ T \times_{W, h} S $$ is an open subscheme of $S$ containing the image of $S_n \to S$ for all $n$. To finish the proof it suffices to show that any open $U \subset S$ containing the image of $S_n \to S$ for some $n \geq 1$ is equal to $S$. This is true because $(A, \Ker(A \to A_n))$ is a henselian pair (More on Algebra, Lemma \ref{more-algebra-lemma-limit-henselian}) and hence every closed point of $S$ is contained in the image of $S_n \to S$.
The functor $DQ_X$ exists because $X$ is quasi-compact and quasi-separated, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-better-coherator}. Since $DQ_X$ is a right adjoint it commutes with products and therefore with derived limits. Hence the equality in the statement of the lemma. \medskip\noindent By Derived Categories of Schemes, Lemma \ref{perfect-lemma-boundedness-better-coherator} the functor $DQ_X$ has bounded cohomological dimension. Hence it suffices to show that $R\lim K_n \in D^-(\mathcal{O}_X)$. To see this, let $U \subset X$ be an affine open. Then there is a canonical exact sequence $$ 0 \to R^1\lim H^{m - 1}(U, K_n) \to H^m(U, R\lim K_n) \to \lim H^m(U, K_n) \to 0 $$ by Cohomology, Lemma \ref{cohomology-lemma-RGamma-commutes-with-Rlim}. Since $U$ is affine and $K_n$ is pseudo-coherent (and hence has quasi-coherent cohomology sheaves by Derived Categories of Schemes, Lemma \ref{perfect-lemma-pseudo-coherent}) we see that $H^m(U, K_n) = H^m(K_n)(U)$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-compare-bounded}. Thus we conclude that it suffices to show that $K_n$ is bounded above independent of $n$. \medskip\noindent Since $K_n$ is pseudo-coherent we have $K_n \in D^-(\mathcal{O}_{X_n})$. Suppose that $a_n$ is maximal such that $H^{a_n}(K_n)$ is nonzero. Of course $a_1 \leq a_2 \leq a_3 \leq \ldots$. Note that $H^{a_n}(K_n)$ is an $\mathcal{O}_{X_n}$-module of finite presentation (Cohomology, Lemma \ref{cohomology-lemma-finite-cohomology}). We have $H^{a_n}(K_{n - 1}) = H^{a_n}(K_n) \otimes_{\mathcal{O}_{X_n}} \mathcal{O}_{X_{n - 1}}$. Since $X_{n - 1} \to X_n$ is a thickening, it follows from Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK}) that if $H^{a_n}(K_n) \otimes_{\mathcal{O}_{X_n}} \mathcal{O}_{X_{n - 1}}$ is zero, then $H^{a_n}(K_n)$ is zero too. Thus $a_n = a_{n - 1}$ for all $n$ and we conclude.
Write $E_n = E|_{X_n}$ and $M_n = R\Gamma(X_n, K_n \otimes^\mathbf{L} E|_{X_n})$. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-pseudo-coherent-direct-image-general} we see that $M_n$ is a pseudo-coherent object of $D(A_n)$ whose formation commutes with base change. Thus the maps $M_n \otimes_{A_n}^\mathbf{L} A_{n - 1} \to M_{n - 1}$ coming from $\varphi_n$ are isomorphisms. By More on Algebra, Lemma \ref{more-algebra-lemma-Rlim-pseudo-coherent-gives-pseudo-coherent} we find that $R\lim M_n$ is pseudo-coherent and that its base change back to $A_n$ recovers $M_n$. On the other hand, the exact functor $R\Gamma(X, -) : D_\QCoh(\mathcal{O}_X) \to D(A)$ of triangulated categories commutes with products and hence with derived limits, whence $$ R\Gamma(X, E \otimes^\mathbf{L} K) = R\lim R\Gamma(X,  E \otimes^\mathbf{L} K_n) = R\lim R\Gamma(X_n, E_n \otimes^\mathbf{L} K_n) = R\lim M_n $$ as desired.
