hcju/stacksqa huggingface dataset:
Combining Limits, Proposition \ref{limits-proposition-approximate} and Lemma \ref{lemma-resolution-property-descends} this reduces to the case where $X$ is Noetherian (small detail omitted). Assume $X$ is Noetherian. Recall that $X \times X$ is covered by the affine opens $U \times V$ for affine opens $U$, $V$ of $X$, see Schemes, Section \ref{schemes-section-fibre-products}. Hence to show that the diagonal $\Delta : X \to X \times X$ is affine, it suffices to show that $U \cap V = \Delta^{-1}(U \times V)$ is affine for all affine opens $U$, $V$ of $X$, see Morphisms, Lemma \ref{morphisms-lemma-characterize-affine}. In particular, it suffices to show that the inclusion morphism $j : U \to X$ is affine if $U$ is an affine open of $X$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-criterion-affine-morphism} it suffices to show that $R^1j_*\mathcal{G} = 0$ for any quasi-coherent $\mathcal{O}_U$-module $\mathcal{G}$. By Proposition \ref{proposition-Noetherian} (this is where we use that we've reduced to the Noetherian case) we can represent $Rj_*\mathcal{G}$ by a complex $\mathcal{H}^\bullet$ of quasi-coherent $\mathcal{O}_X$-modules. Assume $$ H^1(\mathcal{H}^\bullet) = \Ker(\mathcal{H}^1 \to \mathcal{H}^2)/\Im(\mathcal{H}^0 \to \mathcal{H}^1) $$ is nonzero in order to get a contradiction. Then we can find a coherent $\mathcal{O}_X$-module $\mathcal{F}$ and a map $$ \mathcal{F} \longrightarrow \Ker(\mathcal{H}^1 \to \mathcal{H}^2) $$ such that the composition with the projection onto $H^1(\mathcal{H}^\bullet)$ is nonzero. Namely, we can write $\Ker(\mathcal{H}^1 \to \mathcal{H}^2)$ as the filtered union of its coherent submodules by Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type} and then one of these will do the job. Next, we choose a finite locally free $\mathcal{O}_X$-module $\mathcal{E}$ and a surjection $\mathcal{E} \to \mathcal{F}$ using the resolution property of $X$. This produces a map in the derived category $$ \mathcal{E}[-1] \longrightarrow Rj_*\mathcal{G} $$ which is nonzero on cohomology sheaves and hence nonzero in $D(\mathcal{O}_X)$. By adjunction, this is the same thing as a map $$ j^*\mathcal{E}[-1] \to \mathcal{G} $$ nonzero in $D(\mathcal{O}_U)$. Since $\mathcal{E}$ is finite locally free this is the same thing as a nonzero element of $$ H^1(U, j^*\mathcal{E}^\vee \otimes_{\mathcal{O}_U} \mathcal{G}) $$ where $\mathcal{E}^\vee = \SheafHom_{\mathcal{O}_X}(\mathcal{E}, \mathcal{O}_X)$ is the dual finite locally free module. However, this group is zero by Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero} which is the desired contradiction. (If in doubt about the step using duals, please see the more general Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}.)
Let $E$ be a perfect object of $D(\mathcal{O}_X)$. By Lemma \ref{lemma-resolution-property-affine-diagonal} we see that $X$ has affine diagonal. Hence by Proposition \ref{proposition-quasi-compact-affine-diagonal} we can represent $E$ by a complex $\mathcal{F}^\bullet$ of quasi-coherent $\mathcal{O}_X$-modules. Observe that $E$ is in $D^b(\mathcal{O}_X)$ because $X$ is quasi-compact. Hence $\tau_{\geq n}\mathcal{F}^\bullet$ is a bounded below complex of quasi-coherent $\mathcal{O}_X$-modules which represents $E$ if $n \ll 0$. Thus we may apply Lemma \ref{lemma-construct-strictly-perfect} to conclude.
