hcju/stacksqa huggingface dataset:
That parts (1) and (2) are equivalent is immediate from the definition. \medskip\noindent Recall that $\mathcal{O} = \mathcal{O}_{Y, \overline{y}}$ is the filtered colimit of $\mathcal{O}(V)$ over the category of \'etale neighbourhoods $(V, \overline{v}) \to (Y, \overline{y})$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-cofinal-etale}). Moreover, it suffices to consider affine \'etale neighbourhoods $V$. Hence $Y^{sh} = \Spec(\mathcal{O}) = \lim \Spec(\mathcal{O}(V)) = \lim V$. Thus we see that (1) implies (3) by Lemma \ref{lemma-impure-limit}. \medskip\noindent Since an \'etale morphism is locally quasi-finite (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-locally-quasi-finite}) we see that (3) implies (4). \medskip\noindent Finally, assume (4). After replacing $T$ by an open neighbourhood of $t$ we may assume $T \to Y$ is locally quasi-finite. By Lemma \ref{lemma-flat-ascent-impurity} we find an impurity $(T_1 \to Y^{sh}, t_1' \leadsto t_1, \xi_1)$ with $T_1 \to T \times_Y Y^{sh}$ \'etale. Since an \'etale morphism is locally quasi-finite and using Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-quasi-finite} and Morphisms, Lemma \ref{morphisms-lemma-composition-quasi-finite} we see that $T_1 \to Y^{sh}$ is locally quasi-finite. As $\mathcal{O}$ is strictly henselian, we can apply More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-makes-quasi-finite-finite-at-point} to see that after replacing $T_1$ by an open and closed neighbourhood of $t_1$ we may assume that $T_1 \to Y^{sh} = \Spec(\mathcal{O})$ is finite. Let $\theta \in |X^{sh}|$ be the image of $\xi_1$ and let $y' \in \Spec(\mathcal{O})$ be the image of $t_1'$. By Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-base-change-relative-assassin} we see that $\theta \in \text{Ass}_{X^{sh}/Y^{sh}}(\mathcal{F}^{sh})$. Since $\pi : X_{T_1} \to X^{sh}$ is finite, it induces a closed map $|X_{T_1}| \to |X^{sh}|$. Hence the image of $\overline{\{\xi_1\}}$ is $\overline{\{\theta\}}$. It follows that $(Y^{sh} \to Y, y' \leadsto \overline{y}, \theta)$ is an impurity of $\mathcal{F}$ above $y$ and the proof is complete.
It $(T \to Y', t' \leadsto t, \xi)$ is an impurity of $\mathcal{F}_{Y'}$ above $y'$ with $T \to Y'$ quasi-finite at $t$, then $(T \to Y, t' \to t, \xi)$ is an impurity of $\mathcal{F}$ above $y$ with $T \to Y$ quasi-finite at $t$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-quasi-finite}. Hence the lemma follows immediately from the definition of purity.
This is true because impurities go up along a flat base change, see Lemma \ref{lemma-flat-ascent-impurity}. For example part (1) follows because by any impurity $(T \to Y, t' \leadsto t, \xi)$ of $\mathcal{F}$ above $y$ with $T \to Y$ quasi-finite at $t$ by the lemma leads to an impurity $(T_1 \to Y_1, t_1' \leadsto t_1, \xi_1)$ of the pullback $\mathcal{F}_1$ of $\mathcal{F}$ to $X_1 = Y_1 \times_Y X$ over $y_1$ such that $T_1$ is \'etale over $Y_1 \times_Y T$. Hence $T_1 \to Y_1$ is quasi-finite at $t_1$ because \'etale morphisms are locally quasi-finite and compositions of locally quasi-finite morphisms are locally quasi-finite (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-etale-locally-quasi-finite} and \ref{spaces-morphisms-lemma-composition-quasi-finite}). Similarly for part (2).
First we reduce (1) to (2). Namely, let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$ (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-scheme-theoretic-support}). 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 $\mathcal{G}$. In case (1) $Z \to Y$ 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 $(g : T \to Y, t' \leadsto t, \xi)$ be an impurity of $\mathcal{F}$ above $y$. 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.
The question is \'etale local on $Y$ hence we may assume $Y$ is an affine scheme. Then $X$ is quasi-compact, hence we can choose an affine scheme $X'$ and a surjective \'etale morphism $X' \to X$. Then we may apply More on Flatness, Lemma \ref{flat-lemma-existence-complete} to $X' \to Y$, $(X' \to Y)^*\mathcal{F}$, and $y$ to get what we want.