Combinging Lemma \ref{lemma-compute-against-perfect-derived} and Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-enough} we see that $R\Gamma(X, K \otimes^\mathbf{L} E)$ is pseudo-coherent in $D(A)$ for all pseudo-coherent $E$ in $D(\mathcal{O}_X)$. Thus it follows from More on Morphisms, Lemma \ref{more-morphisms-lemma-characterize-pseudo-coherent} that $K$ is pseudo-coherent relative to $A$. Since $X$ is of flat and of finite presentation over $A$, this is the same as being pseudo-coherent on $X$, see More on Morphisms, Lemma \ref{more-morphisms-lemma-check-relative-pseudo-coherence-on-charts}.
Fix $n$. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-computing-sections-as-colim} there exists a system of perfect complexes $E_m$ on $X$ such that $R\Gamma(U, K) = \text{hocolim} R\Gamma(X, K \otimes^\mathbf{L} E_m)$. In fact, this formula holds not just for $K$ but for every object of $D_\QCoh(\mathcal{O}_X)$. Applying this to $K_n$ we obtain \begin{align*} R\Gamma(U_n, K_n) & = R\Gamma(U, K_n) \\ & = \text{hocolim}_m R\Gamma(X, K_n \otimes^\mathbf{L} E_m) \\ & = \text{hocolim}_m R\Gamma(X_n, K_n \otimes^\mathbf{L} E_m|_{X_n}) \end{align*} Using Lemma \ref{lemma-compute-against-perfect-derived} and the fact that $- \otimes_A^\mathbf{L} A_n$ commutes with homotopy colimits we obtain the result.
Given $\mathcal{F}/X/S$ and $U \subset S$ with hypotheses as in the lemma, denote $P$ the property ``$\mathcal{F}$ is flat over $S$ at all generic points of fibres''. It is clear that $P$ is preserved under strict transform, see Divisors, Lemma \ref{divisors-lemma-strict-transform-flat} and Morphisms, Lemma \ref{morphisms-lemma-base-change-module-flat}. It is also clear that $P$ is local on $S$. Hence any and all observations of Remark \ref{remark-successive-blowups} apply to the problem posed by the lemma. \medskip\noindent Consider the function $r : U \to \mathbf{Z}_{\geq 0}$ which assigns to $u \in U$ the integer $$ r(u) = \dim_{\kappa(\xi_u)}(\mathcal{F}_{\xi_u} \otimes \kappa(\xi_u)) $$ where $\xi_u$ is the generic point of the fibre $X_u$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-flat-and-free-at-point-fibre} and the fact that the image of an open in $X_S$ in $S$ is open, we see that $r(u)$ is locally constant. Accordingly $U = U_0 \amalg U_1 \amalg \ldots \amalg U_c$ is a finite disjoint union of open and closed subschemes where $r$ is constant with value $i$ on $U_i$. By Divisors, Lemma \ref{divisors-lemma-separate-disjoint-opens-by-blowing-up} we can find a $U$-admissible blowup to decompose $S$ into the disjoint union of two schemes, the first containing $U_0$ and the second $U_1 \cup \ldots \cup U_c$. Repeating this $c - 1$ more times we may assume that $S$ is a disjoint union $S = S_0 \amalg S_1 \amalg \ldots \amalg S_c$ with $U_i \subset S_i$. Thus we may assume the function $r$ defined above is constant, say with value $r$. \medskip\noindent By Remark \ref{remark-successive-blowups} we see that we may assume that we have an effective Cartier divisor $D \subset S$ whose support is $S \setminus U$. Another application of Remark \ref{remark-successive-blowups} combined with Divisors, Lemma \ref{divisors-lemma-characterize-effective-Cartier-divisor} tells us we may assume that $S = \Spec(R)$ and $D = \Spec(R/(f))$ for some nonzerodivisor $f \in R$. This case is handled by Lemma \ref{lemma-helper-blowup-affine-space}.
We will use basic properties of Fitting ideals, see More on Algebra, Lemma \ref{more-algebra-lemma-fitting-ideal-basics}. Then $IC$ is the Fitting ideal of $C/J \otimes_A C$. Note that $C \to C \otimes_A C$, $c \mapsto 1 \otimes c$ has a section (the multiplication map). By assumption $C \to C \otimes_A C$ is \'etale at every prime in the image of $\Spec(C_h)$ under this section. Hence the multiplication map $C \otimes_A C_h \to C_h$ is \'etale in particular flat, see Algebra, Lemma \ref{algebra-lemma-map-between-etale}. Hence there exists a $C_h$-algebra such that $C \otimes_A C_h \cong C_h \oplus C'$ as $C_h$-algebras, see Algebra, Lemma \ref{algebra-lemma-surjective-flat-finitely-presented}. Thus $(C/J) \otimes_A C_h \cong (C_h/J_h) \oplus C'/I'$ as $C_h$-modules for some ideal $I' \subset C'$. Hence $IC_h = JJ'$ with $J' = \text{Fit}_0(C'/I')$ where we view $C'/J'$ as a $C_h$-module.