By Lemma \ref{lemma-resolution-property-affine-diagonal} we see that $X$ has affine diagonal. Hence by Proposition \ref{proposition-quasi-compact-affine-diagonal} we can represent $\alpha$ by a diagram $$ \mathcal{E}^\bullet \leftarrow \mathcal{H}^\bullet \to \mathcal{F}^\bullet $$ where $\mathcal{H}^\bullet$ is a complex of quasi-coherent $\mathcal{O}_X$-modules and where $\mathcal{H}^\bullet \to \mathcal{E}^\bullet$ is a quasi-isomorphism. For $n \ll 0$ the maps $\mathcal{H}^\bullet \to \mathcal{E}^\bullet$ and $\mathcal{H}^\bullet \to \mathcal{F}^\bullet$ factor through the quasi-isomorphism $\mathcal{H}^\bullet \to \tau_{\geq n}\mathcal{H}^\bullet$ simply because $\mathcal{E}^\bullet$ and $\mathcal{F}^\bullet$ are bounded complexes. Thus we may replace $\mathcal{H}^\bullet$ by $\tau_{\geq n}\mathcal{H}^\bullet$ and assume that $\mathcal{H}^\bullet$ is bounded below. Then we may apply Lemma \ref{lemma-construct-strictly-perfect} to conclude.
By Lemma \ref{lemma-resolution-property-affine-diagonal} we see that $X$ has affine diagonal. Hence by Proposition \ref{proposition-quasi-compact-affine-diagonal} (and the definition of the derived category) there exists a quasi-isomorphism $\gamma^\bullet : \mathcal{G}^\bullet \to \mathcal{E}^\bullet$ where $\mathcal{G}^\bullet$ is a complex of quasi-coherent $\mathcal{O}_X$-modules such that $\alpha^\bullet \circ \gamma^\bullet$ and $\beta^\bullet \circ \gamma^\bullet$ are homotopic maps of complexes. Choose a homotopy $h^i : \mathcal{G}^i \to \mathcal{F}^{i - 1}$ witnessing this fact. Choose $n \ll 0$. Then the map $\gamma^\bullet$ factors canonically over the quotient map $\mathcal{G}^\bullet \to \tau_{\geq n}\mathcal{G}^\bullet$ as $\mathcal{E}^\bullet$ is bounded below. For the exact same reason the maps $h^i$ will factor over the surjections $\mathcal{G}^i \to (\tau_{\geq n}\mathcal{G})^i$. Hence we see that we may replace $\mathcal{G}^\bullet$ by $\tau_{\geq n}\mathcal{G}^\bullet$. Then we may apply Lemma \ref{lemma-construct-strictly-perfect} to conclude.
Recall that $K'_0(R)$ and $K_0(R)$ have been defined in Algebra, Section \ref{algebra-section-K-groups}. \medskip\noindent By More on Algebra, Lemma \ref{more-algebra-lemma-perfect-to-K-group-universal} we have $K_0(R) = K_0(D_{perf}(R))$. By Lemmas \ref{lemma-perfect-affine} and \ref{lemma-affine-compare-bounded} we have $D_{perf}(R) = D_{perf}(\mathcal{O}_X)$. This proves the equality $K_0(R) = K_0(X)$. \medskip\noindent The equality $K'_0(R) = K'_0(X)$ holds because $\textit{Coh}(\mathcal{O}_X)$ is equivalent to the category of finite $R$-modules by Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian}. Moreover it is clear that $K'_0(R)$ is the zeroth K-group of the category of finite $R$-modules from the definitions.
Let $L$ be an object of the left hand side. If $\Spec(A) = U \subset X$ is an affine open, then $L|_U$ corresponds to a perfect complex $L^\bullet$ of $A$-modules with tor-amplitude in $[-1, 0]$, see Lemmas \ref{lemma-affine-compare-bounded}, \ref{lemma-tor-dimension-affine}, and \ref{lemma-perfect-affine}. Then we can consider the invertible $A$-module $\det(L^\bullet)$ constructed in More on Algebra, Lemma \ref{more-algebra-lemma-determinant-two-term-complexes}. If $\Spec(B) = V \subset U$ is another affine open contained in $U$, then $\det(L^\bullet) \otimes_A B = \det(L^\bullet \otimes_A B)$ and hence this construction is compatible with restriction mappings (see Lemma \ref{lemma-quasi-coherence-pullback} and note $A \to B$ is flat). Thus we can glue these invertible modules to obtain an invertible module $\det(L)$ on $X$. The functoriality and canonical sections are constructed in exactly the same manner. Details omitted.
