hcju/stacksqa huggingface dataset:
Let $T = \lim T_i$ be the limits of a directed inverse system of affine schemes. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-finite-presentation} the category of algebraic spaces of finite presentation over $T$ is the colimit of the categories of algebraic spaces of finite presentation over $T_i$. To finish the proof use that flatness and properness descends through the limit, see Limits of Spaces, Lemmas \ref{spaces-limits-lemma-descend-flat} and \ref{spaces-limits-lemma-eventually-proper}.
The functor is an equivalence if we drop ``proper'' from the list of conditions and replace ``of finite presentation'' by ``locally of finite presentation'', see Pushouts of Spaces, Lemma \ref{spaces-pushouts-lemma-equivalence-categories-spaces-pushout-flat}. Thus it suffices to show that given a morphism $X' \to S'$ of an algebraic space to $S'$ which is flat and locally of finite presentation, then $X' \to S'$ is proper if and only if $S \times_{S'} X' \to S$ and $T' \times_{S'} X' \to T'$ are proper. One implication follows from the fact that properness is preserved under base change (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-proper}) and the other from the fact that properness of $S \times_{S'} X' \to S$ implies properness of $X' \to S'$ by More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-thicken-property-morphisms-cartesian}.
The discussion in Artin's Axioms, Section \ref{artin-section-tangent-spaces} only applies to fields of finite type over the base scheme $\Spec(\mathbf{Z})$. Our stack satisfies (RS*) by Lemma \ref{lemma-spaces-RS-star} and we may apply Artin's Axioms, Lemma \ref{artin-lemma-properties-lift-RS-star} to get the vector spaces $T_x(k)$ and $\text{Inf}_x(k)$ mentioned in (2). Moreover, in the finite type case these spaces agree with the ones mentioned in (1) by Artin's Axioms, Remark \ref{artin-remark-compare-deformation-spaces}. With this out of the way we can start the proof. Observe that the first order thickening $\Spec(k) \to \Spec(k[\epsilon]) = \Spec(k[k])$ has conormal module $k$. Hence the formula in Deformation Theory, Lemma \ref{defos-lemma-deform-spaces} describing infinitesimal deformations of $X$ and infinitesimal automorphisms of $X$ become $$ T_x(k) = \Ext^1_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X) \quad\text{and}\quad \text{Inf}_x(k) = \Ext^0_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X) $$ By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-netherlander-fp} and the fact that $X$ is Noetherian, we see that $\NL_{X/k}$ has coherent cohomology sheaves zero except in degrees $0$ and $-1$. By Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-ext-finite} the displayed $\Ext$-groups are finite $k$-vector spaces and the proof is complete.
We check conditions (1) and (2) of Categories, Definition \ref{categories-definition-fibred-groupoids}. \medskip\noindent Condition (1). Let $(X \to S, \mathcal{L})$ be an object of $\Polarizedstack$ and let $(X' \to S') \to (X \to S)$ be a morphism of $\Spacesstack'_{fp, flat, proper}$. Then we let $\mathcal{L}'$ be the pullback of $\mathcal{L}$ to $X'$. Observe that $X, S, S'$ are schemes, hence $X'$ is a scheme as well (as the fibre product of schemes). Then $\mathcal{L}'$ is ample on $X'/S'$ by Morphisms, Lemma \ref{morphisms-lemma-ample-base-change}. In this way we obtain a morphism $(X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})$ lying over $(X' \to S') \to (X \to S)$. \medskip\noindent Condition (2). Consider morphisms $(f, g, \varphi) : (X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})$ and $(a, b, \psi) : (Y \to T, \mathcal{N}) \to (X \to S, \mathcal{L})$ of $\Polarizedstack$. Given a morphism $(k, h) : (Y \to T) \to (X' \to S')$ of $\Spacesstack'_{fp, flat, proper}$ with $(f, g) \circ (k, h) = (a, b)$ we have to show there is a unique morphism $(k, h, \chi) : (Y \to T, \mathcal{N}) \to (X' \to S', \mathcal{L}')$ of $\Polarizedstack$ such that $(f, g, \varphi) \circ (k, h, \chi) = (a, b, \psi)$. We can just take $$ \chi = \psi \circ (k^*\varphi)^{-1} $$ This proves condition (2). A composition of functors defining fibred categories defines a fibred category, see Categories, Lemma \ref{categories-lemma-fibred-over-fibred}. This we see that $\Polarizedstack$ is fibred in groupoids over $\Sch_{fppf}$ (strictly speaking we should check the fibre categories are groupoids and apply Categories, Lemma \ref{categories-lemma-fibred-groupoids}).
