hcju/stacksqa huggingface dataset:
Let $(g : T \to S, t' \leadsto t, \xi)$ be any impurity of $\mathcal{F}$ above $s$. We apply Limits, Lemma \ref{limits-lemma-separate} to $t \in T$ and $Z = \overline{\{\xi\}}$ to obtain 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 we have $Z' \cap X_{a(t)} = \emptyset$, and \item $Z \cap X_V$ maps into $Z'$ via the morphism $X_V \to X_{T'}$. \end{enumerate} As $t'$ specializes to $t$ we may replace $T$ by the open neighbourhood $V$ of $t$. Thus we have a commutative diagram $$ \xymatrix{ X_T \ar[d] \ar[r] & X_{T'} \ar[d] \ar[r] & X \ar[d] \\ T \ar[r]^a & T' \ar[r]^b & S } $$ where $b \circ a = g$. Let $\xi' \in X_{T'}$ denote the image of $\xi$. By Divisors, Lemma \ref{divisors-lemma-base-change-relative-assassin} we see that $\xi' \in \text{Ass}_{X_{T'}/T'}(\mathcal{F}_{T'})$. Moreover, by construction the closure of $\overline{\{\xi'\}}$ is contained in the closed subset $Z'$ which avoids the fibre $X_{a(t)}$. In this way we see that $(T' \to S, a(t') \leadsto a(t), \xi')$ is an impurity of $\mathcal{F}$ above $s$. \medskip\noindent Thus we may assume that $g : T \to S$ is locally of finite presentation. Let $Z = \overline{\{\xi\}}$. By assumption $Z_t = \emptyset$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-empty-generic-fibre} this means that $Z_{t''} = \emptyset$ for $t''$ in an open subset of $\overline{\{t\}}$. Since the fibre of $T \to S$ over $s$ is a Jacobson scheme, see Morphisms, Lemma \ref{morphisms-lemma-ubiquity-Jacobson-schemes} we find that there exist a closed point $t'' \in \overline{\{t\}}$ such that $Z_{t''} = \emptyset$. Then $(g : T \to S, t' \leadsto t'', \xi)$ is the desired impurity.
The notation in the statement means this: Let $p_i : T \to T_i$ be the projection morphisms, let $t_i = p_i(t)$ and $t'_i = p_i(t')$. Finally $\xi_i \in X_{T_i}$ is the image of $\xi$. By Divisors, Lemma \ref{divisors-lemma-base-change-relative-assassin} it is true that $\xi_i$ is a point of the relative assassin of $\mathcal{F}_{T_i}$ over $T_i$. Thus the only point is to show that $\overline{\{\xi_i\}} \cap X_{t_i} = \emptyset$ for some $i$. \medskip\noindent First proof. Let $Z_i = \overline{\{\xi_i\}} \subset X_{T_i}$ and $Z = \overline{\{\xi\}} \subset X_T$ endowed with the reduced induced scheme structure. Then $Z = \lim Z_i$ by Limits, Lemma \ref{limits-lemma-inverse-limit-irreducibles}. Choose a field $k$ and a morphism $\Spec(k) \to T$ whose image is $t$. Then $$ \emptyset = Z \times_T \Spec(k) = (\lim Z_i) \times_{(\lim T_i)} \Spec(k) = \lim Z_i \times_{T_i} \Spec(k) $$ because limits commute with fibred products (limits commute with limits). Each $Z_i \times_{T_i} \Spec(k)$ is quasi-compact because $X_{T_i} \to T_i$ is of finite type and hence $Z_i \to T_i$ is of finite type. Hence $Z_i \times_{T_i} \Spec(k)$ is empty for some $i$ by Limits, Lemma \ref{limits-lemma-limit-nonempty}. Since the image of the composition $\Spec(k) \to T \to T_i$ is $t_i$ we obtain what we want. \medskip\noindent Second proof. Set $Z = \overline{\{\xi\}}$. Apply Limits, Lemma \ref{limits-lemma-separate} to this situation to obtain 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 we have $Z' \cap X_{a(t)} = \emptyset$, and \item $Z \cap X_V$ maps into $Z'$ via the morphism $X_V \to X_{T'}$. \end{enumerate} We may assume $V$ is an affine open of $T$, hence by Limits, Lemmas \ref{limits-lemma-descend-opens} and \ref{limits-lemma-limit-affine} we can find an $i$ and an affine open $V_i \subset T_i$ with $V = f_i^{-1}(V_i)$. By Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} after possibly increasing $i$ a bit we can find a morphism $a_i : V_i \to T'$ such that $a = a_i \circ f_i|_V$. The induced morphism $X_{V_i} \to X_{T'}$ maps $\xi_i$ into $Z'$. As $Z' \cap X_{a(t)} = \emptyset$ we conclude that $(T_i \to S, t'_i \leadsto t_i, \xi_i)$ is an impurity of $\mathcal{F}$ above $s$.