Choose a distinguished triangle $N \to E \to Rj_*(E|_U) \to N[1]$. Then (1) implies $\tau_{\geq a + 1} N = 0$ and (1) is implied by $\tau_{\geq a}N = 0$. Observe that $$ \Hom_{D(\mathcal{O}_X)}(K[-n], Rj_*(E|_U)) = \Hom_{D(\mathcal{O}_U)}(K|_U[-n], E) = 0 $$ for all $n$ as $K|_U = 0$. Thus (2) is equivalent to $\Hom_{D(\mathcal{O}_X)}(K[-n], N) = 0$ for all $n \geq a$.  Observe that there are distinguished triangles $$ K^\bullet(f_1^{e_1}, \ldots, f_i^{e'_i}, \ldots, f_r^{e_r}) \to K^\bullet(f_1^{e_1}, \ldots, f_i^{e'_i + e''_i}, \ldots, f_r^{e_r}) \to K^\bullet(f_1^{e_1}, \ldots, f_i^{e''_i}, \ldots, f_r^{e_r}) \to \ldots $$ of Koszul complexes, see More on Algebra, Lemma \ref{more-algebra-lemma-koszul-mult}. Hence $\Hom_{D(\mathcal{O}_X)}(K[-n], N) = 0$ for all $n \geq a$ is equivalent to $\Hom_{D(\mathcal{O}_X)}(K_e[-n], N) = 0$ for all $n \geq a$ and all $e \geq 1$ with $K_e$ as in Lemma \ref{lemma-represent-cohomology-class-on-closed}. Since $N|_U = 0$, that lemma implies that this in turn is equivalent to $H^n(X, N) = 0$ for $n \geq a$. We conclude that (2) is equivalent to $\tau_{\geq a}N = 0$ since $N$ is determined by the complex of $A$-modules $R\Gamma(X, N)$, see Lemma \ref{lemma-affine-compare-bounded}. Thus we find that our lemma is true.
Choose a distinguished triangle $E \to Rj_*(E|_U) \to N \to E[1]$. Then (1) implies $\tau_{\leq a - 1}N = 0$ and (1) is implied by $\tau_{\leq a}N = 0$. Observe that $$ \Hom_{D(\mathcal{O}_X)}(K[-n], Rj_*(E|_U)) = \Hom_{D(\mathcal{O}_U)}(K|_U[-n], E) = 0 $$ for all $n$ as $K|_U = 0$. Thus (2) is equivalent to $\Hom_{D(\mathcal{O}_X)}(K[-n], N) = 0$ for all $n \leq a$.  Observe that there are distinguished triangles $$ K^\bullet(f_1^{e_1}, \ldots, f_i^{e'_i}, \ldots, f_r^{e_r}) \to K^\bullet(f_1^{e_1}, \ldots, f_i^{e'_i + e''_i}, \ldots, f_r^{e_r}) \to K^\bullet(f_1^{e_1}, \ldots, f_i^{e''_i}, \ldots, f_r^{e_r}) \to \ldots $$ of Koszul complexes, see More on Algebra, Lemma \ref{more-algebra-lemma-koszul-mult}. Hence $\Hom_{D(\mathcal{O}_X)}(K[-n], N) = 0$ for all $n \leq a$ is equivalent to $\Hom_{D(\mathcal{O}_X)}(K_e[-n], N) = 0$ for all $n \leq a$ and all $e \geq 1$ with $K_e$ as in Lemma \ref{lemma-represent-cohomology-class-on-closed}. Since $N|_U = 0$, that lemma implies that this in turn is equivalent to $H^n(X, N) = 0$ for $n \leq a$. We conclude that (2) is equivalent to $\tau_{\leq a}N = 0$ since $N$ is determined by the complex of $A$-modules $R\Gamma(X, N)$, see Lemma \ref{lemma-affine-compare-bounded}. Thus we find that our lemma is true.