Omitted. Hints: Argue exactly as in the proof of Lemma \ref{lemma-extend-spaces-to-spaces} and use Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} to descent the invertible sheaf in the construction of the quasi-inverse functor. The relative ampleness property descends by Descent on Spaces, Lemma \ref{spaces-descent-lemma-descending-property-ample}.
By Lemmas \ref{lemma-spaces-stack} and \ref{lemma-polarized-stack} the statement makes sense. To prove it, we choose a scheme $S$ and an object $\xi = (X \to S)$ of $\Spacesstack'_{fp, flat, proper}$ over $S$. We have to show that $$ \mathcal{X} = (\Sch/S)_{fppf} \times_{\xi, \Spacesstack'_{fp, flat, proper}} \Polarizedstack $$ is an algebraic stack over $S$. Observe that an object of $\mathcal{X}$ is given by a pair $(T/S, \mathcal{L})$ where $T$ is a scheme over $S$ and $\mathcal{L}$ is an invertible $\mathcal{O}_{X_T}$-module which is ample on $X_T/T$. Morphisms are defined in the obvious manner. In particular, we see immediately that we have an inclusion $$ \mathcal{X} \subset \Picardstack_{X/S} $$ of categories over $(\Sch/S)_{fppf}$, inducing equality on morphism sets. Since $\Picardstack_{X/S}$ is an algebraic stack by Proposition \ref{proposition-pic} it suffices to show that the inclusion above is representable by open immersions. This is exactly the content of Descent on Spaces, Lemma \ref{spaces-descent-lemma-ample-in-neighbourhood}.
Let $I$ be a directed set and let $(A_i, \varphi_{ii'})$ be a system of rings over $I$. Set $S = \Spec(A)$ and $S_i = \Spec(A_i)$. We have to show that on fibre categories we have $$ \Polarizedstack_S = \colim \Polarizedstack_{S_i} $$ We know that the category of schemes of finite presentation over $S$ is the colimit of the category of schemes of finite presentation over $S_i$, see Limits, Lemma \ref{limits-lemma-descend-finite-presentation}. Moreover, given $X_i \to S_i$ of finite presentation, with limit $X \to S$, then the category of invertible $\mathcal{O}_X$-modules $\mathcal{L}$ is the colimit of the categories of invertible $\mathcal{O}_{X_i}$-modules $\mathcal{L}_i$, see Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation} and \ref{limits-lemma-descend-invertible-modules}. If $X \to S$ is proper and flat, then for sufficiently large $i$ the morphism $X_i \to S_i$ is proper and flat too, see Limits, Lemmas \ref{limits-lemma-eventually-proper} and \ref{limits-lemma-descend-flat-finite-presentation}. Finally, if $\mathcal{L}$ is ample on $X$ then $\mathcal{L}_i$ is ample on $X_i$ for $i$ sufficiently large, see Limits, Lemma \ref{limits-lemma-limit-ample}. Putting everything together finishes the proof.
By More on Morphisms, Lemma \ref{more-morphisms-lemma-equivalence-categories-schemes-over-pushout-flat} there is an equivalence $$ \textit{flat-lfp}_{S'} \longrightarrow \textit{flat-lfp}_S \times_{\textit{flat-lfp}_T} \textit{flat-lfp}_{T'} $$ where $\textit{flat-lfp}_S$ signifies the category of schemes flat and locally of finite presentation over $S$. Let $X'/S'$ on the left hand side correspond to the triple $(X/S, Y'/T', \varphi)$ on the right hand side. Set $Y = T \times_{T'} Y'$ which is isomorphic with $T \times_S X$ via $\varphi$. Then More on Morphisms, Lemma \ref{more-morphisms-lemma-scheme-over-pushout-flat-modules} shows that we have an equivalence $$ \textit{QCoh-flat}_{X'/S'} \longrightarrow \textit{QCoh-flat}_{X/S} \times_{\textit{QCoh-flat}_{Y/T}} \textit{QCoh-flat}_{Y'/T'} $$ where $\textit{QCoh-flat}_{X/S}$ signifies the category of quasi-coherent $\mathcal{O}_X$-modules flat over $S$. Since $X \to S$, $Y \to T$, $X' \to S'$, $Y' \to T'$ are flat, this will in particular apply to invertible modules to give an equivalence of categories $$ \textit{Pic}(X') \longrightarrow \textit{Pic}(X) \times_{\textit{Pic}(Y)} \textit{Pic}(Y') $$ where $\textit{Pic}(X)$ signifies the category of