hcju/stacksqa huggingface dataset:
Apply Lemmas \ref{lemma-compute-what-it-should-be}, \ref{lemma-compute-against-perfect}, \ref{lemma-relative-pseudo-coherence}, \ref{lemma-compute-over-affine}, \ref{lemma-finitely-presented}, and \ref{lemma-proper-support} to get an open subscheme $W \subset X$ containing all points lying over $\Spec(A_n)$ and a finitely presented $\mathcal{O}_W$-module $\mathcal{F}$ whose support is proper over $A$ with $\mathcal{F}_n = \mathcal{F} \otimes_{\mathcal{O}_W} \mathcal{O}_{X_n}$ for all $n \geq 1$. (This makes sense as $X_n \subset W$.) By Lemma \ref{lemma-proper-pure} we see that $\mathcal{F}$ is universally pure relative to $\Spec(A)$. By Theorem \ref{theorem-flat-dimension-n-representable} (for explanation, see Lemma \ref{lemma-when-universal-flattening}) there exists a universal flattening $S' \to \Spec(A)$ of $\mathcal{F}$ and moreover the morphism $S' \to \Spec(A)$ is a monomorphism of finite presentation. Since the base change of $\mathcal{F}$ to $\Spec(A_n)$ is $\mathcal{F}_n$ we find that $\Spec(A_n) \to \Spec(A)$ factors (uniquely) through $S'$ for each $n$. By Lemma \ref{lemma-monomorphism-isomorphism} we see that $S' = \Spec(A)$. This means that $\mathcal{F}$ is flat over $A$. Finally, since the scheme theoretic support $Z$ of $\mathcal{F}$ is proper over $\Spec(A)$, the morphism $Z \to X$ is closed. Hence the pushforward $(W \to X)_*\mathcal{F}$ is supported on $W$ and has all the desired properties.
Apply Lemmas \ref{lemma-compute-what-it-should-be-derived}, \ref{lemma-compute-against-perfect-derived}, \ref{lemma-relative-pseudo-coherence-derived} to get a pseudo-coherent object $K$ of $D(\mathcal{O}_X)$. Choosing affine opens in Lemma \ref{lemma-compute-over-affine-derived} it follows immediately that $K$ restricts to $K_n$ over $X_n$.
We first reduce to the Noetherian case. We strongly urge the reader to skip this paragraph. There exists a closed immersion $X \to X'$ with $X' \to S$ of finite presentation and separated. See Limits, Proposition \ref{limits-proposition-separated-closed-in-finite-presentation}. If we find a compactification of $X'$ over $S$, then taking the scheme theoretic image of $X$ in this will give a compactification of $X$ over $S$. Thus we may assume $X \to S$ is separated and of finite presentation. We may write $S = \lim S_i$ as a directed limit of a system of Noetherian schemes with affine transition morphisms. See Limits, Proposition \ref{limits-proposition-approximate}. We can choose an $i$ and a morphism $X_i \to S_i$ of finite presentation whose base change to $S$ is $X \to S$, see Limits, Lemma \ref{limits-lemma-descend-finite-presentation}. After increasing $i$ we may assume $X_i \to S_i$ is separated, see Limits, Lemma \ref{limits-lemma-descend-separated-finite-presentation}. If we can find a compactification of $X_i$ over $S_i$, then the base change of this to $S$ will be a compactification of $X$ over $S$. This reduces us to the case discussed in the next paragraph. \medskip\noindent Assume $S$ is Noetherian. We can choose a finite affine open covering $X = \bigcup_{i = 1, \ldots, n} U_i$ such that $U_1 \cap \ldots \cap U_n$ is dense in $X$. This follows from Properties, Lemma \ref{properties-lemma-point-and-maximal-points-affine} and the fact that $X$ is quasi-compact with finitely many irreducible components. For each $i$ we can choose an $n_i \geq 0$ and an immersion $U_i \to \mathbf{A}^{n_i}_S$ by Morphisms, Lemma \ref{morphisms-lemma-quasi-affine-finite-type-over-S}. Hence $U_i$ has a compactification over $S$ for $i = 1, \ldots, n$ by taking the scheme theoretic image in $\mathbf{P}^{n_i}_S$. Applying Lemma \ref{lemma-two-compactifications} $(n - 1)$ times we conclude that the theorem is true.