We may replace $B$ by a principal localization at an element $g \in B$, $g \not \in \mathfrak q$. Thus we may assume that $B$ is standard \'etale, see Algebra, Proposition \ref{algebra-proposition-etale-locally-standard}. Thus we may assume $B$ is a localization of $C = A[x]/(f)$ for some monic $f \in A[x]$ of some degree $d$. Say $B = C_h$ for some $h \in C$. Choose elements $h_1, \ldots, h_n \in C$ which generate $J$ over $B$. The condition $V(J) \subset V(aB)$ signifies that $a^m = \sum b_i h_i$ in $B$ for some large $m$. Set $h_{n + 1} = a^m$. As in Lemma \ref{lemma-trick-fitting-ideal} we take $I = \text{Fit}_0(C/(h_1, \ldots, h_{r + 1}))$. Since the module $C/(h_1, \ldots, h_{r + 1})$ is annihilated by $a^m$ we see that $a^{dm} \in I$ which implies that $V(I) \subset V(a)$.
Since $X \to S$ is quasi-compact and quasi-separated by assumption, the strict transform of $X$ with respect to a blowing up $S' \to S$ is also quasi-compact and quasi-separated. Hence to prove the lemma it suffices to find a $U$-admissible blowup such that the strict transform is flat and locally of finite presentation. Let $X = W_1 \cup \ldots \cup W_n$ be a finite affine open covering. If we can find a $U$-admissible blowup $S_i \to S$ such that the strict transform of $W_i$ is flat and locally of finite presentation, then there exists a $U$-admissible blowing up $S' \to S$ dominating all $S_i \to S$ which does the job (see Divisors, Lemma \ref{divisors-lemma-dominate-admissible-blowups}; see also Remark \ref{remark-successive-blowups}). Hence we may assume $X$ is affine. \medskip\noindent Assume $X$ is affine. By Morphisms, Lemma \ref{morphisms-lemma-quasi-affine-finite-type-over-S} we can choose an immersion $j : X \to \mathbf{A}^n_S$ over $S$. Let $V \subset \mathbf{A}^n_S$ be a quasi-compact open subscheme such that $j$ induces a closed immersion $i : X \to V$ over $S$. Apply Theorem \ref{theorem-flatten-module} to $V \to S$ and the quasi-coherent module $i_*\mathcal{O}_X$ to obtain a $U$-admissible blowup $S' \to S$ such that the strict transform of $i_*\mathcal{O}_X$ is flat over $S'$ and of finite presentation over $\mathcal{O}_{V \times_S S'}$. Let $X'$ be the strict transform of $X$ with respect to $S' \to S$. Let $i' : X' \to V \times_S S'$ be the induced morphism. Since taking strict transform commutes with pushforward along affine morphisms (Divisors, Lemma \ref{divisors-lemma-strict-transform-affine}), we see that $i'_*\mathcal{O}_{X'}$ is flat over $S$ and of finite presentation as a $\mathcal{O}_{V \times_S S'}$-module. This implies the lemma.
By Lemma \ref{lemma-flat-after-blowing-up} we may assume that $X \to S$ is flat and of finite presentation. After replacing $S$ by a $U$-admissible blowup if necessary, we may assume that $U \subset S$ is scheme theoretically dense. Then $f$ is finite by Lemma \ref{lemma-proper-flat-finite-over-dense-open}. Hence $f$ is finite locally free by Morphisms, Lemma \ref{morphisms-lemma-finite-flat}.
The discussion in Remark \ref{remark-successive-blowups} applies. Thus we may do a first $U$-admissible blowup and assume the complement $S \setminus U$ is the support of an effective Cartier divisor $D$. In particular $U$ is scheme theoretically dense in $S$. Next, we do another $U$-admissible blowup to get to the situation where $X \to S$ is flat and of finite presentation, see Lemma \ref{lemma-flat-after-blowing-up}. In this case the result follows from Lemma \ref{lemma-zariski}.