invertible $\mathcal{O}_X$-modules. There is a small point here: one has to show that if an object $\mathcal{F}'$ of $\textit{QCoh-flat}_{X'/S'}$ pulls back to invertible modules on $X$ and $Y'$, then $\mathcal{F}'$ is an invertible $\mathcal{O}_{X'}$-module. It follows from the cited lemma that $\mathcal{F}'$ is an $\mathcal{O}_{X'}$-module of finite presentation. By More on Morphisms, Lemma \ref{more-morphisms-lemma-flat-and-free-at-point-fibre} it suffices to check the restriction of $\mathcal{F}'$ to fibres of $X' \to S'$ is invertible. But the fibres of $X' \to S'$ are the same as the fibres of $X \to S$ and hence these restrictions are invertible. \medskip\noindent Having said the above we obtain an equivalence of categories if we drop the assumption (for the category of objects over $S$) that $X \to S$ be proper and the assumption that $\mathcal{L}$ be ample. Now it is clear that if $X' \to S'$ is proper, then $X \to S$ and $Y' \to T'$ are proper (Morphisms, Lemma \ref{morphisms-lemma-base-change-proper}). Conversely, if $X \to S$ and $Y' \to T'$ are proper, then $X' \to S'$ is proper by More on Morphisms, Lemma \ref{more-morphisms-lemma-thicken-property-morphisms-cartesian}. Similarly, if $\mathcal{L}'$ is ample on $X'/S'$, then $\mathcal{L}'|_X$ is ample on $X/S$ and $\mathcal{L}'|_{Y'}$ is ample on $Y'/T'$ (Morphisms, Lemma \ref{morphisms-lemma-ample-base-change}). Finally, if $\mathcal{L}'|_X$ is ample on $X/S$ and $\mathcal{L}'|_{Y'}$ is ample on $Y'/T'$, then $\mathcal{L}'$ is ample on $X'/S'$ by More on Morphisms, Lemma \ref{more-morphisms-lemma-thicken-property-relatively-ample}.
This follows from Artin's Axioms, Lemma \ref{artin-lemma-SGE-implies-openness-versality} and Lemmas \ref{lemma-polarized-diagonal}, \ref{lemma-polarized-RS-star}, \ref{lemma-polarized-limits}, and \ref{lemma-polarized-strong-effectiveness}. For the ``usual'' proof of this fact, please see the discussion in the remark following this proof.
Using the embedding (\ref{equation-curves-over-proper-spaces}), the description of the image, and the corresponding fact for $\Spacesstack'_{fp, flat, proper}$ (Lemma \ref{lemma-spaces-fibred-in-groupoids}) this reduces to the following statement: Given a morphism $$ \xymatrix{ X' \ar[r] \ar[d] & X \ar[d] \\ S' \ar[r] & S } $$ in $\Spacesstack'_{fp, flat, proper}$ (recall that this implies in particular the diagram is cartesian) if $X \to S$ has relative dimension $\leq 1$, then $X' \to S'$ has relative dimension $\leq 1$. This follows from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}.
Using the embedding (\ref{equation-curves-over-proper-spaces}), the description of the image, and the corresponding fact for $\Spacesstack'_{fp, flat, proper}$ (Lemma \ref{lemma-spaces-stack}) this reduces to the following statement: Given an object $X \to S$ of $\Spacesstack'_{fp, flat, proper}$ and an fppf covering $\{S_i \to S\}_{i \in I}$ the following are equivalent: \begin{enumerate} \item $X \to S$ has relative dimension $\leq 1$, and \item for each $i$ the base change $X_i \to S_i$ has relative dimension $\leq 1$. \end{enumerate} This follows from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}.
Using the embedding (\ref{equation-curves-over-proper-spaces}), the description of the image, and the corresponding fact for $\Spacesstack'_{fp, flat, proper}$ (Lemma \ref{lemma-spaces-limits}) this reduces to the following statement: Let $T = \lim T_i$ be the limits of a directed inverse system of affine schemes. Let $i \in I$ and let $X_i \to T_i$ be an object of $\Spacesstack'_{fp, flat, proper}$ over $T_i$. Assume that $T \times_{T_i} X_i \to T$ has relative dimension $\leq 1$. Then for some $i' \geq i$ the morphism $T_{i'} \times_{T_i} X_i \to T_i$ has relative dimension $\leq 1$. This follows from Limits of Spaces, Lemma \ref{spaces-limits-lemma-eventually-relative-dimension}.