As an \'etale morphism is a flat morphism (see Morphisms, Lemma \ref{morphisms-lemma-etale-flat}) the implication ``$\Leftarrow$'' follows from Algebra, Lemma \ref{algebra-lemma-composition-flat}. For the converse assume that $\mathcal{F}$ is flat at $x$ over $S$. Denote $\tilde x \in X \times_S T$ the point lying over $x$ in $X$ and over the image of $x$ in $T$ in $T$. Then $(X \times_S T \to X)^*\mathcal{F}$ is flat at $\tilde x$ over $T$ via $\text{pr}_2 : X \times_S T \to T$, see Morphisms, Lemma \ref{morphisms-lemma-base-change-module-flat}. The diagonal $\Delta_{T/S} : T \to T \times_S T$ is an open immersion; combine Morphisms, Lemmas \ref{morphisms-lemma-diagonal-unramified-morphism} and \ref{morphisms-lemma-etale-smooth-unramified}. So $X$ is identified with open subscheme of $X \times_S T$, the restriction of $\text{pr}_2$ to this open is the given morphism $X \to T$, the point $\tilde x$ corresponds to the point $x$ in this open, and $(X \times_S T \to X)^*\mathcal{F}$ restricted to this open is $\mathcal{F}$. Whence we see that $\mathcal{F}$ is flat at $x$ over $T$.
We may replace $S$ by an affine neighbourhood of $s$ and after that $T$ by an affine neighbourhood of $t$. Set $\mathcal{F} = (\Spec(\mathcal{O}_{T, t}) \to T)_*\widetilde M$. This is a quasi-coherent sheaf (see Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} or argue directly) on $T$ whose stalk at $t$ is $M$ (details omitted). Apply Lemma \ref{lemma-etale-flat-up-down}.
By More on Algebra, Lemma \ref{more-algebra-lemma-dumb-properties-henselization} the local ring maps $\mathcal{O}_{S, s} \to \mathcal{O}_{S, s}^h \to \mathcal{O}_{S, s}^{sh}$ are faithfully flat. Hence (3) $\Rightarrow$ (2) $\Rightarrow$ (1) and (5) $\Rightarrow$ (4) follow from Algebra, Lemma \ref{algebra-lemma-composition-flat}. By faithful flatness the equivalences (6) $\Leftrightarrow$ (5) and (5) $\Leftrightarrow$ (3) follow from Algebra, Lemma \ref{algebra-lemma-flatness-descends}. Thus it suffices to show that (1) $\Rightarrow$ (2) $\Rightarrow$ (3) and (4) $\Rightarrow$ (5). To prove these we may assume $S$ is an affine scheme. \medskip\noindent Assume (1). By Lemma \ref{lemma-etale-flat-up-down-local-ring} we see that $M^{sh}$ is flat over $\mathcal{O}_{T, t}$ for any \'etale neighbourhood $(T, t) \to (S, s)$. Since $\mathcal{O}_{S, s}^h$ and $\mathcal{O}_{S, s}^{sh}$ are directed colimits of local rings of the form $\mathcal{O}_{T, t}$ (see Algebra, Lemmas \ref{algebra-lemma-henselization-different} and \ref{algebra-lemma-strict-henselization-different}) we conclude that $M^{sh}$ is flat over $\mathcal{O}_{S, s}^h$ and $\mathcal{O}_{S, s}^{sh}$ by Algebra, Lemma \ref{algebra-lemma-colimit-rings-flat}. Thus (1) implies (2) and (3). Of course this implies also (2) $\Rightarrow$ (3) by replacing $\mathcal{O}_{S, s}$ by $\mathcal{O}_{S, s}^h$. The same argument applies to prove (4) $\Rightarrow$ (5).