Using the triangle $ \tau_{< a} E \to E \to \tau_{\geq a} E \to$ we see that the equivalence follows if we can show $$ \Hom_{D(\mathcal{O}_X)}(P[-i], \tau_{< a} E) = \Hom_{D(\mathcal{O}_X)}(P, (\tau_{< a} E)[i]) = 0  $$ for $i \gg 0$. As $P$ is perfect this is true by Lemma \ref{lemma-ext-from-perfect-into-bounded-QCoh}.
By Lemma \ref{lemma-affine-compare-bounded} we may assume that $E$ is given by a complex of quasi-coherent sheaves $\mathcal{F}^\bullet$. Let $M^\bullet = \Gamma(X, \mathcal{F}^\bullet)$ be the corresponding complex of $A$-modules. By Lemmas \ref{lemma-alternating-cech-complex-complex} and \ref{lemma-alternating-cech-complex-complex-computes-cohomology} we have quasi-isomorphisms $$ \colim_e \text{Tot}(\Hom_A(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet)) \longrightarrow \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}^\bullet)) \longrightarrow R\Gamma(U, \mathcal{F}^\bullet) $$ Taking $H^0$ on both sides we obtain $$ \colim_e \Hom_{D(A)}(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet) = H^0(U, E) $$ Since $\Hom_{D(A)}(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet) = \Hom_{D(\mathcal{O}_X)}(I_e, E)$ by Lemma \ref{lemma-affine-compare-bounded} the lemma follows.
Consider the commutative diagram $$ \xymatrix{ D^-(\textit{Coh}(\mathcal{O}_X)) \ar[r] \ar[d] & D^-_{\textit{Coh}}(\mathcal{O}_X) \ar[d] \\ D^-(\QCoh(\mathcal{O}_X)) \ar[r] & D^-_\QCoh(\mathcal{O}_X) } $$ By Lemma \ref{lemma-coh-to-qcoh} the left vertical arrow is fully faithful. By Proposition \ref{proposition-Noetherian} the bottom arrow is an equivalence. By construction the right vertical arrow is fully faithful. We conclude that the top horizontal arrow is fully faithful. If $K$ is an object of $D^-_{\textit{Coh}}(\mathcal{O}_X)$ then the object $K'$ of $D^-(\QCoh(\mathcal{O}_X))$ which corresponds to it by Proposition \ref{proposition-Noetherian} will have coherent cohomology sheaves. Hence $K'$ is in the essential image of the left vertical arrow by Lemma \ref{lemma-coh-to-qcoh} and we find that the top horizontal arrow is essentially surjective. This finishes the proof for the bounded above case. The bounded case follows immediately from the bounded above case.
If $K$ is a perfect object of $D(\mathcal{O}_X)$ with dual $K^\vee$ (Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}) we have $$ \Hom_{D(\mathcal{O}_X)}(K, M) = H^0(X, K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} M) $$ functorially in $M$. Since $K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} -$ commutes with direct sums and since $H^0(X, -)$ commutes with direct sums on $D_\QCoh(\mathcal{O}_X)$ by Lemma \ref{lemma-quasi-coherence-pushforward-direct-sums} we conclude that $K$ is compact in $D_\QCoh(\mathcal{O}_X)$. \medskip\noindent Conversely, let $K$ be a compact object of $D_\QCoh(\mathcal{O}_X)$. To show that $K$ is perfect, it suffices to show that $K|_U$ is perfect for every affine open $U \subset X$, see Cohomology, Lemma \ref{cohomology-lemma-perfect-independent-representative}. Observe that $j : U \to X$ is a quasi-compact and separated morphism. Hence $Rj_* : D_\QCoh(\mathcal{O}_U) \to D_\QCoh(\mathcal{O}_X)$ commutes with direct sums, see Lemma \ref{lemma-quasi-coherence-pushforward-direct-sums}. Thus the adjointness of restriction to $U$ and $Rj_*$ implies that $K|_U$ is a compact object of $D_\QCoh(\mathcal{O}_U)$. Hence we reduce to the case that $X$ is affine. \medskip\noindent Assume $X = \Spec(A)$ is affine. By Lemma \ref{lemma-affine-compare-bounded} the problem is translated into the same problem for $D(A)$. For $D(A)$ the result is More on Algebra, Proposition \ref{more-algebra-proposition-perfect-is-compact}.