Choose a scheme $V$, a point $v \in V$, and an \'etale morphism $V \to Y$ mapping $v$ to $y$. Choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. Then $|U_v| \to F$ is an open continuous map of topological spaces as $|U| \to |X|$ is continuous and open. Hence the result follows from the case of schemes which is More on Flatness, Lemma \ref{flat-lemma-open-in-fibre-where-flat}.
Choose a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r]_g & V \ar[d] \\ X \ar[r]^f & Y } $$ where $U$ and $V$ are schemes and the vertical arrows are surjective \'etale. Choose $u \in U$ mapping to $x$. Let $\mathcal{E} = \mathcal{F}|_U$ and $\mathcal{H} = \mathcal{G}|_V$. Let $v \in V$ be the image of $u$. Then $x \in \text{WeakAss}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G})$ if and only if $u \in \text{WeakAss}_X(\mathcal{E} \otimes_{\mathcal{O}_X} g^*\mathcal{H})$ by Divisors on Spaces, Definition \ref{spaces-divisors-definition-weakly-associated}. Similarly, $y \in \text{WeakAss}_Y(\mathcal{G})$ if and only if $v \in \text{WeakAss}_V(\mathcal{H})$. Finally, we have $x \in \text{Ass}_{X/Y}(\mathcal{F})$ if and only if $u \in \text{Ass}_{U_v}(\mathcal{E}|_{U_v})$ by Divisors on Spaces, Definition \ref{spaces-divisors-definition-relative-weak-assassin}. Observe that flatness of $\mathcal{F}$ at $x$ is equivalent to flatness of $\mathcal{E}$ at $u$, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-flat-module}. The equivalence for $g : U \to V$, $\mathcal{E}$, $\mathcal{H}$, $u$, and $v$ is More on Flatness, Lemma \ref{flat-lemma-bourbaki-finite-type-general-base-at-point}.
Choose an \'etale morphism $(Y', y') \to (Y, y)$ where $Y'$ is an affine scheme. Then $X \times_Y Y'$ is quasi-compact. Choose an affine scheme $X'$ and a surjective \'etale morphism $X' \to X \times_Y Y'$. Picture $$ \xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ Y & Y' \ar[l] } $$ Then $\mathcal{F}' = g^*\mathcal{F}$ is flat over $Y'$ at all points of $X'_{y'}$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-module-flat}. Hence we can apply the lemma in the case of schemes (More on Flatness, Lemma \ref{flat-lemma-finite-type-flat-along-fibre-free-variant}) to the morphism $X' \to Y'$, the quasi-coherent sheaf $g^*\mathcal{F}$, and the point $y'$. This gives an \'etale morphism $(Y'', y'') \to (Y', y')$ and a commutative diagram $$ \xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] & X'' \ar[l]^{g'} \ar[d] \\ Y & Y' \ar[l] & \Spec(\mathcal{O}_{Y'', y''}) \ar[l] } $$ To get what we want we take $(Y'', y'') \to (Y, y)$ and $g \circ g' : X'' \to X$.
Observe that $|X_s| \subset |X|$ is the set of points of $|X|$ lying over $s \in S$, see Decent Spaces, Lemma \ref{decent-spaces-lemma-topology-fibre}. Let $t \in |X_s|$ be a specialization of $x'$ in $|X|$. Choose an affine scheme $U$ and a point $u \in U$ and an \'etale morphism $\varphi : U \to X$ mapping $u$ to $t$. By Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-specialization} we can choose a specialization $u' \leadsto u$ with $u'$ mapping to $x'$. Set $g = f \circ \varphi$. Observe that $s' = g(u') = f(x')$ specializes to $s$. By our definition of $\text{Ass}_{X/S}(\mathcal{F})$ we have $u' \in \text{Ass}_{U/S}(\varphi^*\mathcal{F})$. By the schemes version of this lemma (More on Flatness, Lemma \ref{flat-lemma-associated-point-specializes}) we see that there is a specialization $u' \leadsto u$ with $u \in \text{Ass}_{U_s}(\varphi^*\mathcal{F}_s) = \text{Ass}_{U/S}(\varphi^*\mathcal{F}) \cap U_s$. Hence $x = \varphi(u) \in \text{Ass}_{X/S}(\mathcal{F})$ lies over $s$ and the lemma is proved.
By Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian} the map $|X'_Z| \to |X_Z| \times_{|X|} |X'|$ is surjective as $X'_Z$ is equal to $X_Z \times_X X'$. By Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-base-change-relative-assassin} the map $|X_Z| \to |X|$ sends $\text{Ass}_{X_Z/Z}(\mathcal{F}_Z)$ into $\text{Ass}_{X/Y}(\mathcal{F})$. The lemma follows.
The assumptions on $X \to Y$ and $X' \to Y$ guarantee that we may apply the material in Sections \ref{section-impure} and \ref{section-pure} to these morphisms and the sheaves $\mathcal{F}$ and $g^*\mathcal{F}$. Since $g$ is \'etale we see that $\text{Ass}_{X'/Y}(g^*\mathcal{F})$ is the inverse image of $\text{Ass}_{X/Y}(\mathcal{F})$ and the same remains true after base change. \medskip\noindent Proof of (1). Assume $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$. Suppose that $(T \to Y, t' \leadsto t, \xi)$ is an impurity of $\mathcal{F}$ above $y$. Since $\text{Ass}_{X_T/T}(\mathcal{F}_T) \subset g_T(|X'_T|)$ by Lemma \ref{lemma-contains-relative-ass-after-base-change} we can choose a point $\xi' \in |X'_T|$ mapping to $\xi$. By the above we see that $(T \to Y, t' \leadsto t, \xi')$ is an impurity of $g^*\mathcal{F}$ above $y'$. This implies (1) is true. \medskip\noindent Proof of (2). This follows from the fact that $g(|X'|)$ is open in $|X|$ and the fact that by purity every point of $\text{Ass}_{X/Y}(\mathcal{F})$ specializes to a point of $F$. \medskip\noindent Proof of (3). This follows from the fact that $g(|X'|)$ is open in $|X|$ and the fact that by purity combined with Lemma \ref{lemma-associated-point-specializes} every point of $\text{Ass}_{X/Y}(\mathcal{F})$ specializes to a point of $\text{Ass}_{X/Y}(\mathcal{F}) \cap F$.
Consider the morphism $\Spec(\mathcal{O}_{Y, \overline{y}}) \to Y$. This is a flat morphism from the spectrum of a stricly henselian local ring which maps the closed point to $y$. By Lemma \ref{lemma-flat-descend-pure} we reduce to the case described in the next paragraph. \medskip\noindent Assume $Y$ is the spectrum of a strictly henselian local ring $R$ with closed point $y$. By Lemma \ref{lemma-finite-type-flat-along-fibre-free-variant} there exists an \'etale morphism $g : X' \to X$ with $g(|X'|) \supset |X_y|$, with $X'$ affine, and with $\Gamma(X', g^*\mathcal{F})$ a free $R$-module. Then $g^*\mathcal{F}$ is universally pure relative to $Y$, see More on Flatness, Lemma \ref{flat-lemma-affine-locally-projective-pure}. Hence it suffices to prove that $g(|X'|)$ contains $\text{Ass}_{X/Y}(\mathcal{F})$ by Lemma \ref{lemma-pure-on-top} part (1). This in turn follows from Lemma \ref{lemma-pure-on-top} part (2).
The final assertion means that $\mathcal{F}_Z \to (g_Z)_*g_Z^*\mathcal{F}_Z$ is injective for any morphism $Z \to Y$. Since the assumption on the relative assassin is preserved by base change (Lemma \ref{lemma-contains-relative-ass-after-base-change}) it suffices to prove the injectivity of the displayed arrow. \medskip\noindent Let $\mathcal{K} = \Ker(\mathcal{F} \to g_*g^*\mathcal{F})$. 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 on Spaces, Lemma \ref{spaces-divisors-lemma-weakly-ass-zero}. We have $\text{WeakAss}_X(\mathcal{K}) \subset \text{WeakAss}_X(\mathcal{F})$, see Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-ses-weakly-ass}. As $\mathcal{F}$ is flat we see from Lemma \ref{lemma-bourbaki-finite-type-general-base} that $\text{WeakAss}_X(\mathcal{F}) \subset \text{Ass}_{X/Y}(\mathcal{F})$. By assumption any point $x$ of $\text{Ass}_{X/Y}(\mathcal{F})$ is the image of some $x' \in |X'|$. Since $g$ is flat the local ring map $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{X', \overline{x}'}$ is faithfully flat, hence the map $$ \mathcal{F}_{\overline{x}} \longrightarrow (g^*\mathcal{F})_{\overline{x}'} = \mathcal{F}_{\overline{x}} \otimes_{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{X', \overline{x}'} $$ is injective (see Algebra, Lemma \ref{algebra-lemma-faithfully-flat-universally-injective}). Since the displayed arrow factors through $\mathcal{F}_{\overline{x}} \to (g_*g^*\mathcal{F})_{\overline{x}}$, we conclude that $\mathcal{K}_{\overline{x}} = 0$. Hence $x$ cannot be a weakly associated point of $\mathcal{K}$ and we win.
