Lemma 108.12.1. The diagonal of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ over $B$ is affine and of finite presentation.
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 $T$. $\square$
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