Let $(g : T \to S, t' \leadsto t, \xi)$ be an impurity of $\mathcal{F}$ above $s$ such that $g$ is quasi-finite at $t$. After shrinking $T$ we may assume that $g$ is locally of finite type. Apply More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point} to $T \to S$ and $t \mapsto s$. This gives us a diagram $$ \xymatrix{ T \ar[d] & T \times_S U \ar[l] \ar[d] & V \ar[l] \ar[ld] \\ S & U \ar[l] } $$ where $(U, u) \to (S, s)$ is an elementary \'etale neighbourhood and $V \subset T \times_S U$ is an open neighbourhood of $v = (t, u)$ such that $V \to U$ is finite and such that $v$ is the unique point of $V$ lying over $u$. Since the morphism $V \to T$ is \'etale hence flat we see that there exists a specialization $v' \leadsto v$ such that $v' \mapsto t'$. Note that $\kappa(t') \subset \kappa(v')$ is finite separable. Pick any point $\zeta \in X_{v'}$ mapping to $\xi \in X_{t'}$. By Divisors, Lemma \ref{divisors-lemma-base-change-relative-assassin} we see that $\zeta \in \text{Ass}_{X_V/V}(\mathcal{F}_V)$. Moreover, the closure $\overline{\{\zeta\}}$ does not meet the fibre $X_v$ as by assumption the closure $\overline{\{\xi\}}$ does not meet $X_t$. In other words $(V \to S, v' \leadsto v, \zeta)$ is an impurity of $\mathcal{F}$ above $S$. \medskip\noindent Next, let $u' \in U'$ be the image of $v'$ and let $\theta \in X_U$ be the image of $\zeta$. Then $\theta \mapsto u'$ and $u' \leadsto u$. By Divisors, Lemma \ref{divisors-lemma-base-change-relative-assassin} we see that $\theta \in \text{Ass}_{X_U/U}(\mathcal{F})$. Moreover, as $\pi : X_V \to X_U$ is finite we see that $\pi\big(\overline{\{\zeta\}}\big) = \overline{\{\pi(\zeta)\}}$. Since $v$ is the unique point of $V$ lying over $u$ we see that $X_u \cap \overline{\{\pi(\zeta)\}} = \emptyset$ because $X_v \cap \overline{\{\zeta\}} = \emptyset$. In this way we conclude that $(U \to S, u' \leadsto u, \theta)$ is an impurity of $\mathcal{F}$ above $s$ and we win.
We may replace $S$ by an affine neighbourhood of $s$. By Lemma \ref{lemma-impure-finite-presentation} we may assume that we have an impurity $(g : T \to S, t' \leadsto t, \xi)$ of such that $g$ is locally of finite type and $t$ a closed point of the fibre of $g$ above $s$. We may replace $T$ by the reduced induced scheme structure on $\overline{\{t'\}}$. Let $Z = \overline{\{\xi\}} \subset X_T$. By assumption $Z_t = \emptyset$ and the image of $Z \to T$ contains $t'$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-relative-assassin-in-neighbourhood} there exists a nonempty open $V \subset Z$ such that for any $w \in f(V)$ any generic point $\xi'$ of $V_w$ is in $\text{Ass}_{X_T/T}(\mathcal{F}_T)$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-nonempty-generic-fibre} there exists a nonempty open $W \subset T$ with $W \subset f(V)$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-finite-quasi-section-meeting-nearby-open-X} there exists a closed subscheme $T' \subset T$ such that $t \in T'$, $T' \to S$ is quasi-finite at $t$, and there exists a point $z \in T' \cap W$, $z \leadsto t$ which does not map to $s$. Choose any generic point $\xi'$ of the nonempty scheme $V_z$. Then $(T' \to S, z \leadsto t, \xi')$ is the desired impurity.
