## 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 3074–3095 (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}
```

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