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Tag 07WI

Chapter 87: Criteria for Representability > Section 87.5: Limit preserving on objects

Lemma 87.5.6. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\textit{Sch}/S)_{fppf}$. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces as in Algebraic Stacks, Definition 84.10.1. If

  1. $f$ is representable by algebraic spaces,
  2. $\mathcal{Y} \to (\textit{Sch}/S)_{fppf}$ is limit preserving on objects,
  3. for an affine scheme $V$ locally of finite presentation over $S$ and $y \in \mathcal{Y}_V$ the resulting morphism of algebraic spaces $f_y : F_y \to V$, see Algebraic Stacks, Equation (84.9.1.1), has property $\mathcal{P}$.

Then $f$ has property $\mathcal{P}$.

Proof. Let $V$ be a scheme over $S$ and $y \in \mathcal{Y}_V$. We have to show that $F_y \to V$ has property $\mathcal{P}$. Since $\mathcal{P}$ is fppf local on the base we may assume that $V$ is an affine scheme which maps into an affine open $\mathop{\rm Spec}(\Lambda) \subset S$. Thus we can write $V = \mathop{\rm lim}\nolimits V_i$ with each $V_i$ affine and of finite presentation over $\mathop{\rm Spec}(\Lambda)$, see Algebra, Lemma 10.126.2. Then $y$ comes from an object $y_i$ over $V_i$ for some $i$ by assumption (2). By assumption (3) the morphism $F_{y_i} \to V_i$ has property $\mathcal{P}$. As $\mathcal{P}$ is stable under arbitrary base change and since $F_y = F_{y_i} \times_{V_i} V$ we conclude that $F_y \to V$ has property $\mathcal{P}$ as desired. $\square$

    The code snippet corresponding to this tag is a part of the file criteria.tex and is located in lines 615–632 (see updates for more information).

    \begin{lemma}
    \label{lemma-check-property-limit-preserving}
    Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism
    of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{P}$
    be a property of morphisms of algebraic spaces as in
    Algebraic Stacks, Definition
    \ref{algebraic-definition-relative-representable-property}. If
    \begin{enumerate}
    \item $f$ is representable by algebraic spaces,
    \item $\mathcal{Y} \to (\Sch/S)_{fppf}$ is limit preserving on objects,
    \item for an affine scheme $V$ locally of finite presentation over $S$ and
    $y \in \mathcal{Y}_V$ the resulting morphism of algebraic spaces
    $f_y : F_y \to V$, see Algebraic Stacks, Equation
    (\ref{algebraic-equation-representable-by-algebraic-spaces}),
    has property $\mathcal{P}$.
    \end{enumerate}
    Then $f$ has property $\mathcal{P}$.
    \end{lemma}
    
    \begin{proof}
    Let $V$ be a scheme over $S$ and $y \in \mathcal{Y}_V$. We have to show
    that $F_y \to V$ has property $\mathcal{P}$. Since $\mathcal{P}$ is
    fppf local on the base we may assume that $V$ is an affine scheme which
    maps into an affine open $\Spec(\Lambda) \subset S$. Thus we can write
    $V = \lim V_i$ with each $V_i$ affine and of finite presentation over
    $\Spec(\Lambda)$, see Algebra, Lemma \ref{algebra-lemma-ring-colimit-fp}.
    Then $y$ comes from an object $y_i$ over $V_i$ for some $i$ by assumption (2).
    By assumption (3) the morphism $F_{y_i} \to V_i$ has property $\mathcal{P}$.
    As $\mathcal{P}$ is stable under arbitrary base change and since
    $F_y = F_{y_i} \times_{V_i} V$ we conclude that $F_y \to V$ has property
    $\mathcal{P}$ as desired.
    \end{proof}

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