We may replace $S$ by an affine neighbourhood of $s$. Say $S = \Spec(A)$ and $s$ corresponds to the prime $\mathfrak p \subset A$. Then $\mathcal{O}_{S, s}^h = \colim_{(T, t)} \Gamma(T, \mathcal{O}_T)$ where the limit is over the opposite of the cofiltered category of affine elementary \'etale neighbourhoods $(T, t)$ of $(S, s)$, see More on Morphisms, Lemma \ref{more-morphisms-lemma-describe-henselization} and its proof. Hence $S^h = \lim_i T_i$ and we win by Lemma \ref{lemma-impure-limit}.
As an \'etale morphism is locally quasi-finite it is clear that (2) implies (3). We have seen that (3) implies (2) in Lemma \ref{lemma-quasi-finite-impurity-elementary}. We have seen that (1) implies (2) in Lemma \ref{lemma-impurity-on-henselization}. Finally, if $(T \to S, t' \leadsto t, \xi)$ is an impurity of $\mathcal{F}$ above $s$ such that $(T, t) \to (S, s)$ is an elementary \'etale neighbourhood, then we can choose a factorization $S^h \to T \to S$ of the structure morphism $S^h \to S$. Choose any point $s' \in S^h$ mapping to $t'$ and choose any $\xi' \in X_{s'}$ mapping to $\xi \in X_{t'}$. Then $(S^h \to S, s' \leadsto s, \xi')$ is an impurity of $\mathcal{F}$ above $s$. We omit the details.
It $(T \to S', t' \leadsto t, \xi)$ is an impurity of $\mathcal{F}_{S'}$ above $s'$ with $T \to S'$ quasi-finite at $t$, then $(T \to S, t' \to t, \xi)$ is an impurity of $\mathcal{F}$ above $s$ with $T \to S$ quasi-finite at $t$, see Morphisms, Lemma \ref{morphisms-lemma-composition-quasi-finite}. Hence the lemma follows immediately from the characterization (2) of purity given following Definition \ref{definition-pure}.
Let $(T \to S, t' \leadsto t, \xi)$ be an impurity of $\mathcal{F}$ above $s$. Set $T_1 = T \times_S S'$, and let $t_1$ be the unique point of $T_1$ mapping to $t$ and $s'$. Since $T_1 \to T$ is flat at $t_1$, see Morphisms, Lemma \ref{morphisms-lemma-base-change-flat}, there exists a specialization $t'_1 \leadsto t_1$ lying over $t' \leadsto t$, see Algebra, Section \ref{algebra-section-going-up}. Choose a point $\xi_1 \in X_{t'_1}$ which corresponds to a generic point of $\Spec(\kappa(t'_1) \otimes_{\kappa(t')} \kappa(\xi))$, see Schemes, Lemma \ref{schemes-lemma-points-fibre-product}. By Divisors, Lemma \ref{divisors-lemma-base-change-relative-assassin} we see that $\xi_1 \in \text{Ass}_{X_{T_1}/T_1}(\mathcal{F}_{T_1})$. As the Zariski closure of $\{\xi_1\}$ in $X_{T_1}$ maps into the Zariski closure of $\{\xi\}$ in $X_T$ we conclude that this closure is disjoint from $X_{t_1}$. Hence $(T_1 \to S', t'_1 \leadsto t_1, \xi_1)$ is an impurity of $\mathcal{F}_{S'}$ above $s'$. In other words we have proved the contrapositive to part (2) of the lemma. Finally, if $(T, t) \to (S, s)$ is an elementary \'etale neighbourhood, then $(T_1, t_1) \to (S', s')$ is an elementary \'etale neighbourhood too, and in this way we see that (1) holds.