Using the embedding (\ref{equation-curves-over-proper-spaces}), the description of the image, and the corresponding fact for $\Spacesstack'_{fp, flat, proper}$ (Lemma \ref{lemma-spaces-RS-star}) this reduces to the following statement: given a morphism $X' \to S'$ of an algebraic space to $S'$ which is of finite presentation, flat, proper then $X' \to S'$ has relative dimension $\leq 1$ if and only if $S \times_{S'} X' \to S$ and $T' \times_{S'} X' \to T'$ have relative dimension $\leq 1$. One implication follows from the fact that having relative dimension $\leq 1$ is preserved under base change (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-dimension-fibre-after-base-change}). The other follows from the fact that having relative dimension $\leq 1$ is checked on the fibres and that the fibres of $X' \to S'$ (over points of the scheme $S'$) are the same as the fibres of $S \times_{S'} X' \to S$ since $S \to S'$ is a thickening by More on Morphisms, Lemma \ref{more-morphisms-lemma-pushout-along-thickening}.
For definitions of the notions in the lemma, please see Artin's Axioms, Section \ref{artin-section-formal-objects}. Let $(A, \mathfrak m, \kappa)$ be a Noetherian complete local ring. Let $(X_n \to \Spec(A/\mathfrak m^n))$ be a formal object of $\Curvesstack$ over $A$. By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-formal-algebraic-space-proper-reldim-1} there exists a projective morphism $X \to \Spec(A)$ and a compatible system of ismomorphisms $X \times_{\Spec(A)} \Spec(A/\mathfrak m^n) \cong X_n$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-check-flatness-on-infinitesimal-nbhds} we see that $X \to \Spec(A)$ is flat. By More on Morphisms, Lemma \ref{more-morphisms-lemma-dimension-fibres-proper-flat} we see that $X \to \Spec(A)$ has relative dimension $\leq 1$. This proves the lemma.
Since (\ref{equation-curves-over-proper-spaces}) is a fully faithful embedding of categories it suffices to show the following: given an object $X \to S$ of $\Spacesstack'_{fp, flat, proper}$ there exists an open and closed subscheme $U \subset S$ such that a morphism $S' \to S$ factors through $U$ if and only if the base change $X' \to S'$ of $X \to S$ has relative dimension $\leq 1$. This follows immediately from More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-dimension-fibres-proper-flat}.
We show that $p$ is fibred in groupoids by checking conditions (1) and (2) of Categories, Definition \ref{categories-definition-fibred-groupoids}. Given an object $(T', g', E')$ of $\Complexesstack_{X/B}$ and a morphism $h : T \to T'$ of schemes over $S$ we can set $g = h \circ g'$ and $E = L(h')^*E'$ where $h' : X_T \to X_{T'}$ is the base change of $h$. Then it is clear that we obtain a morphism $(T, g, E) \to (T', g', E')$ of $\Complexesstack_{X/B}$ lying over $h$. This proves (1). For (2) suppose we are given morphisms $$ (h_1, \varphi_1) : (T_1, g_1, E_1) \to (T, g, E) \quad\text{and}\quad (h_2, \varphi_2) : (T_2, g_2, E_2) \to (T, g, E) $$ of $\Complexesstack_{X/B}$ and a morphism $h : T_1 \to T_2$ such that $h_2 \circ h = h_1$. Then we can let $\varphi$ be the composition $$ L(h')^*E_2 \xrightarrow{L(h')^*\varphi_2^{-1}} L(h')^*L(h_2)^*E = L(h_1)^*E \xrightarrow{\varphi_1} E_1 $$ to obtain the morphism $(h, \varphi) : (T_1, g_1, E_1) \to (T_2, g_2, E_2)$ that witnesses the truth of condition (2).
