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Tag 07XM

Chapter 88: Artin's axioms > Section 88.11: 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). $\square$

    The code snippet corresponding to this tag is a part of the file artin.tex and is located in lines 1334–1349 (see updates for more information).

    \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}

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