First we reduce (1) to (2). Namely, let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$. Let $i : Z \to X$ be the corresponding closed immersion and write $\mathcal{F} = i_*\mathcal{G}$ for some finite type quasi-coherent $\mathcal{O}_Z$-module $\mathcal{G}$, see Morphisms, Section \ref{morphisms-section-support}. In case (1) $Z \to S$ is proper by assumption. Thus by Lemma \ref{lemma-supported-on-closed} case (1) reduces to case (2). \medskip\noindent Assume $f$ is proper. Let $(g : T \to S, t' \leadsto t, \xi)$ be an impurity of $\mathcal{F}$ above $s \in S$. Since $f$ is proper, it is universally closed. Hence $f_T : X_T \to T$ is closed. Since $f_T(\xi) = t'$ this implies that $t \in f(\overline{\{\xi\}})$ which is a contradiction.
Let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$, see Morphisms, Definition \ref{morphisms-definition-scheme-theoretic-support}. Then $Z \to S$ is a separated, finite type morphism of schemes with finite fibres. Hence it is separated and quasi-finite, see Morphisms, Lemma \ref{morphisms-lemma-quasi-finite}. By Lemma \ref{lemma-supported-on-closed} it suffices to prove the lemma for $Z \to S$ and the sheaf $\mathcal{F}$ viewed as a finite type quasi-coherent module on $Z$. Hence we may assume that $X \to S$ is separated and quasi-finite and that $\text{Supp}(\mathcal{F}) = X$. \medskip\noindent It follows from Lemma \ref{lemma-proper-pure} and Morphisms, Lemma \ref{morphisms-lemma-finite-proper} that (2) implies (3). Trivially (3) implies (1). Assume (1) holds. We will prove that (2) holds. It is clear that we may assume $S$ is affine. By More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite} we can find a diagram $$ \xymatrix{ X \ar[rd]_f \ar[rr]_j & & T \ar[ld]^\pi \\ & S & } $$ with $\pi$ finite and $j$ a quasi-compact open immersion. If we show that $j$ is closed, then $j$ is a closed immersion and we conclude that $f = \pi \circ j$ is finite. To show that $j$ is closed it suffices to show that specializations lift along $j$, see Schemes, Lemma \ref{schemes-lemma-quasi-compact-closed}. Let $x \in X$, set $t' = j(x)$ and let $t' \leadsto t$ be a specialization. We have to show $t \in j(X)$. Set $s' = f(x)$ and $s = \pi(t)$ so $s' \leadsto s$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical} we can find an elementary \'etale neighbourhood $(U, u) \to (S, s)$ and a decomposition $$ T_U = T \times_S U = V \amalg W $$ into open and closed subschemes, such that $V \to U$ is finite and there exists a unique point $v$ of $V$ mapping to $u$, and such that $v$ maps to $t$ in $T$. As $V \to T$ is \'etale, we can lift generalizations, see Morphisms, Lemmas \ref{morphisms-lemma-generalizations-lift-flat} and \ref{morphisms-lemma-etale-flat}. Hence there exists a specialization $v' \leadsto v$ such that $v'$ maps to $t' \in T$. In particular we see that $v' \in X_U \subset T_U$. Denote $u' \in U$ the image of $t'$. Note that $v' \in \text{Ass}_{X_U/U}(\mathcal{F})$ because $X_{u'}$ is a finite discrete set and $X_{u'} = \text{Supp}(\mathcal{F}_{u'})$. As $\mathcal{F}$ is pure relative to $S$ we see that $v'$ must specialize to a point in $X_u$. Since $v$ is the only point of $V$ lying over $u$ (and since no point of $W$ can be a specialization of $v'$) we see that $v \in X_u$. Hence $t \in X$.
Let $\xi \in X$ with $s' = f(\xi)$ and $s' \leadsto s$ a specialization of $S$. If $\xi$ is an associated point of $X_{s'}$, then $\xi$ is the unique generic point because $X_{s'}$ is an integral scheme. Let $\xi_0$ be the unique generic point of $X_s$. As $X \to S$ is flat we can lift $s' \leadsto s$ to a specialization $\xi' \leadsto \xi_0$ in $X$, see Morphisms, Lemma \ref{morphisms-lemma-generalizations-lift-flat}. The $\xi \leadsto \xi'$ because $\xi$ is the generic point of $X_{s'}$ hence $\xi \leadsto \xi_0$. This means that $(\text{id}_S, s' \to s, \xi)$ is not an impurity of $\mathcal{O}_X$ above $s$. Since the assumption that $f$ is finite type, flat with geometrically integral fibres is preserved under base change, we see that there doesn't exist an impurity after any base change. In this way we see that $X$ is universally $S$-pure.