If you can parse the statement of the proposition, then please skip this first paragraph. For some of the definitions used, please see Derived Categories, Definition \ref{derived-definition-triangulated-subcategory} (triangulated subcategory), Derived Categories, Definition \ref{derived-definition-saturated} (saturated triangulated subcategory), Derived Categories, Definition \ref{derived-definition-localization} (multiplicative system compatible with the triangulated structure), and Categories, Definition \ref{categories-definition-saturated-multiplicative-system} (saturated multiplicative system). Observe that $D_{perf}(\mathcal{O}_X)$ is a saturated triangulated subcategory of $D(\mathcal{O}_X)$ by Cohomology, Lemmas \ref{cohomology-lemma-two-out-of-three-perfect} and \ref{cohomology-lemma-summands-perfect}. Also, note that $K^b(\mathcal{A})$ is a triangulated category, see Derived Categories, Lemma \ref{derived-lemma-bounded-triangulated-subcategories}. \medskip\noindent It is clear that the functor sends distinguished triangles to distinguished triangles, i.e., is exact. Then $S$ is a saturated multiplicative system compatible with the triangulated structure on $K^b(\mathcal{A})$ by Derived Categories, Lemma \ref{derived-lemma-triangle-functor-localize}. Hence the localization $S^{-1}K^b(\mathcal{A})$ exists and is a triangulated category by Derived Categories, Proposition \ref{derived-proposition-construct-localization}. We get an exact factorization $S^{-1}K^b(\mathcal{A}) \to D_{perf}(\mathcal{O}_X)$ by Derived Categories, Lemma \ref{derived-lemma-universal-property-localization}. By Lemmas \ref{lemma-resolution-property-perfect-complex}, \ref{lemma-resolution-property-map-perfect-complex}, and \ref{lemma-resolution-property-homotopy-map-perfect-complex} this functor is an equivalence. Then finally the functor $S^{-1}K^b(\mathcal{A}) \to D_{perf}(\mathcal{O}_X)$ is an equivalence of triangulated categories (in the sense that distinguished triangles correspond) by Derived Categories, Lemma \ref{derived-lemma-exact-equivalence}.
Condition (3) means that if $V \to Y$ is a surjective \'etale morphism where $V$ is a scheme, then the weakly associated points of $V$ are locally finite on the scheme $V$. (Recall that the weakly associated points of $V$ are exactly the inverse image of the weakly associated points of $Y$ by Divisors on Spaces, Definition \ref{spaces-divisors-definition-weakly-associated}.) Having said this the question is \'etale local on $X$ and $Y$, hence we may assume $X$ and $Y$ are schemes. Thus the result follows from More on Flatness, Theorem \ref{flat-theorem-finite-type-flat}.
Assume (2). Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. Choose a point $u \in U$ mapping to $x$. Let $v \in V$ be the image of $u$. We will deduce the result from the corresponding result for $\mathcal{F}|_U = (U \to X)^*\mathcal{F}$ and the point $u$. $U_v$. This works because $\text{Ass}_{U/V}(\mathcal{F}|_U) \cap |U_v|$ is equal to $\text{Ass}_{U_v}(\mathcal{F}|_{U_v})$ and equal to the inverse image of $F \cap \text{Ass}_{X/Y}(\mathcal{F})$. Since the map $|U_v| \to F$ is continuous we see that specializations in $|U_v|$ map to specializations in $F$, hence condition (2) is inherited by $U \to V$, $\mathcal{F}|_U$, and the point $u$. Thus More on Flatness, Theorem \ref{flat-theorem-check-flatness-at-associated-points} applies and we conclude that (1) holds. \medskip\noindent If $Y$ is decent, then we can represent $y$ by a quasi-compact monomorphism $\Spec(k) \to Y$ (by definition of decent spaces, see Decent Spaces, Definition \ref{decent-spaces-definition-very-reasonable}). Then $F = |X_k|$, see Decent Spaces, Lemma \ref{decent-spaces-lemma-topology-fibre}. If in addition $X$ is decent (or more generally if $f$ is decent, see Decent Spaces, Definition \ref{decent-spaces-definition-relative-conditions} and Decent Spaces, Lemma \ref{decent-spaces-lemma-property-for-morphism-out-of-property}), then $X_y$ is a decent space too. Furthermore, specializations in $F$ can be lifted to specializations in $U_v \to X_y$, see Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-specialization}. Having said this it is clear that the reverse implication holds, because it holds in the case of schemes.
