The Stacks project

Proof. Write $B(T)$ for the discrete category whose objects are the $S$-morphisms $T \to B$. Let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be a filtered limit of affine schemes over $S$. Assigning to an object $(T, h, E)$ of $\mathcal{C}\! \mathit{omplexes}_{X/B, T}$ the object $h$ of $B(T)$ gives us a commutative diagram of fibre categories

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 70.3.11 that the bottom horizontal arrow is an equivalence. This means that we may assume $T = \mathop{\mathrm{lim}}\nolimits T_ i$ 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 $X_ T = \mathop{\mathrm{colim}}\nolimits X_ i$. By More on Morphisms of Spaces, Lemma 76.52.9 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 99.16.2, then after increasing $i$ we have that $E_ i$ satisfies condition (2) of Lemma 99.16.2. Let $W \subset |T_ i|$ be the open constructed in Lemma 99.16.1 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 32.4.10 Then $i'$ works by construction of $W$. $\square$