Let $\{T_i \to T\}_{i \in I}$ be an fpqc covering of schemes over $B$. Set $X_i = X_{T_i} = X \times_S T_i$ and $u_i = u_{T_i}$. Note that $\{X_i \to X_T\}_{i \in I}$ is an fpqc covering of $X_T$, see Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-fpqc}. In particular, for every $x \in |X_T|$ there exists an $i \in I$ and an $x_i \in |X_i|$ mapping to $x$. Since $\mathcal{O}_{X_T, \overline{x}} \to \mathcal{O}_{X_i, \overline{x_i}}$ is flat, hence faithfully flat (see Morphisms of Spaces, Section \ref{spaces-morphisms-section-flat}). we conclude that $(u_i)_{x_i}$ is injective, surjective, bijective, or zero if and only if $(u_T)_x$ is injective, surjective, bijective, or zero. The lemma follows.
let $v : \mathcal{H} \to \mathcal{E}$ be a map of quasi-coherent modules on an algebraic space $Y$ and let $\varphi : Y' \to Y$ be a surjective flat morphism of algebraic spaces, then $v$ is an isomorphism, injective, surjective, or zero if and only if $\varphi^*v$ is an isomorphism, injective, surjective, or zero. Namely, for every $y \in |Y|$ there exists a $y' \in |Y'|$ and the map of local rings $\mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{Y', \overline{y'}}$ is faithfully flat (see Morphisms of Spaces, Section \ref{spaces-morphisms-section-flat}). Of course, to check for injectivity or being zero it suffices to look at the points in the support of $\mathcal{H}$, and to check for surjectivity it suffices to look at points in the support of $\mathcal{E}$. Moreover, under the finite type assumptions as in the statement of the lemma, taking the supports commutes with base change, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-support-finite-type}. Thus the lemma is clear.
Let $T$ be a scheme over $B$. Denote $i_T : Z_T \to X_T$ the base change of $i$ and $\mathcal{H}_T$ the pullback of $\mathcal{H}$ to $Z_T$. Observe that $(i^*\mathcal{F})_T = i_T^*\mathcal{F}_T$ and $i_{T, *}\mathcal{H}_T = (i_*\mathcal{H})_T$. The first statement follows from commutativity of pullbacks and the second from Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-affine-base-change}. Hence we see that $u_T$ and $v_T$ are adjoint maps as well. Thus $u_T = 0$ if and only if $v_T = 0$. This proves (1). In case (2) we see that $u_T$ is surjective if and only if $v_T$ is surjective because $u_T$ factors as $$ \mathcal{F}_T \to i_{T, *}i_T^*\mathcal{F}_T \xrightarrow{i_{T, *}v_T} i_{T, *}\mathcal{H}_T $$ and the fact that $i_{T, *}$ is an exact functor fully faithfully embedding the category of quasi-coherent modules on $Z_T$ into the category of quasi-coherent $\mathcal{O}_{X_T}$-modules. See Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-i-star-equivalence}.
By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-affine-base-change} we have $g_{T, *}u_T = u'_T$. Moreover, $g_{T, *} : \QCoh(\mathcal{O}_{X_T}) \to \QCoh(\mathcal{O}_X)$ is a faithful, exact functor reflecting isomorphisms, injective maps, and surjective maps.
Part (1) follows from the following statement: If $T' \to T$ is a surjective flat morphism of algebraic spaces over $Y$, then $\mathcal{F}_{T'}$ is flat over $T'$ if and only if $\mathcal{F}_T$ is flat over $T$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-module-flat}. Part (2) follows from Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-flat} if $f$ is also quasi-separated (i.e., $f$ is of finite presentation). For the general case, first reduce to the case where the base is affine and then cover $X$ by finitely many affines to reduce to the quasi-separated case. Details omitted.
