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Tag 01JS

Chapter 25: Schemes > Section 25.17: Fibre products of schemes

Bare-hands construction of fiber products: an affine open cover of a fiber product of schemes can be assembled from compatible affine open covers of the pieces.

Lemma 25.17.4. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. Let $S = \bigcup U_i$ be any affine open covering of $S$. For each $i \in I$, let $f^{-1}(U_i) = \bigcup_{j \in J_i} V_j$ be an affine open covering of $f^{-1}(U_i)$ and let $g^{-1}(U_i) = \bigcup_{k \in K_i} W_k$ be an affine open covering of $g^{-1}(U_i)$. Then $$ X \times_S Y = \bigcup\nolimits_{i \in I} \bigcup\nolimits_{j \in J_i, ~k \in K_i} V_j \times_{U_i} W_k $$ is an affine open covering of $X \times_S Y$.

Proof. See discussion above the lemma. $\square$

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 3073–3094 (see updates for more information).

    \begin{lemma}
    \label{lemma-affine-covering-fibre-product}
    \begin{slogan}
    Bare-hands construction of fiber products: an affine open cover of a
    fiber product of schemes can be assembled from compatible
    affine open covers of the pieces.
    \end{slogan}
    Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes
    with the same target. Let $S = \bigcup U_i$ be any affine open
    covering of $S$. For each $i \in I$, let
    $f^{-1}(U_i) = \bigcup_{j \in J_i} V_j$ be an affine open covering
    of $f^{-1}(U_i)$ and let
    $g^{-1}(U_i) = \bigcup_{k \in K_i} W_k$ be an affine open covering
    of $g^{-1}(U_i)$. Then
    $$
    X \times_S Y =
    \bigcup\nolimits_{i \in I}
    \bigcup\nolimits_{j \in J_i, \ k \in K_i}
    V_j \times_{U_i} W_k
    $$
    is an affine open covering of $X \times_S Y$.
    \end{lemma}
    
    \begin{proof}
    See discussion above the lemma.
    \end{proof}

    Comments (1)

    Comment #1060 by Charles Rezk on October 4, 2014 a 2:21 pm UTC

    Suggested slogan: Bare-hands construction of fiber products: an affine open cover of a fiber product of schemes can be assembled from compatible affine open covers of the pieces.

    There are also 2 comments on Section 25.17: Schemes.

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