Consider two objects $x = (T, g, E)$ and $y = (T, g', E')$ of $\mathcal{X}$ over a scheme $T$. We have to show that $\mathit{Isom}_\mathcal{X}(x, y)$ is an algebraic space over $T$, see Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-diagonal}. If for $h : T' \to T$ the restrictions $x|_{T'}$ and $y|_{T'}$ are isomorphic in the fibre category $\mathcal{X}_{T'}$, then $g \circ h = g' \circ h$. Hence there is a transformation of presheaves $$ \mathit{Isom}_\mathcal{X}(x, y) \longrightarrow \text{Equalizer}(g, g') $$ Since the diagonal of $B$ is representable (by schemes) this equalizer is a scheme. Thus we may replace $T$ by this equalizer and $E$ and $E'$ by their pullbacks. Thus we may assume $g = g'$. \medskip\noindent Assume $g = g'$. After replacing $B$ by $T$ and $X$ by $X_T$ we arrive at the following problem. Given $E, E' \in D(\mathcal{O}_X)$ satisfying conditions (1), (2) of Lemma \ref{lemma-complexes} we have to show that $\mathit{Isom}(E, E')$ is an algebraic space. Here $\mathit{Isom}(E, E')$ is the functor $$ (\Sch/B)^{opp} \to \textit{Sets},\quad T \mapsto \{\varphi : E_T \to E'_T \text{ isomorphism in }D(\mathcal{O}_{X_T})\} $$ where $E_T$ and $E'_T$ are the derived pullbacks of $E$ and $E'$ to $X_T$. Now, let $W \subset B$, resp.\ $W' \subset B$ be the open subspace of $B$ associated to $E, E'$, resp.\ to $E', E$ by Lemma \ref{lemma-complexes-open-neg-exts-vanishing}. Clearly, if there exists an isomorphism $E_T \to E'_T$ as in the definition of $\mathit{Isom}(E, E')$, then we see that $T \to B$ factors into both $W$ and $W'$ (because we have condition (1) for $E$ and $E'$ and we'll obviously have $E_t \cong E'_t$ so no nonzero maps $E_t[i] \to E_t$ or $E'_t[i] \to E_t$ over the fibre $X_t$ for $i > 0$. Thus we may replace $B$ by the open $W \cap W'$. In this case the functor $H = \SheafHom(E, E')$ $$ (\Sch/B)^{opp} \to \textit{Sets},\quad T \mapsto \Hom_{\mathcal{O}_{X_T}}(E_T, E'_T) $$ is an algebraic space affine and of finite presentation over $B$ by Lemma \ref{lemma-complexes-open-neg-exts-vanishing}. The same is true for $H' = \SheafHom(E', E)$, $I = \SheafHom(E, E)$, and $I' = \SheafHom(E', E')$. Therefore we can repeat the argument of the proof of Proposition \ref{proposition-isom} to see that $$ \mathit{Isom}(E, E') = (H' \times_B H) \times_{c, I \times_B I', \sigma} B $$ for some morphisms $c$ and $\sigma$. Thus $\mathit{Isom}(E, E')$ is an algebraic space.
To prove that $\Complexesstack_{X/B}$ is a stack in groupoids, we have to show that the presheaves $\mathit{Isom}$ are sheaves and that descent data are effective. The statement on $\mathit{Isom}$ follows from Lemma \ref{lemma-complexes-diagonal}, see Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-diagonal}. Let us prove the statement on descent data. \medskip\noindent Suppose that $\{a_i : T_i \to T\}$ is an fppf covering of schemes over $S$. Let $(\xi_i, \varphi_{ij})$ be a descent datum for $\{T_i \to T\}$ with values in $\Complexesstack_{X/B}$. For each $i$ we can write $\xi_i = (T_i, g_i, E_i)$. Denote $\text{pr}_0 : T_i \times_T T_j \to T_i$ and $\text{pr}_1 : T_i \times_T T_j \to T_j$ the projections. The condition that $\xi_i|_{T_i \times_T T_j} \cong \xi_j|_{T_i \times_T T_j}$ implies in particular that $g_i \circ \text{pr}_0 = g_j \circ \text{pr}_1$. Thus there exists a unique morphism $g : T \to B$ such that $g_i = g \circ a_i$, see Descent on Spaces, Lemma \ref{spaces-descent-lemma-fpqc-universal-effective-epimorphisms}. Denote $X_T = T \times_{g, B} X$. Set $X_i = X_{T_i} = T_i \times_{g_i, B} X = T_i \times_{a_i, T} X_T$ and $$ X_{ij} = X_{T_i} \times_{X_T} X_{T_j} = X_i \times_{X_T} X_j $$ with projections $\text{pr}_i$ and $\text{pr}_j$ to $X_i$ and $X_j$. Observe