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Tag 0DPV

98.12. 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 89.16.12.

Lemma 98.12.1. 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 89.16.5. 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 89.16.1 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 89.4.3). 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 98.12.2. 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 98.12.1. To prove that $\mathcal{C}\!\mathit{omplexes}_{X/B} \to B$ is locally of finite presentation, we have to show that $\mathcal{C}\!\mathit{oh}_{X/B} \to B$ is limit preserving, see Limits of Stacks, Proposition 92.3.8. This follows from Quot, Lemma 89.16.8 (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 $\Cohstack_{X/B} \to B$
    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}

    Comments (2)

    Comment #3013 by arp on December 5, 2017 a 2:17 am UTC

    $\mathcal{C}\!\mathit{oh}_{X/B} \to B$ should be $\mathcal{C}\!\mathit{omplexes}_{X/B} \to B$

    Comment #3014 by arp on December 5, 2017 a 2:19 am UTC

    whoops, previous comment was intended for tag 0DPX

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