The discussion in Remark \ref{remark-successive-blowups} applies. Thus we may do a first $U$-admissible blowup and assume the complement $S \setminus U$ is the support of an effective Cartier divisor $D$. In particular $U$ is scheme theoretically dense in $S$. Choose another $U$-admissible blowup $S' \to S$ such that the strict transform $X'$ of $X$ is an open subscheme of $S'$, see Lemma \ref{lemma-zariski-after-blowup}. Since $X' \to S'$ is proper, and $U \subset S'$ is dense, we see that $X' = S'$. Some details omitted.
We may and do replace $X$ by the scheme theoretic image of $s$. Then $X \to W$ is an isomorphism over $U$, see Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-image-of-partial-section}. By Lemma \ref{lemma-dominate-modification-by-blowup} there exists a $U$-admissible blowup $W' \to W$ and an extension $W' \to X$ of $s$. We finish the proof by applying Divisors, Lemma \ref{divisors-lemma-extend-admissible-blowups} to extend $W' \to W$ to a $U$-admissible blowup of $S$.
To prove part (a) we have to check conditions (1), (2), (3) of Categories, Definition \ref{categories-definition-codirected}. Condition (1) holds exactly because we assumed that $X$ is compactifyable. Let $j_i : X \to \overline{X}_i$, $i = 1, 2$ be two compactifications. Then we can consider the scheme theoretic image $\overline{X}$ of $(j_1, j_2) : X \to \overline{X}_1 \times_S \overline{X}_2$. This determines a third compactification $j : X \to \overline{X}$ which dominates both $j_i$: $$ \xymatrix{ (X, \overline{X}_1) & (X, \overline{X}) \ar[l] \ar[r] & (X, \overline{X}_2) } $$ Thus (2) holds. Let $f_1, f_2 : \overline{X}_1 \to \overline{X}_2$ be two morphisms between compactifications $j_i : X \to \overline{X}_i$, $i = 1, 2$. Let $\overline{X} \subset \overline{X}_1$ be the equalizer of $f_1$ and $f_2$. As $\overline{X}_2 \to S$ is separated, we see that $\overline{X}$ is a closed subscheme of $\overline{X}_1$ and hence proper over $S$. Moreover, we obtain an open immersion $X \to \overline{X}$ because $f_1|_X = f_2|_X = \text{id}_X$. The morphism $(X \to \overline{X}) \to (j_1 : X \to \overline{X}_1)$ given by the closed immersion $\overline{X} \to \overline{X}_1$ equalizes $f_1$ and $f_2$ which proves condition (3). \medskip\noindent Proof of (b). Let $j : X \to \overline{X}$ be a compactification. If $\overline{X}'$ denotes the scheme theoretic closure of $X$ in $\overline{X}$, then $X$ is dense and scheme theoretically dense in $\overline{X}'$ by Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-immersion}. This proves the first condition of Categories, Definition \ref{categories-definition-initial}. Since we have already shown the category of compactifications of $X$ is cofiltered, the second condition of Categories, Definition \ref{categories-definition-initial} follows from the first (we omit the solution to this categorical exercise). \medskip\noindent Proof of (c). After replacing $\overline{X}'$ with the scheme theoretic closure of $j'(X)$ (which doesn't change the underlying topological space) this follows from Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-image-of-partial-section}.
Let $j : X \to \overline{X}$ be a compactification of $X$ over $S$. Then we let $\overline{X}'$ be the scheme theoretic image of $(j, f) : X \to \overline{X} \times_S Y$. The morphism $\overline{X}' \to Y$ is proper because $\overline{X} \times_S Y \to Y$ is proper as a base change of $\overline{X} \to S$. On the other hand, since $Y$ is separated over $S$, the morphism $(1, f) : X \to X \times_S Y$ is a closed immersion (Schemes, Lemma \ref{schemes-lemma-semi-diagonal}) and hence $X \to \overline{X}'$ is an open immersion by Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-image-of-partial-section} applied to the ``partial section'' $s = (j, f)$ to the projection $\overline{X} \times_S Y \to \overline{X}$.