Lemma 100.12.1. The diagonal of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ over $B$ is affine and of finite presentation.

## 100.12 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 91.16.12.

**Proof.**
The representability of the diagonal (by algebraic spaces) was shown in Quot, Lemma 91.16.5. 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

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

By Quot, Lemma 91.16.1 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

where $c(\varphi ', \varphi ) = (\varphi \circ \varphi ', \varphi ' \circ \varphi )$ and $\sigma = (\text{id}, \text{id})$ (compare with the proof of Quot, Proposition 91.4.3). 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 100.12.2. 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 100.12.1. 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 94.3.8. This follows from Quot, Lemma 91.16.8 (small detail omitted).
$\square$

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