By More on Algebra, Lemma \ref{more-algebra-lemma-dumb-properties-henselization} the local ring maps $\mathcal{O}_{S, s} \to \mathcal{O}_{S, s}^h \to \mathcal{O}_{S, s}^{sh}$ are faithfully flat. Hence (3) $\Rightarrow$ (2) $\Rightarrow$ (1) and (5) $\Rightarrow$ (4) follow from More on Algebra, Lemma \ref{more-algebra-lemma-flat-push-tor-amplitude}. By faithful flatness the equivalences (6) $\Leftrightarrow$ (5) and (5) $\Leftrightarrow$ (3) follow from More on Algebra, Lemma \ref{more-algebra-lemma-flat-descent-tor-amplitude}. Thus it suffices to show that (1) $\Rightarrow$ (3), (2) $\Rightarrow$ (3), and (4) $\Rightarrow$ (5). \medskip\noindent Assume (1). In particular $M^{sh}$ has vanishing cohomology in degrees $< a$ and $> b$. Hence we can represent $M^{sh}$ by a complex $P^\bullet$ of free $\mathcal{O}_{X, x}^{sh}$-modules with $P^i = 0$ for $i > b$ (see for example the very general Derived Categories, Lemma \ref{derived-lemma-subcategory-left-resolution}). Note that $P^n$ is flat over $\mathcal{O}_{S, s}$ for all $n$. Consider $\Coker(d_P^{a - 1})$. By More on Algebra, Lemma \ref{more-algebra-lemma-last-one-flat} this is a flat $\mathcal{O}_{S, s}$-module. Hence by Lemma \ref{lemma-flat-up-down-henselization} this is a flat $\mathcal{O}_{S, s}^{sh}$-module. Thus $\tau_{\geq a}P^\bullet$ is a complex of flat $\mathcal{O}_{S, s}^{sh}$-modules representing $M^{sh}$ in $D(\mathcal{O}_{S, s}^{sh}$ and we find that $M^{sh}$ has tor amplitude in $[a, b]$, see More on Algebra, Lemma \ref{more-algebra-lemma-tor-amplitude}. Thus (1) implies (3). Of course this implies also (2) $\Rightarrow$ (3) by replacing $\mathcal{O}_{S, s}$ by $\mathcal{O}_{S, s}^h$. The same argument applies to prove (4) $\Rightarrow$ (5).
The implication ``$\Leftarrow$'' follows immediately from Divisors, Lemma \ref{divisors-lemma-weakly-ass-pullback}. Assume $t \in \text{WeakAss}(g^*\mathcal{G})$. Let $\Spec(A) \subset S$ be an affine open neighbourhood of $s$. Let $\mathcal{G}$ be the quasi-coherent sheaf associated to the $A$-module $M$. Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$. As $g$ is finite flat we have $g^{-1}(\Spec(A)) = \Spec(B)$ for some finite flat $A$-algebra $B$. Note that $g^*\mathcal{G}$ is the quasi-coherent $\mathcal{O}_{\Spec(B)}$-module associated to the $B$-module $M \otimes_A B$ and $g_*g^*\mathcal{G}$ is the quasi-coherent $\mathcal{O}_{\Spec(A)}$-module associated to the $A$-module $M \otimes_A B$. By Algebra, Lemma \ref{algebra-lemma-finite-flat-local} we have $B_{\mathfrak p} \cong A_{\mathfrak p}^{\oplus n}$ for some integer $n \geq 0$. Note that $n \geq 1$ as we assumed there exists at least one point of $T$ lying over $s$. Hence we see by looking at stalks that $$ s \in \text{WeakAss}(\mathcal{G}) \Leftrightarrow s \in \text{WeakAss}(g_*g^*\mathcal{G}) $$ Now the assumption that $t \in \text{WeakAss}(g^*\mathcal{G})$ implies that $s \in \text{WeakAss}(g_*g^*\mathcal{G})$ by Divisors, Lemma \ref{divisors-lemma-weakly-associated-finite} and hence by the above $s \in  \text{WeakAss}(\mathcal{G})$.
After replacing $S$ and $U$ by affine neighbourhoods of $s$ and $u$ we may assume that $g$ is a standard \'etale morphism of affines, see Morphisms, Lemma \ref{morphisms-lemma-etale-locally-standard-etale}. Thus we may assume $S = \Spec(A)$ and $X = \Spec(A[x, 1/g]/(f))$, where $f$ is monic and $f'$ is invertible in $A[x, 1/g]$. Note that $A[x, 1/g]/(f) = (A[x]/(f))_g$ is also the localization of the finite free $A$-algebra $A[x]/(f)$. Hence we may think of $U$ as an open subscheme of the scheme $T = \Spec(A[x]/(f))$ which is finite locally free over $S$. This reduces us to Lemma \ref{lemma-finite-flat-weak-assassin-up-down} above.