Let $x' \leadsto t$ be a specialization with $t \in X_s$. We may replace $X$ by an affine neighbourhood of $t$ and $S$ by an affine neighbourhood of $s$. Choose a closed immersion $i : X \to \mathbf{A}^n_S$. Then it suffices to prove the lemma for the module $i_*\mathcal{F}$ on $\mathbf{A}^n_S$ and the point $i(x')$. Hence we may assume $X \to S$ is of finite presentation. \medskip\noindent Let $x' \leadsto t$ be a specialization with $t \in X_s$. Set $A = \mathcal{O}_{S, s}$, $B = \mathcal{O}_{X, t}$, and $N = \mathcal{F}_t$. Note that $B$ is essentially of finite presentation over $A$ and that $N$ is a finite $B$-module flat over $A$. Also $N$ is a finitely presented $B$-module by Lemma \ref{lemma-finite-type-flat-at-point-local}. Let $\mathfrak q' \subset B$ be the prime ideal corresponding to $x'$ and let $\mathfrak p' \subset A$ be the prime ideal corresponding to $s'$. The assumption $x' \in \text{Ass}_{X/S}(\mathcal{F})$ means that $\mathfrak q'$ is an associated prime of $N \otimes_A \kappa(\mathfrak p')$. Let $\Sigma \subset B$ be the multiplicative subset of elements which are not zerodivisors on $N/\mathfrak m_A N$. By Lemma \ref{lemma-homothety-universally-injective} the map $N \to \Sigma^{-1}N$ is universally injective. In particular, we see that $N \otimes_A \kappa(\mathfrak p') \to \Sigma^{-1}N \otimes_A \kappa(\mathfrak p')$ is injective which implies that $\mathfrak q'$ is an associated prime of $\Sigma^{-1}N \otimes_A \kappa(\mathfrak p')$ and hence $\mathfrak q'$ is in the image of $\Spec(\Sigma^{-1}B) \to \Spec(B)$. Thus Lemma \ref{lemma-homothety-spectrum} implies that $\mathfrak q' \subset \mathfrak q$ for some prime $\mathfrak q \in \text{Ass}_B(N/\mathfrak m_A N)$ (which in particular implies that $\mathfrak m_A = A \cap \mathfrak q$). If $x \in X_s$ denotes the point corresponding to $\mathfrak q$, then $x \in \text{Ass}_{X_s}(\mathcal{F}_s)$ and $x' \leadsto x$ as desired.
Since the morphism $S' \to S$ is \'etale, we see that if $\mathcal{F}$ is pure along $X_s$, then $\mathcal{F} \times_S S'$ is pure along $X_s$, see Lemma \ref{lemma-quasi-finite-base-change}. Since purity satisfies flat descent, see Lemma \ref{lemma-flat-descend-pure}, we see that if $\mathcal{F} \times_S S'$ is pure along $X_{s'}$, then $\mathcal{F}$ is pure along $X_s$. Hence we may replace $S$ by $S'$ and assume that $S = S'$ so that $g : X' \to X$ is an \'etale morphism between schemes of finite type over $S$. Moreover, we may replace $S$ by $\Spec(\mathcal{O}_{S, s})$ and assume that $S$ is local. \medskip\noindent First, assume that $\mathcal{F}$ is pure along $X_s$. In this case every point of $\text{Ass}_{X/S}(\mathcal{F})$ specializes to a point of $X_s$ by purity. Hence by Lemma \ref{lemma-associated-point-specializes} we see that every point of $\text{Ass}_{X/S}(\mathcal{F})$ specializes to a point of $\text{Ass}_{X_s}(\mathcal{F}_s)$. Thus every point of $\text{Ass}_{X/S}(\mathcal{F})$ is in the image of $g$ (as the image is open and contains $\text{Ass}_{X_s}(\mathcal{F}_s)$). \medskip\noindent Conversely, assume that $g(X')$ contains $\text{Ass}_{X/S}(\mathcal{F})$. Let $S^h = \Spec(\mathcal{O}_{S, s}^h)$ be the henselization of $S$ at $s$. Denote $g^h : (X')^h \to X^h$ the base change of $g$ by $S^h \to S$, and denote $\mathcal{F}^h$ the pullback of $\mathcal{F}$ to $X^h$. By Divisors, Lemma \ref{divisors-lemma-base-change-relative-assassin} and Remark \ref{divisors-remark-base-change-relative-assassin} the relative assassin $\text{Ass}_{X^h/S^h}(\mathcal{F}^h)$ is the inverse image of $\text{Ass}_{X/S}(\mathcal{F})$ via the projection $X^h \to X$. As we have assumed that $g(X')$ contains $\text{Ass}_{X/S}(\mathcal{F})$ we conclude that the base change $g^h((X')^h) = g(X') \times_S S^h$ contains $\text{Ass}_{X^h/S^h}(\mathcal{F}^h)$. In this way we reduce to the case where $S$ is the spectrum of a henselian local ring. Let $x \in \text{Ass}_{X/S}(\mathcal{F})$. To finish the proof of the lemma we have to show that $x$ specializes to a point of $X_s$, see criterion (4) for purity in discussion following Definition \ref{definition-pure}. By assumption there exists a $x' \in X'$ such that $g(x') = x$. As $g : X' \to X$ is \'etale, we see that $x' \in \text{Ass}_{X'/S}(g^*\mathcal{F})$, see Lemma \ref{lemma-etale-weak-assassin-up-down} (applied to the morphism of fibres $X'_w \to X_w$ where $w \in S$ is the image of $x'$). Since $g^*\mathcal{F}$ is pure along $X'_s$ we see that $x' \leadsto y$ for some $y \in X'_s$. Hence $x = g(x') \leadsto g(y)$ and $g(y) \in X_s$ as desired.
We first make a preliminary remark. Suppose that $(S', s') \to (S, s)$ is an elementary \'etale neighbourhood. Denote $\mathcal{F}'$ the pullback of $\mathcal{F}$ to $X' = X \times_S S'$. By the discussion following Definition \ref{definition-pure} we see that $\mathcal{F}'$ is pure along $X'_{s'}$. Moreover, $\mathcal{F}'$ is flat over $S'$ along $X'_{s'}$. Then it suffices to prove that $\mathcal{F}'$ is universally pure along $X'_{s'}$. Namely, given any morphism $(T, t) \to (S, s)$ of pointed schemes the fibre product $(T', t') = (T \times_S S', (t, s'))$ is flat over $(T, t)$ and hence if $\mathcal{F}_{T'}$ is pure along $X_{t'}$ then $\mathcal{F}_T$ is pure along $X_t$ by Lemma \ref{lemma-flat-descend-pure}. Thus during the proof we may always replace $(s, S)$ by an elementary \'etale neighbourhood. We may also replace $S$ by $\Spec(\mathcal{O}_{S, s})$ due to the local nature of the problem. \medskip\noindent Choose an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram $$ \xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & \Spec(\mathcal{O}_{S', s'}) \ar[l] } $$ such that $X' \to X \times_S \Spec(\mathcal{O}_{S', s'})$ is \'etale, $X_s = g((X')_{s'})$, the scheme $X'$ is affine, and such that $\Gamma(X', g^*\mathcal{F})$ is a free $\mathcal{O}_{S', s'}$-module, see Lemma \ref{lemma-finite-type-flat-along-fibre-free-variant}. Note that $X' \to \Spec(\mathcal{O}_{S', s'})$ is of finite type (as a quasi-compact morphism which is the composition of an \'etale morphism and the base change of a finite type morphism). By our preliminary remarks in the first paragraph of the proof we may replace $S$ by $\Spec(\mathcal{O}_{S', s'})$. Hence we may assume there exists a commutative diagram $$ \xymatrix{ X \ar[dr] & & X' \ar[ll]^g \ar[ld] \\ & S & } $$ of schemes of finite type over $S$, where $g$ is \'etale, $X_s \subset g(X')$, with $S$ local with closed point $s$, with $X'$ affine, and with $\Gamma(X', g^*\mathcal{F})$ a free $\Gamma(S, \mathcal{O}_S)$-module. Note that in this case $g^*\mathcal{F}$ is universally pure over $S$, see Lemma \ref{lemma-affine-locally-projective-pure}. \medskip\noindent In this situation we apply Lemma \ref{lemma-criterion} to deduce that $\text{Ass}_{X/S}(\mathcal{F}) \subset g(X')$ from our assumption that $\mathcal{F}$ is pure along $X_s$ and flat over $S$ along $X_s$. By Divisors, Lemma \ref{divisors-lemma-base-change-relative-assassin} and Remark \ref{divisors-remark-base-change-relative-assassin} we see that for any morphism of pointed schemes $(T, t) \to (S, s)$ we have $$ \text{Ass}_{X_T/T}(\mathcal{F}_T) \subset (X_T \to X)^{-1}(\text{Ass}_{X/S}(\mathcal{F})) \subset g(X') \times_S T = g_T(X'_T). $$ Hence by Lemma \ref{lemma-criterion} applied to the base change of our displayed diagram to $(T, t)$ we conclude that $\mathcal{F}_T$ is pure along $X_t$ as desired.
Let $s \in U$. Using Lemma \ref{lemma-finite-presentation-flat-along-fibre} we can find an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram $$ \xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ S & S' \ar[l] } $$ such that $g$ is \'etale, $X_s \subset g(X')$, the schemes $X'$, $S'$ are affine, and such that $\Gamma(X', g^*\mathcal{F})$ is a projective $\Gamma(S', \mathcal{O}_{S'})$-module. Note that $g^*\mathcal{F}$ is universally pure over $S'$, see Lemma \ref{lemma-affine-locally-projective-pure}. Set $W' \subset X \times_S S'$ equal to the image of the \'etale morphism $X' \to X \times_S S'$. Note that $W$ is open and quasi-compact over $S'$. Set $$ E = \{t \in S' \mid \text{Ass}_{X_t}(\mathcal{F}_t) \subset W' \}. $$ By More on Morphisms, Lemma \ref{more-morphisms-lemma-relative-assassin-constructible} $E$ is a constructible subset of $S'$. By Lemma \ref{lemma-criterion} we see that $\Spec(\mathcal{O}_{S', s'}) \subset E$. By Morphisms, Lemma \ref{morphisms-lemma-constructible-containing-open} we see that $E$ contains an open neighbourhood $V'$ of $s'$. Applying Lemma \ref{lemma-criterion} once more we see that for any point $s_1$ in the image of $V'$ in $S$ the sheaf $\mathcal{F}$ is pure along $X_{s_1}$. Since $S' \to S$ is \'etale the image of $V'$ in $S$ is open and we win.
Let $\mathcal{K} = \Ker(\varphi)$. Our goal is to prove that $\mathcal{K} = 0$. In order to do this it suffices to prove that $\text{WeakAss}_X(\mathcal{K}) = \emptyset$, see Divisors, Lemma \ref{divisors-lemma-weakly-ass-zero}. We have $\text{WeakAss}_X(\mathcal{K}) \subset \text{WeakAss}_X(\mathcal{F})$, see Divisors, Lemma \ref{divisors-lemma-ses-weakly-ass}. As $\mathcal{F}$ is flat we see from Lemma \ref{lemma-bourbaki-finite-type-general-base} that $\text{WeakAss}_X(\mathcal{F}) \subset \text{Ass}_{X/S}(\mathcal{F})$. By purity any point $x'$ of $\text{Ass}_{X/S}(\mathcal{F})$ is a generalization of a point of $X_s$, and hence is the specialization of a point $x \in \text{Ass}_{X_s}(\mathcal{F}_s)$, by Lemma \ref{lemma-associated-point-specializes}. Hence the injectivity of $\varphi_x$ implies the injectivity of $\varphi_{x'}$, whence $\mathcal{K}_{x'} = 0$.