Let $\mathcal{K} = \Ker(u)$ and denote $v : \mathcal{K} \to \mathcal{F}$ the inclusion. By Lemma \ref{lemma-relate-zero-iso} we see that $F_{u, iso} = F_{v, zero}$. By Lemma \ref{lemma-F-zero-closed-pure} applied to $v$ we see that $F_{u, iso} = F_{v, zero}$ is representable by a closed subspace of $B$. Note that $\mathcal{K}$ is of finite type if $\mathcal{G}$ is of finite presentation, see Modules on Sites, Lemma \ref{sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation}. Hence we also obtain the final statement of the lemma.
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 subspace $W \subset X$ 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. In particular $S'$ is a scheme (this follows from the proof of the theorem but it also follows a postoriori by Morphisms of Spaces, Proposition \ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}). 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 More on Flatness, Lemma \ref{flat-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 $U$ in Lemma \ref{lemma-compute-over-affine-derived} it follows immediately that $K$ restricts to $K_n$ over $X_n$.
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 on Spaces, Lemma \ref{spaces-divisors-lemma-base-change-relative-assassin} we have $\xi_i \in \text{Ass}_{X_{T_i}/T_i}(\mathcal{F}_{T_i})$. Thus the only point is to show that $t_i \not \in f_{T_i}(\overline{\{\xi_i\}})$ for some $i$. \medskip\noindent Let $Z_i \subset X_{T_i}$ be the reduced induced scheme structure on $\overline{\{\xi_i\}} \subset |X_{T_i}|$ and let $Z \subset X_T$ be the reduced induced scheme structure on $\overline{\{\xi\}} \subset |X_T|$. Then $Z = \lim Z_i$ by Limits of Spaces, Lemma \ref{spaces-limits-lemma-inverse-limit-irreducibles} (the lemma applies because each $X_{T_i}$ is decent). 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 of Spaces, Lemma \ref{spaces-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.
Choose an \'etale morphism $T_1 \to Y_1 \times_Y T$ where $T_1$ is a scheme and let $t_1 \in T_1$ be a point mapping to $y_1$ and $t$. It is possible to find a pair $(T_1, t_1)$ like this by Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}. The morphism of schemes $T_1 \to T$ is flat at $t_1$ (use Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-flat} and the definition of flat morphisms of algebraic spaces) there exists a specialization $t'_1 \leadsto t_1$ lying over $t' \leadsto t$, see Morphisms, Lemma \ref{morphisms-lemma-generalizations-lift-flat}. Choose a point $\xi_1 \in |X_{T_1}|$ mapping to $t'_1$ and $\xi$ with $\xi_1 \in \text{Ass}_{X_{T_1}/T_1}(\mathcal{F}_{T_1})$. point of $\Spec(\kappa(t'_1) \otimes_{\kappa(t')} \kappa(\xi))$. This is possible by Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-base-change-relative-assassin}. As the closure $Z_1$ of $\{\xi_1\}$ in $|X_{T_1}|$ maps into the closure of $\{\xi\}$ in $|X_T|$ we conclude that the image of $Z_1$ in $|T_1|$ cannot contain $t_1$. Hence $(T_1 \to Y_1, t'_1 \leadsto t_1, \xi_1)$ is an impurity of $\mathcal{F}_1$ above $Y_1$.