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Tag 07XK

88.11. Limit preserving

The morphism $p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}$ is limit preserving on objects, as defined in Criteria for Representability, Section 87.5, if the functor of the definition below is essentially surjective. However, the example in Examples, Section 100.46 shows that this isn't equivalent to being limit preserving.

Definition 88.11.1. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\textit{Sch}/S)_{fppf}$. We say $\mathcal{X}$ is limit preserving if for every affine scheme $T$ over $S$ which is a limit $T = \mathop{\rm lim}\nolimits T_i$ of a directed inverse system of affine schemes $T_i$ over $S$, we have an equivalence $$ \mathop{\rm colim}\nolimits \mathcal{X}_{T_i} \longrightarrow \mathcal{X}_T $$ of fibre categories.

We spell out what this means. First, given objects $x, y$ of $\mathcal{X}$ over $T_i$ we should have $$ \mathop{\rm Mor}\nolimits_{\mathcal{X}_T}(x|_T, y|_T) = \mathop{\rm colim}\nolimits_{i' \geq i} \mathop{\rm Mor}\nolimits_{\mathcal{X}_{T_i'}}(x|_{T_i'}, y|_{T_i'}) $$ and second every object of $\mathcal{X}_T$ is isomorphic to the restriction of an object over $T_i$ for some $i$. Note that the first condition means that the presheaves $\mathit{Isom}_\mathcal{X}(x, y)$ (see Stacks, Definition 8.2.2) are limit preserving.

Lemma 88.11.2. Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\textit{Sch}/S)_{fppf}$.

  1. If $\mathcal{X} \to (\textit{Sch}/S)_{fppf}$ and $\mathcal{Z} \to (\textit{Sch}/S)_{fppf}$ are limit preserving on objects and $\mathcal{Y}$ is limit preserving, then $\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to (\textit{Sch}/S)_{fppf}$ is limit preserving on objects.
  2. If $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ are limit preserving, then so is $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$.

Proof. This is formal. Proof of (1). Let $T = \mathop{\rm lim}\nolimits_{i \in I} T_i$ be the directed limit of affine schemes $T_i$ over $S$. We will prove that the functor $\mathop{\rm colim}\nolimits \mathcal{X}_{T_i} \to \mathcal{X}_T$ is essentially surjective. Recall that an object of the fibre product over $T$ is a quadruple $(T, x, z, \alpha)$ where $x$ is an object of $\mathcal{X}$ lying over $T$, $z$ is an object of $\mathcal{Z}$ lying over $T$, and $\alpha : p(x) \to q(z)$ is a morphism in the fibre category of $\mathcal{Y}$ over $T$. By assumption on $\mathcal{X}$ and $\mathcal{Z}$ we can find an $i$ and objects $x_i$ and $z_i$ over $T_i$ such that $x_i|_T \cong T$ and $z_i|_T \cong z$. Then $\alpha$ corresponds to an isomorphism $p(x_i)|_T \to q(z_i)|_T$ which comes from an isomorphism $\alpha_{i'} : p(x_i)|_{T_{i'}} \to q(z_i)|_{T_{i'}}$ by our assumption on $\mathcal{Y}$. After replacing $i$ by $i'$, $x_i$ by $x_i|_{T_{i'}}$, and $z_i$ by $z_i|_{T_{i'}}$ we see that $(T_i, x_i, z_i, \alpha_i)$ is an object of the fibre product over $T_i$ which restricts to an object isomorphic to $(T, x, z, \alpha)$ over $T$ as desired.

We omit the arguments showing that $\mathop{\rm colim}\nolimits \mathcal{X}_{T_i} \to \mathcal{X}_T$ is fully faithful in (2). $\square$

Lemma 88.11.3. Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. Then the following are equivalent

  1. $\mathcal{X}$ is a stack in setoids and $\mathcal{X} \to (\textit{Sch}/S)_{fppf}$ is limit preserving on objects,
  2. $\mathcal{X}$ is a stack in setoids and limit preserving,
  3. $\mathcal{X}$ is representable by an algebraic space locally of finite presentation.