The equivalence of (1) and (2) is the definition, see Divisors, Definition \ref{divisors-definition-weakly-associated}. The implications (2) $\Rightarrow$ (3) $\Rightarrow$ (4) follows from Divisors, Lemma \ref{divisors-lemma-weakly-ass-pullback} applied to the flat (More on Algebra, Lemma \ref{more-algebra-lemma-dumb-properties-henselization}) morphisms $$ \Spec(\mathcal{O}_{S, s}^{sh}) \to \Spec(\mathcal{O}_{S, s}^h) \to \Spec(\mathcal{O}_{S, s}) $$ and the closed points. To prove (4) $\Rightarrow$ (2) we may replace $S$ by an affine neighbourhood. Suppose that $x \in \mathcal{F}_s \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^{sh}$ is an element whose annihilator has radical equal to $\mathfrak m_s^{sh}$. (See Algebra, Lemma \ref{algebra-lemma-weakly-ass-local}.) Since $\mathcal{O}_{S, s}^{sh}$ is equal to the limit of $\mathcal{O}_{U, u}$ over \'etale neighbourhoods $f : (U, u) \to (S, s)$ by Algebra, Lemma \ref{algebra-lemma-strict-henselization-different} we may assume that $x$ is the image of some $x' \in \mathcal{F}_s \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{U, u}$. The local ring map $\mathcal{O}_{U, u} \to \mathcal{O}_{S, s}^{sh}$ is faithfully flat (as it is the strict henselization), hence universally injective (Algebra, Lemma \ref{algebra-lemma-faithfully-flat-universally-injective}). It follows that the annihilator of $x'$ is the inverse image of the annihilator of $x$. Hence the radical of this annihilator is equal to $\mathfrak m_u$. Thus $u$ is a weakly associated point of $f^*\mathcal{F}$. By Lemma \ref{lemma-etale-weak-assassin-up-down} we see that $s$ is a weakly associated point of $\mathcal{F}$.
Say the morphism $f : X \to S$ is given by the ring map $A \to B$ and that $\mathcal{F}$ is the quasi-coherent sheaf associated to the $B$-module $M$. By Morphisms, Lemma \ref{morphisms-lemma-locally-finite-type-characterize} we know that $A \to B$ is a finite type ring map, and by Properties, Lemma \ref{properties-lemma-finite-type-module} we know that $M$ is a finite $B$-module. In particular the support of $\mathcal{F}$ is the closed subscheme of $\Spec(B)$ cut out by the annihilator $I = \{x \in B \mid xm = 0\ \forall m \in M\}$ of $M$, see Algebra, Lemma \ref{algebra-lemma-support-closed}. Let $\mathfrak q \subset B$ be the prime ideal corresponding to $x$ and let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$. Note that $X_s = \Spec(B \otimes_A \kappa(\mathfrak p))$ and that $\mathcal{F}_s$ is the quasi-coherent sheaf associated to the $B \otimes_A \kappa(\mathfrak p)$ module $M \otimes_A \kappa(\mathfrak p)$. By Morphisms, Lemma \ref{morphisms-lemma-support-finite-type} the support of $\mathcal{F}_s$ is equal to $V(I(B \otimes_A \kappa(\mathfrak p)))$. Since $B \otimes_A \kappa(\mathfrak p)$ is of finite type over $\kappa(\mathfrak p)$ there exist finitely many elements $f_1, \ldots, f_m \in I$ such that $$ I(B \otimes_A \kappa(\mathfrak p)) = (f_1, \ldots, f_n)(B \otimes_A \kappa(\mathfrak p)). $$ Denote $i : Z \to X$ the closed subscheme cut out by $(f_1, \ldots, f_m)$, in a formula $Z = \Spec(B/(f_1, \ldots, f_m))$. Since $M$ is annihilated by $I$ we can think of $M$ as an $B/(f_1, \ldots, f_m)$-module. In other words, $\mathcal{F}$ is the pushforward of a finite type module on $Z$. As $Z_s = \text{Supp}(\mathcal{F}_s)$ by construction, this proves the lemma.