Let $U \to X$ be a surjective \'etale morphism of algebraic spaces where $U = \coprod U_i$ is a disjoint union of affine schemes (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}). By Lemma \ref{lemma-iso-go-up} we see that we may replace $X$ by $U$. In other words, we may assume that $X = \coprod X_i$ is a disjoint union of affine schemes $X_i$. Suppose that we can prove the lemma for $u_i = u|_{X_i}$. Then we find a closed subscheme $Z_i \subset S$ such that $T \to S$ factors through $Z_i$ and $u_{i, Z_i}$ is zero. If $Z_i = \Spec(R/I_i) \subset \Spec(R) = S$, then taking $Z = \Spec(R/\sum I_i)$ works. Thus we may assume that $X = \Spec(A)$ is affine. \medskip\noindent Choose a finite affine open covering $T = T_1 \cup \ldots \cup T_m$. It is clear that we may replace $T$ by $\coprod_{j = 1, \ldots, m} T_j$. Hence we may assume $T$ is affine. Say $T = \Spec(R')$. Let $u : M \to N$ be the homomorphisms of $A$-modules corresponding to $u : \mathcal{F} \to \mathcal{G}$. Then $N$ is a flat $R$-module as $\mathcal{G}$ is flat over $S$. The assumption of the lemma means that the composition $$ M \otimes_R R' \to N \otimes_R R' $$ is zero. Let $z \in M$. By Lazard's theorem (Algebra, Theorem \ref{algebra-theorem-lazard}) and the fact that $\otimes$ commutes with colimits we can find free $R$-module $F_z$, an element $\tilde z \in F_z$, and a map $F_z \to N$ such that $u(z)$ is the image of $\tilde z$ and $\tilde z$ maps to zero in $F_z \otimes_R R'$. Choose a basis $\{e_{z, \alpha}\}$ of $F_z$ and write $\tilde z = \sum f_{z, \alpha} e_{z, \alpha}$ with $f_{z, \alpha} \in R$. Let $I \subset R$ be the ideal generated by the elements $f_{z, \alpha}$ with $z$ ranging over all elements of $M$. By construction $I$ maps to zero in $R'$ and the elements $\tilde z$ map to zero in $F_z/IF_z$ whence in $N/IN$. Thus $Z = \Spec(R/I)$ is a solution to the problem in this case. \medskip\noindent Assume $\mathcal{F}$ is of finite type with quasi-compact scheme theoretic support. Write $Z = \Spec(R/I)$. Write $I = \bigcup I_\lambda$ as a filtered union of finitely generated ideals. Set $Z_\lambda = \Spec(R/I_\lambda)$, so $Z = \colim Z_\lambda$. Since $u_Z$ is zero, we see that $u_{Z_\lambda}$ is zero for some $\lambda$ by Lemma \ref{lemma-iso-limits}. This finishes the proof of the lemma.
To check whether $u_s$ is zero, is \'etale local on the fibre $X_s$. Hence we may pick a point $x \in |X_s| \subset |X|$ and check in an \'etale neighbourhood. Choose $$ \xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (S', s') \ar[l] } $$ as in Proposition \ref{proposition-finite-presentation-flat-at-point}. Let $T' \subset S'$ be the inverse image of $T$. Observe that $s'$ is in the closure of $T'$ because $S' \to S$ is open. Hence we reduce to the algebra problem described in the next paragraph. \medskip\noindent We have an $R$-module map $u : M \to N$ such that $N$ is projective as an $R$-module and such that $u_t : M \otimes_R \kappa(t) \to N \otimes_R \kappa(t)$ is zero for each $t \in T$. Problem: show that $u_s = 0$. Let $I \subset R$ be the ideal defined in Lemma \ref{lemma-F-zero-module-map}. Then $I$ maps to zero in $\kappa(t)$ for all $t \in T$. Hence $T \subset V(I)$. Since $s$ is in the closure of $T$ we see that $s \in V(I)$. Hence $u_s = 0$.
