Lemma 108.12.1. The diagonal of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ over $B$ is affine and of finite presentation.
108.12 Properties of moduli of complexes on a proper morphism
Let $f : X \to B$ be a morphism of algebraic spaces which is proper, flat, and of finite presentation. Then the stack $\mathcal{C}\! \mathit{omplexes}_{X/B}$ parametrizing relatively perfect complexes with vanishing negative self-exts is algebraic. See Quot, Theorem 99.16.12.
Proof. The representability of the diagonal by algebraic spaces was shown in Quot, Lemma 99.16.5. From the proof we find that we have to show: given a scheme $T$ over $B$ and objects $E, E' \in D(\mathcal{O}_{X_ T})$ such that $(T, E)$ and $(T, E')$ are objects of the fibre category of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ over $T$, then $\mathit{Isom}(E, E') \to T$ is affine and of finite presentation. Here $\mathit{Isom}(E, E')$ is the functor
\[ (\mathit{Sch}/T)^{opp} \to \textit{Sets},\quad T' \mapsto \{ \varphi : E_{T'} \to E'_{T'} \text{ isomorphism in }D(\mathcal{O}_{X_{T'}})\} \]
where $E_{T'}$ and $E'_{T'}$ are the derived pullbacks of $E$ and $E'$ to $X_{T'}$. Consider the functor $H = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, E')$ defined by the rule
\[ (\mathit{Sch}/T)^{opp} \to \textit{Sets},\quad T' \mapsto \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X_{T'}}}(E_ T, E'_ T) \]
By Quot, Lemma 99.16.1 this is an algebraic space affine and of finite presentation over $T$. The same is true for $H' = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (E', E)$, $I = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, E)$, and $I' = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (E', E')$. Therefore we see that
\[ \mathit{Isom}(E, E') = (H' \times _ T H) \times _{c, I \times _ T I', \sigma } T \]
where $c(\varphi ', \varphi ) = (\varphi \circ \varphi ', \varphi ' \circ \varphi )$ and $\sigma = (\text{id}, \text{id})$ (compare with the proof of Quot, Proposition 99.4.3). Thus $\mathit{Isom}(E, E')$ is affine over $T$ as a fibre product of schemes affine over $T$. Similarly, $\mathit{Isom}(E, E')$ is of finite presentation over $T$. $\square$
Lemma 108.12.2. The morphism $\mathcal{C}\! \mathit{omplexes}_{X/B} \to B$ is quasi-separated and locally of finite presentation.
Proof. To check $\mathcal{C}\! \mathit{omplexes}_{X/B} \to B$ is quasi-separated we have to show that its diagonal is quasi-compact and quasi-separated. This is immediate from Lemma 108.12.1. To prove that $\mathcal{C}\! \mathit{omplexes}_{X/B} \to B$ is locally of finite presentation, we have to show that $\mathcal{C}\! \mathit{omplexes}_{X/B} \to B$ is limit preserving, see Limits of Stacks, Proposition 102.3.8. This follows from Quot, Lemma 99.16.8 (small detail omitted). $\square$
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