Let $V \subset S$ be an affine neighbourhood of $s$. Let $U \subset f^{-1}(V)$ be an affine neighbourhood of $x$. Then it suffices to prove the lemma for $f|_U : U \to V$ and $\mathcal{F}|_U$. Hence in the rest of the proof we assume that $X$ and $S$ are affine. \medskip\noindent First, suppose that $X_s = \text{Supp}(\mathcal{F}_s)$, in particular $n = \dim_x(X_s)$. Apply More on Morphisms, Lemmas \ref{more-morphisms-lemma-local-local-structure-finite-type} and \ref{more-morphisms-lemma-local-local-structure-finite-type-affine}. This gives us a commutative diagram $$ \xymatrix{ X \ar[dd] & X' \ar[l]^g \ar[d]^\pi \\ & Y' \ar[d]^h  \\ S & S' \ar[l]_e } $$ and point $x' \in X'$. We set $Z' = X'$, $i = \text{id}$, and $\mathcal{G} = g^*\mathcal{F}$ to obtain a solution in this case. \medskip\noindent In general choose a closed immersion $Z \to X$ and a sheaf $\mathcal{G}$ on $Z$ as in Lemma \ref{lemma-sheaf-lives-on-subscheme}. Applying the result of the previous paragraph to $Z \to S$ and $\mathcal{G}$ we obtain a diagram $$ \xymatrix{ X \ar[dd]_f & Z \ar[l] \ar[dd]_{f|_Z} & Z' \ar[l]^g \ar[d]^\pi \\ & & Y' \ar[d]^h \\ S & S \ar@{=}[l] & S' \ar[l]_e } $$ and point $z' \in Z'$ satisfying all the required properties. We will use Lemma \ref{lemma-lift-etale} to embed $Z'$ into a scheme \'etale over $X$. We cannot apply the lemma directly as we want $X'$ to be a scheme over $S'$. Instead we consider the morphisms $$ \xymatrix{ Z' \ar[r] & Z \times_S S' \ar[r] & X \times_S S' } $$ The first morphism is \'etale by Morphisms, Lemma \ref{morphisms-lemma-etale-permanence}. The second is a closed immersion as a base change of a closed immersion. Finally, as $X$, $S$, $S'$, $Z$, $Z'$ are all affine we may apply Lemma \ref{lemma-lift-etale} to get an \'etale morphism of affine schemes $X' \to X \times_S S'$ such that $$ Z' = (Z \times_S S') \times_{(X \times_S S')} X' = Z \times_X X'. $$ As $Z \to X$ is a closed immersion of finite presentation, so is $Z' \to X'$. Let $x' \in X'$ be the point corresponding to $z' \in Z'$. Then the completed diagram $$ \xymatrix{ X \ar[dd] & X' \ar[dd] \ar[l] & Z' \ar[l]^i \ar[d]^\pi \\ & & Y' \ar[d]^h \\ S & S' \ar[l]_e & S' \ar@{=}[l] } $$ is a solution of the original problem.
Assume $f$ locally of finite presentation. Then $Z' \to S$ is locally of finite presentation as a composition of such, see Morphisms, Lemma \ref{morphisms-lemma-composition-finite-presentation}. Note that $Y' \to S$ is also locally of finite presentation as a composition of a smooth and an \'etale morphism. Hence Morphisms, Lemma \ref{morphisms-lemma-finite-presentation-permanence} implies $\pi$ is locally of finite presentation. Since $\pi$ is finite we conclude that it is also separated and quasi-compact, hence $\pi$ is actually of finite presentation. \medskip\noindent To prove the equivalence of (1), (2), and (3) we also consider: (4) $g^*\mathcal{F}$ is a $\mathcal{O}_{X'}$-module of finite presentation in a neighbourhood of $x'$. The pullback of a module of finite presentation is of finite presentation, see Modules, Lemma \ref{modules-lemma-pullback-finite-presentation}. Hence (1) $\Rightarrow$ (4). The \'etale morphism $g$ is open, see Morphisms, Lemma \ref{morphisms-lemma-etale-open}. Hence for any open neighbourhood $U' \subset X'$ of $x'$, the image $g(U')$ is an open neighbourhood of $x$ and the map $\{U' \to g(U')\}$ is an \'etale covering. Thus (4) $\Rightarrow$ (1) by Descent, Lemma \ref{descent-lemma-finite-presentation-descends}. Using Descent, Lemma \ref{descent-lemma-finite-finitely-presented-module} and some easy topological arguments (see More on Morphisms, Lemma \ref{more-morphisms-lemma-finite-morphism-single-point-in-fibre}) we see that (4) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3). \medskip\noindent To prove the equivalence of (a), (b), (c) consider the ring maps $$ \mathcal{O}_{X, x} \to \mathcal{O}_{X', x'} \to \mathcal{O}_{Z', z'} \leftarrow \mathcal{O}_{Y', y'} $$ The first ring map is faithfully flat. Hence $\mathcal{F}_x$ is of finite presentation over $\mathcal{O}_{X, x}$ if and only if $g^*\mathcal{F}_{x'}$ is of finite presentation over $\mathcal{O}_{X', x'}$, see Algebra, Lemma \ref{algebra-lemma-descend-properties-modules}. The second ring map is surjective (hence finite) and finitely presented by assumption, hence $g^*\mathcal{F}_{x'}$ is of finite presentation over $\mathcal{O}_{X', x'}$ if and only if $\mathcal{G}_{z'}$ is of finite presentation over $\mathcal{O}_{Z', z'}$, see Algebra, Lemma \ref{algebra-lemma-finite-finitely-presented-extension}. Because $\pi$ is finite, of finite presentation, and $\pi^{-1}(\{y'\}) = \{x'\}$ the ring homomorphism $\mathcal{O}_{Y', y'} \leftarrow \mathcal{O}_{Z', z'}$ is finite and of finite presentation, see More on Morphisms, Lemma \ref{more-morphisms-lemma-finite-morphism-single-point-in-fibre}. Hence $\mathcal{G}_{z'}$ is of finite presentation over $\mathcal{O}_{Z', z'}$ if and only if $\pi_*\mathcal{G}_{y'}$ is of finite presentation over $\mathcal{O}_{Y', y'}$, see Algebra, Lemma \ref{algebra-lemma-finite-finitely-presented-extension}.
To prove the equivalence of (1), (2), and (3) we also consider: (4) $g^*\mathcal{F}$ is flat over $S$ in a neighbourhood of $x'$. We will use Lemma \ref{lemma-etale-flat-up-down} to equate flatness over $S$ and $S'$ without further mention. The \'etale morphism $g$ is flat and open, see Morphisms, Lemma \ref{morphisms-lemma-etale-open}. Hence for any open neighbourhood $U' \subset X'$ of $x'$, the image $g(U')$ is an open neighbourhood of $x$ and the map $U' \to g(U')$ is surjective and flat. Thus (4) $\Leftrightarrow$ (1) by Morphisms, Lemma \ref{morphisms-lemma-flat-permanence}. Note that $$ \Gamma(X', g^*\mathcal{F}) = \Gamma(Z', \mathcal{G}) = \Gamma(Y', \pi_*\mathcal{G}) $$ Hence the flatness of $g^*\mathcal{F}$, $\mathcal{G}$ and $\pi_*\mathcal{G}$ over $S'$ are all equivalent (this uses that $X'$, $Z'$, $Y'$, and $S'$ are all affine). Some omitted topological arguments (compare More on Morphisms, Lemma \ref{more-morphisms-lemma-finite-morphism-single-point-in-fibre}) regarding affine neighbourhoods now show that (4) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3). \medskip\noindent To prove the equivalence of (a), (b), (c) consider the commutative diagram of local ring maps $$ \xymatrix{ \mathcal{O}_{X', x'} \ar[r]_\iota & \mathcal{O}_{Z', z'} & \mathcal{O}_{Y', y'} \ar[l]^\alpha & \mathcal{O}_{S', s'} \ar[l]^\beta \\ \mathcal{O}_{X, x} \ar[u]^\gamma & & & \mathcal{O}_{S, s} \ar[lll]_\varphi \ar[u]_\epsilon } $$ We will use Lemma \ref{lemma-etale-flat-up-down-local-ring} to equate flatness over $\mathcal{O}_{S, s}$ and $\mathcal{O}_{S', s'}$ without further mention. The map $\gamma$ is faithfully flat. Hence $\mathcal{F}_x$ is flat over $\mathcal{O}_{S, s}$ if and only if $g^*\mathcal{F}_{x'}$ is flat over $\mathcal{O}_{S', s'}$, see Algebra, Lemma \ref{algebra-lemma-flatness-descends-more-general}. As $\mathcal{O}_{S', s'}$-modules the modules $g^*\mathcal{F}_{x'}$, $\mathcal{G}_{z'}$, and $\pi_*\mathcal{G}_{y'}$ are all isomorphic, see More on Morphisms, Lemma \ref{more-morphisms-lemma-finite-morphism-single-point-in-fibre}. This finishes the proof.
