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Tag 03N0

Chapter 34: Descent > Section 34.10: Fpqc coverings are universal effective epimorphisms

Lemma 34.10.2. Let $\{f_i : T_i \to T\}_{i \in I}$ be a fpqc covering. Suppose that for each $i$ we have an open subset $W_i \subset T_i$ such that for all $i, j \in I$ we have $\text{pr}_0^{-1}(W_i) = \text{pr}_1^{-1}(W_j)$ as open subsets of $T_i \times_T T_j$. Then there exists a unique open subset $W \subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$.

Proof. Apply Lemma 34.10.1 to the map $\coprod_{i \in I} T_i \to T$. It implies there exists a subset $W \subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$, namely $W = \bigcup f_i(W_i)$. To see that $W$ is open we may work Zariski locally on $T$. Hence we may assume that $T$ is affine. Using the definition of a fpqc covering, this reduces us to the case where $\{f_i : T_i \to T\}$ is a standard fpqc covering. In this case we may apply Morphisms, Lemma 28.24.11 to the morphism $\coprod T_i \to T$ to conclude that $W$ is open. $\square$

    The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 3228–3236 (see updates for more information).

    \begin{lemma}
    \label{lemma-open-fpqc-covering}
    Let $\{f_i : T_i \to T\}_{i \in I}$ be a fpqc covering.
    Suppose that for each $i$ we have an open subset $W_i \subset T_i$
    such that for all $i, j \in I$ we have
    $\text{pr}_0^{-1}(W_i) = \text{pr}_1^{-1}(W_j)$ as open
    subsets of $T_i \times_T T_j$. Then there exists a unique open subset
    $W \subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$.
    \end{lemma}
    
    \begin{proof}
    Apply
    Lemma \ref{lemma-equiv-fibre-product}
    to the map $\coprod_{i \in I} T_i \to T$.
    It implies there exists a subset $W \subset T$ such that
    $W_i = f_i^{-1}(W)$ for each $i$, namely $W = \bigcup f_i(W_i)$.
    To see that $W$ is open we may work Zariski locally on $T$.
    Hence we may assume that $T$ is affine. Using the definition
    of a fpqc covering, this reduces us to the case where
    $\{f_i : T_i \to T\}$ is a standard fpqc covering. In this case we
    may apply
    Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology}
    to the morphism
    $\coprod T_i \to T$ to conclude that $W$ is open.
    \end{proof}

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