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 26.17.4. 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. $\square$

Comment #1060 by Charles Rezk on

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.

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