Fibre products of affines are affine, see Schemes, Lemma \ref{schemes-lemma-fibre-product-affines}. Base change preserves closed immersions, morphisms of finite presentation, finite morphisms, smooth morphisms, morphisms with geometrically irreducible fibres, and morphisms of relative dimension $n$, see Morphisms, Lemmas \ref{morphisms-lemma-base-change-closed-immersion}, \ref{morphisms-lemma-base-change-finite-presentation}, \ref{morphisms-lemma-base-change-finite}, \ref{morphisms-lemma-base-change-smooth}, \ref{morphisms-lemma-base-change-relative-dimension-d}, and More on Morphisms, Lemma \ref{more-morphisms-lemma-base-change-fibres-geometrically-irreducible}. We have $i'_*\mathcal{G}' \cong \mathcal{F}'$ because pushforward along the finite morphism $i$ commutes with base change, see Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-base-change}. We have $\dim(\text{Supp}(\mathcal{F}_s)) = \dim(\text{Supp}(\mathcal{F}'_{s'}))$ by Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-after-base-change} because $$ \text{Supp}(\mathcal{F}_s) \times_s s' = \text{Supp}(\mathcal{F}'_{s'}). $$ This proves the lemma.
By Lemma \ref{lemma-base-change-one-step} $(Z', Y', i', \pi', \mathcal{G}')$ is a one step d\'evissage of $\mathcal{F}'/X'/S'$ over $s'$. Properties (1) -- (4) of Definition \ref{definition-one-step-devissage-at-x} hold for $(Z', Y', i', \pi', \mathcal{G}', z', y')$ as the assumption that $\kappa(s) = \kappa(s')$ insures that the fibres $X'_{s'}$, $Z'_{s'}$, and $Y'_{s'}$ are isomorphic to $X_s$, $Z_s$, and $Y_s$.
Part (1) is immediate from Lemma \ref{lemma-base-change-one-step-at-x} and the fact that the inverse image of a standard open under a morphism of affine schemes is a standard open, see Algebra, Lemma \ref{algebra-lemma-spec-functorial}. \medskip\noindent Let $W \subset Y$ as in (2). Because $Y \to S$ is smooth it is open, see Morphisms, Lemma \ref{morphisms-lemma-smooth-open}. Hence we can find a standard open neighbourhood $S'$ of $s$ contained in the image of $W$. Then the fibres of $W_{S'} \to S'$ are nonempty open subschemes of the fibres of $Y \to S$ over $S'$ and hence geometrically irreducible too. Setting $Y' = W_{S'}$ and $Z' = \pi^{-1}(Y')$ we see that $Z' \subset Z$ is a standard open neighbourhood of $z$. Let $\overline{h} \in \Gamma(Z, \mathcal{O}_Z)$ be a function such that $Z' = D(\overline{h})$. As $i : Z \to X$ is a closed immersion, we can find a function $h \in \Gamma(X, \mathcal{O}_X)$ such that $i^\sharp(h) = \overline{h}$. Take $X' = D(h) \subset X$. In this way we obtain a standard shrinking as in (2). \medskip\noindent Let $U \subset X$ be as in (3). We may after shrinking $U$ assume that $U$ is a standard open. By More on Morphisms, Lemma \ref{more-morphisms-lemma-finite-morphism-single-point-in-fibre} there exists a standard open $W \subset Y$ neighbourhood of $y$ such that $\pi^{-1}(W) \subset i^{-1}(U)$. Apply (2) to get a standard shrinking $X', S', Z', Y'$ with $Y' = W_{S'}$. Since $Z' \subset \pi^{-1}(W) \subset i^{-1}(U)$ we may replace $X'$ by $X' \cap U$ (still a standard open as $U$ is also standard open) without violating any of the conditions defining a standard shrinking. Hence we win.