## Tag `0DPV`

## 99.12. 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 90.16.12.

Lemma 99.12.1. 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 90.16.5. 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 90.16.1 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 90.4.3). 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 99.12.2. 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 99.12.1. 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 93.3.8. This follows from Quot, Lemma 90.16.8 (small detail omitted). $\square$

The code snippet corresponding to this tag is a part of the file `moduli.tex` and is located in lines 1992–2068 (see updates for more information).

```
\section{Properties of moduli of complexes on a proper morphism}
\label{section-complexes}
\noindent
Let $f : X \to B$ be a morphism of algebraic spaces which is proper,
flat, and of finite presentation. Then the stack
$\Complexesstack_{X/B}$ parametrizing relatively perfect complexes
with vanishing negative self-exts is algebraic. See
Quot, Theorem \ref{quot-theorem-complexes-algebraic}.
\begin{lemma}
\label{lemma-complexes-diagonal-affine-fp}
The diagonal of $\Complexesstack_{X/B}$ over $B$ is affine
and of finite presentation.
\end{lemma}
\begin{proof}
The representability of the diagonal (by algebraic spaces)
was shown in Quot, Lemma \ref{quot-lemma-complexes-diagonal}.
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 $\Complexesstack_{X/B}$ over $T$, then
$\mathit{Isom}(E, E') \to T$
is affine and of finite presentation.
Here $\mathit{Isom}(E, E')$ is the functor
$$
(\Sch/T)^{opp} \to \textit{Sets},\quad
T' \mapsto \{\varphi : E_{T'} \to E'_{T'}
\text{ isomorphism in }D(\mathcal{O}_{X_{T'}})\}
$$
where $E_{T'}$ and $E'_{T'}$ are the derived pullbacks of $E$ and $E'$
to $X_{T'}$. Consider the functor $H = \SheafHom(E, E')$ defined
by the rule
$$
(\Sch/T)^{opp} \to \textit{Sets},\quad
T' \mapsto \Hom_{\mathcal{O}_{X_{T'}}}(E_T, E'_T)
$$
By Quot, Lemma \ref{quot-lemma-complexes-open-neg-exts-vanishing}
this is an algebraic space affine and of finite presentation over $T$.
The same is true for $H' = \SheafHom(E', E)$, $I = \SheafHom(E, E)$, and
$I' = \SheafHom(E', E')$. 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 \ref{quot-proposition-isom}). 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 $T$.
\end{proof}
\begin{lemma}
\label{lemma-complexes-qs-lfp}
The morphism $\Complexesstack_{X/B} \to B$ is quasi-separated and
locally of finite presentation.
\end{lemma}
\begin{proof}
To check $\Complexesstack_{X/B} \to B$ is quasi-separated we have to
show that its diagonal is quasi-compact and quasi-separated.
This is immediate from Lemma \ref{lemma-complexes-diagonal-affine-fp}.
To prove that $\Complexesstack_{X/B} \to B$ is locally of finite
presentation, we have to show that $\Complexesstack_{X/B} \to B$
is limit preserving, see
Limits of Stacks, Proposition
\ref{stacks-limits-proposition-characterize-locally-finite-presentation}.
This follows from Quot, Lemma \ref{quot-lemma-complexes-limits}
(small detail omitted).
\end{proof}
\input{chapters}
```

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