Proof. Under each of the three assumptions $\mathcal{X}$ is representable by an algebraic space $X$ over $S$, see Algebraic Stacks, Proposition 84.13.3. It is clear that (1) and (2) are equivalent as a functor between setoids is an equivalence if and only if it is surjective on isomorphism classes. Finally, (1) and (3) are equivalent by Limits of Spaces, Proposition 61.3.9. $\square$

Lemma 88.11.4. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\textit{Sch}/S)_{fppf}$. Assume $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces and $\mathcal{X}$ is limit preserving. Then $\Delta$ is locally of finite type.

Proof. We apply Criteria for Representability, Lemma 87.5.6. Let $V$ be an affine scheme $V$ of finite type over $S$ and let $\theta$ be an object of $\mathcal{X} \times \mathcal{X}$ over $V$. Let $F_\theta$ be an algebraic space representing $\mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}, \theta} (\textit{Sch}/V)_{fppf}$ and let $f_\theta : F_\theta \to V$ be the canonical morphism (see Algebraic Stacks, Section 84.9). It suffices to show that $F_\theta \to V$ has the corresponding properties. By Lemmas 88.11.2 and 88.11.3 we see that $F_\theta \to S$ is locally of finite presentation. It follows that $F_\theta \to V$ is locally of finite type by Morphisms of Spaces, Lemma 58.23.6. $\square$

    The code snippet corresponding to this tag is a part of the file artin.tex and is located in lines 1297–1431 (see updates for more information).

    \section{Limit preserving}
    \label{section-limits}
    
    \noindent
    The morphism $p : \mathcal{X} \to (\Sch/S)_{fppf}$ is limit preserving
    on objects, as defined in Criteria for Representability, Section
    \ref{criteria-section-limit-preserving}, if the functor of the definition
    below is essentially surjective. However, the example
    in Examples, Section \ref{examples-section-limit-preserving}
    shows that this isn't equivalent to being limit preserving.
    
    \begin{definition}
    \label{definition-limit-preserving}
    Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids
    over $(\Sch/S)_{fppf}$. We say $\mathcal{X}$ is {\it limit preserving}
    if for every affine scheme $T$ over $S$ which is a limit $T = \lim T_i$
    of a directed inverse system of affine schemes $T_i$ over $S$, we have
    an equivalence
    $$
    \colim \mathcal{X}_{T_i} \longrightarrow \mathcal{X}_T
    $$
    of fibre categories.
    \end{definition}
    
    \noindent
    We spell out what this means. First, given objects $x, y$ of $\mathcal{X}$
    over $T_i$ we should have
    $$
    \Mor_{\mathcal{X}_T}(x|_T, y|_T) =
    \colim_{i' \geq i} \Mor_{\mathcal{X}_{T_i'}}(x|_{T_i'}, y|_{T_i'})
    $$
    and second every object of $\mathcal{X}_T$ is isomorphic to the restriction
    of an object over $T_i$ for some $i$. Note that the first condition means
    that the presheaves $\mathit{Isom}_\mathcal{X}(x, y)$ (see
    Stacks, Definition \ref{stacks-definition-mor-presheaf})
    are limit preserving.
    