By Lemma \ref{lemma-finite-type-flat-pure-is-universal} the module $\mathcal{G}$ is universally pure relative to $B$. In order to prove that $F_{zero}$ is an algebraic space, it suffices to show that $F_{zero} \to B$ is representable, see Spaces, Lemma \ref{spaces-lemma-representable-over-space}. Let $B' \to B$ be a morphism where $B'$ is a scheme and let $u' : \mathcal{F}' \to \mathcal{G}'$ be the pullback of $u$ to $X' = X_{B'}$. Then the associated functor $F'_{zero}$ equals $F_{zero} \times_B B'$. This reduces us to the case that $B$ is a scheme. \medskip\noindent Assume $B$ is a scheme. We will show that $F_{zero}$ is representable by a closed subscheme of $B$. By Lemma \ref{lemma-iso-sheaf} and Descent, Lemmas \ref{descent-lemma-closed-immersion} and \ref{descent-lemma-descent-data-sheaves} the question is local for the \'etale topology on $B$. Let $b \in B$. We first replace $B$ by an affine neighbourhood of $b$. Choose a diagram $$ \xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ B & B' \ar[l] } $$ and $b' \in B'$ mapping to $b \in B$ as in Lemma \ref{lemma-finite-presentation-flat-along-fibre}. As we are working \'etale locally, we may replace $B$ by $B'$ and assume that we have a diagram $$ \xymatrix{ X \ar[rd] & & X' \ar[ll]^g \ar[ld] \\ & B } $$ with $B$ and $X'$ affine such that $\Gamma(X', g^*\mathcal{G})$ is a projective $\Gamma(B, \mathcal{O}_B)$-module and $g(|X'|) \supset |X_b|$. Let $U \subset X$ be the open subspace with $|U| = g(|X'|)$. By Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-relative-assassin-constructible} the set $$ E = \{t \in B : \text{Ass}_{X_t}(\mathcal{G}_t) \subset |U_t|\} = \{t \in B : \text{Ass}_{X/B}(\mathcal{G}) \cap |X_t| \subset |U_t|\} $$ is constructible in $B$. By Lemma \ref{lemma-pure-on-top} part (2) we see that $E$ contains $\Spec(\mathcal{O}_{B, b})$. By Morphisms, Lemma \ref{morphisms-lemma-constructible-containing-open} we see that $E$ contains an open neighbourhood of $b$. Hence after replacing $B$ by a smaller affine neighbourhood of $b$ we may assume that $\text{Ass}_{X/B}(\mathcal{G}) \subset g(|X'|)$. \medskip\noindent From Lemma \ref{lemma-universally-separating} it follows that $u : \mathcal{F} \to \mathcal{G}$ is injective if and only if $g^*u : g^*\mathcal{F} \to g^*\mathcal{G}$ is injective, and the same remains true after any base change. Hence we have reduced to the case where, in addition to the assumptions in the theorem, $X \to B$ is a morphism of affine schemes and $\Gamma(X, \mathcal{G})$ is a projective $\Gamma(B, \mathcal{O}_B)$-module. This case follows immediately from Lemma \ref{lemma-F-zero-module-map}. \medskip\noindent We still have to show that $F_{zero} \to B$ is of finite presentation if $\mathcal{F}$ is of finite type. This follows from Lemma \ref{lemma-iso-limits} combined with Limits of Spaces, Proposition \ref{spaces-limits-proposition-characterize-locally-finite-presentation}.
If $f$ is of finite presentation, then this follows immediately from Lemmas \ref{lemma-F-zero-closed-pure} and \ref{lemma-proper-pure}. This is the only case of interest and we urge the reader to skip the rest of the proof, which deals with the possibility (allowed by the assumptions in this lemma) that $f$ is not quasi-separated or quasi-compact. \medskip\noindent Let $i : Z \to X$ be the closed subspace cut out by the zeroth fitting ideal of $\mathcal{G}$ (Divisors on Spaces, Section \ref{spaces-divisors-section-fitting-ideals}). Then $Z \to B$ is proper by assumption (see Derived Categories of Spaces, Section \ref{spaces-perfect-section-proper-over-base}). On the other hand $i$ is of finite presentation (Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-fitting-ideal-of-finitely-presented} and Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-finite-presentation}). There exists a quasi-coherent $\mathcal{O}_Z$-module $\mathcal{H}$ of finite type with $i_*\mathcal{H} = \mathcal{G}$ (Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-on-subscheme-cut-out-by-Fit-0}). In fact $\mathcal{H}$ is of finite presentation as an $\mathcal{O}_Z$-module by Algebra, Lemma \ref{algebra-lemma-finitely-presented-over-subring} (details omitted). Then $F_{zero}$ is the same as the functor $F_{zero}$ for the map $i^*\mathcal{F} \to \mathcal{H}$ adjoint to $u$, see Lemma \ref{lemma-relate-zero-affine}. The sheaf $\mathcal{H}$ is flat relative to $B$ because the same is true for $\mathcal{G}$ (check on stalks; details omitted). Moreover, note that if $\mathcal{F}$ is of finite type, then $i^*\mathcal{F}$ is of finite type. Hence we have reduced the lemma to the case discussed in the first paragraph of the proof.