## Tag `01JR`

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

Lemma 25.17.3. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. Let $X \times_S Y$, $p$, $q$ be the fibre product. Suppose that $U \subset S$, $V \subset X$, $W \subset Y$ are open subschemes such that $f(V) \subset U$ and $g(W) \subset U$. Then the canonical morphism $V \times_U W \to X \times_S Y$ is an open immersion which identifies $V \times_U W$ with $p^{-1}(V) \cap q^{-1}(W)$.

Proof.Let $T$ be a scheme Suppose $a : T \to V$ and $b : T \to W$ are morphisms such that $f \circ a = g \circ b$ as morphisms into $U$. Then they agree as morphisms into $S$. By the universal property of the fibre product we get a unique morphism $T \to X \times_S Y$. Of course this morphism has image contained in the open $p^{-1}(V) \cap q^{-1}(W)$. Thus $p^{-1}(V) \cap q^{-1}(W)$ is a fibre product of $V$ and $W$ over $U$. The result follows from the uniqueness of fibre products, see Categories, Section 4.6. $\square$

The code snippet corresponding to this tag is a part of the file `schemes.tex` and is located in lines 3041–3051 (see updates for more information).

```
\begin{lemma}
\label{lemma-open-fibre-product}
Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes
with the same target. Let $X \times_S Y$, $p$, $q$ be the fibre product.
Suppose that $U \subset S$,
$V \subset X$, $W \subset Y$ are open subschemes
such that $f(V) \subset U$ and $g(W) \subset U$.
Then the canonical morphism
$V \times_U W \to X \times_S Y$ is an open immersion
which identifies $V \times_U W$ with $p^{-1}(V) \cap q^{-1}(W)$.
\end{lemma}
\begin{proof}
Let $T$ be a scheme
Suppose $a : T \to V$ and $b : T \to W$ are morphisms
such that $f \circ a = g \circ b$ as morphisms into $U$.
Then they agree as morphisms into $S$.
By the universal property of the fibre product we get
a unique morphism $T \to X \times_S Y$. Of course this morphism
has image contained in the open $p^{-1}(V) \cap q^{-1}(W)$.
Thus $p^{-1}(V) \cap q^{-1}(W)$ is a fibre product of
$V$ and $W$ over $U$. The result follows from the uniqueness
of fibre products, see Categories, Section
\ref{categories-section-fibre-products}.
\end{proof}
```

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