    \begin{lemma}
    \label{lemma-fibre-product-limit-preserving}
    Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and
    $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories
    fibred in groupoids over $(\Sch/S)_{fppf}$.
    \begin{enumerate}
    \item If $\mathcal{X} \to (\Sch/S)_{fppf}$ and
    $\mathcal{Z} \to (\Sch/S)_{fppf}$ are limit preserving on objects and
    $\mathcal{Y}$ is limit preserving, then
    $\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to (\Sch/S)_{fppf}$ is
    limit preserving on objects.
    \item If $\mathcal{X}$, $\mathcal{Y}$,
    and $\mathcal{Z}$ are limit preserving, then so
    is $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    This is formal. Proof of (1). Let $T = \lim_{i \in I} T_i$ be the directed
    limit of affine schemes $T_i$ over $S$. We will prove that the functor
    $\colim \mathcal{X}_{T_i} \to \mathcal{X}_T$ is essentially surjective.
    Recall that an object of the fibre product over $T$ is a quadruple
    $(T, x, z, \alpha)$ where $x$ is an object of $\mathcal{X}$ lying over $T$,
    $z$ is an object of $\mathcal{Z}$ lying over $T$, and
    $\alpha : p(x) \to q(z)$ is a morphism in the fibre category of
    $\mathcal{Y}$ over $T$. By assumption on $\mathcal{X}$ and $\mathcal{Z}$
    we can find an $i$ and objects $x_i$ and $z_i$ over $T_i$ such that
    $x_i|_T \cong T$ and $z_i|_T \cong z$. Then $\alpha$ corresponds to
    an isomorphism $p(x_i)|_T \to q(z_i)|_T$ which comes from an isomorphism
    $\alpha_{i'} : p(x_i)|_{T_{i'}} \to q(z_i)|_{T_{i'}}$ by our assumption on
    $\mathcal{Y}$. After replacing $i$ by $i'$, $x_i$ by $x_i|_{T_{i'}}$, and
    $z_i$ by $z_i|_{T_{i'}}$ we see that $(T_i, x_i, z_i, \alpha_i)$
    is an object of the fibre product over $T_i$ which restricts to
    an object isomorphic to $(T, x, z, \alpha)$ over $T$ as desired.
    
    \medskip\noindent
    We omit the arguments showing that $\colim \mathcal{X}_{T_i} \to \mathcal{X}_T$
    is fully faithful in (2).
    \end{proof}
    
    \begin{lemma}
    \label{lemma-limit-preserving-algebraic-space}
    Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$.
    Then the following are equivalent
    \begin{enumerate}
    \item $\mathcal{X}$ is a stack in setoids and
    $\mathcal{X} \to (\Sch/S)_{fppf}$ is limit preserving on objects,
    \item $\mathcal{X}$ is a stack in setoids and limit preserving,
    \item $\mathcal{X}$ is representable by an algebraic space
    locally of finite presentation.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Under each of the three assumptions $\mathcal{X}$ is representable
    by an algebraic space $X$ over $S$, see Algebraic Stacks, Proposition
    \ref{algebraic-proposition-algebraic-stack-no-automorphisms}.
    It is clear that (1) and (2) are equivalent as a functor between
    setoids is an equivalence if and only if it is surjective on isomorphism
    classes. Finally, (1) and (3) are equivalent by
    Limits of Spaces, Proposition
    \ref{spaces-limits-proposition-characterize-locally-finite-presentation}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-diagonal}
    Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred
    in groupoids over $(\Sch/S)_{fppf}$. Assume
    $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is
    representable by algebraic spaces and $\mathcal{X}$ is limit preserving.
    Then $\Delta$ is locally of finite type.
    \end{lemma}
    
    \begin{proof}
    We apply Criteria for Representability, Lemma
    \ref{criteria-lemma-check-property-limit-preserving}.
    Let $V$ be an affine scheme $V$ of finite type over $S$
    and let $\theta$ be an object of $\mathcal{X} \times \mathcal{X}$
    over $V$. Let $F_\theta$ be an algebraic space representing
    $\mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}, \theta}
    (\Sch/V)_{fppf}$ and let $f_\theta : F_\theta \to V$ be the canonical morphism
    (see Algebraic Stacks, Section
    \ref{algebraic-section-morphisms-representable-by-algebraic-spaces}).
    It suffices to show that
    $F_\theta \to V$ has the corresponding properties. By
    Lemmas \ref{lemma-fibre-product-limit-preserving} and
    \ref{lemma-limit-preserving-algebraic-space}
    we see that $F_\theta \to S$ is locally of finite presentation.
    It follows that $F_\theta \to V$ is locally of finite type
    by Morphisms of Spaces, Lemma
    \ref{spaces-morphisms-lemma-permanence-finite-type}.
    \end{proof}

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