that the pullback of $(T_i, g_i, E_i)$ by $\text{pr}_0 : T_i \times_T T_j \to T_i$ is given by $(T_i \times_T T_j, g_i \circ \text{pr}_0, L\text{pr}_i^*E_i)$. Hence a descent datum for $\{T_i \to T\}$ in $\Complexesstack_{X/B}$ is given by the objects $(T_i, g \circ a_i, E_i)$ and for each pair $i, j$ an isomorphism in $D\mathcal{O}_{X_{ij}})$ $$ \varphi_{ij} : L\text{pr}_i^*E_i \longrightarrow L\text{pr}_j^*E_j $$ satisfying the cocycle condition over the pullback of $X$ to $T_i \times_T T_j \times_T T_k$. Using the vanishing of negative Exts provided by (b) of Lemma \ref{lemma-complexes}, we may apply Simplicial Spaces, Lemma \ref{spaces-simplicial-lemma-fppf-glue-neg-ext-zero} to obtain descent\footnote{To check this, first consider the {\v C}ech fppf hypercovering $a : T_\bullet \to T$ with $T_n = \coprod_{i_0 \ldots i_n} T_{i_0} \times_T \ldots \times_T T_{i_n}$ and then set $U_\bullet = T_\bullet \times_{a, T} X_T$. Then $U_\bullet \to X_T$ is an fppf hypercovering to which we may apply Simplicial Spaces, Lemma \ref{spaces-simplicial-lemma-fppf-glue-neg-ext-zero}.} for these complexes. In other words, we find there exists an object $E$ in $D_\QCoh(\mathcal{O}_{X_T})$ restricting to $E_i$ on $X_{T_i}$ compatible with $\varphi_{ij}$. Recall that being $T$-perfect signifies being pseudo-coherent and having locally finite tor dimension over $f^{-1}\mathcal{O}_T$. Thus $E$ is $T$-perfect by an application of More on Morphisms of Spaces, Lemmas \ref{spaces-more-morphisms-lemma-pseudo-coherent-descends-fpqc} and \ref{spaces-more-morphisms-lemma-tor-amplitude-descends-fppf}. Finally, we have to check condition (2) from Lemma \ref{lemma-complexes} for $E$. This immediately follows from the description of the open $W$ in Lemma \ref{lemma-complexes-open-neg-exts-vanishing} and the fact that (2) holds for $E_i$ on $X_{T_i}/T_i$.
Write $B(T)$ for the discrete category whose objects are the $S$-morphisms $T \to B$. Let $T = \lim T_i$ be a filtered limit of affine schemes over $S$. Assigning to an object $(T, h, E)$ of $\Complexesstack_{X/B, T}$ the object $h$ of $B(T)$ gives us a commutative diagram of fibre categories $$ \xymatrix{ \colim \Complexesstack_{X/B, T_i} \ar[r] \ar[d] & \Complexesstack_{X/B, T} \ar[d] \\ \colim B(T_i) \ar[r] & B(T) } $$ We have to show the top horizontal arrow is an equivalence. Since we have assume that $B$ is locally of finite presentation over $S$ we see from Limits of Spaces, Remark \ref{spaces-limits-remark-limit-preserving} that the bottom horizontal arrow is an equivalence. This means that we may assume $T = \lim T_i$ be a filtered limit of affine schemes over $B$. Denote $g_i : T_i \to B$ and $g : T \to B$ the corresponding morphisms. Set $X_i = T_i \times_{g_i, B} X$ and $X_T = T \times_{g, B} X$. Observe that $X_T = \colim X_i$. By More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-descend-relatively-perfect} the category of $T$-perfect objects of $D(\mathcal{O}_{X_T})$ is the colimit of the categories of $T_i$-perfect objects of $D(\mathcal{O}_{X_{T_i}})$. Thus all we have to prove is that given an $T_i$-perfect object $E_i$ of $D(\mathcal{O}_{X_{T_i}})$ such that the derived pullback $E$ of $E_i$ to $X_T$ satisfies condition (2) of Lemma \ref{lemma-complexes}, then after increasing $i$ we have that $E_i$ satisfies condition (2) of Lemma \ref{lemma-complexes}. Let $W \subset |T_i|$ be the open constructed in Lemma \ref{lemma-complexes-open-neg-exts-vanishing} for $E_i$ and $E_i$. By assumption on $E$ we find that $T \to T_i$ factors through $T$. Hence there is an $i' \geq i$ such that $T_{i'} \to T_i$ factors through $W$, see Limits, Lemma \ref{limits-lemma-limit-contained-in-constructible} Then $i'$ works by construction of $W$.