## Tag `0356`

Chapter 25: Schemes > Section 25.12: Reduced schemes

Lemma 25.12.6. Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $Y$ be a reduced scheme. A morphism $f : Y \to X$ factors through $Z$ if and only if $f(Y) \subset Z$ (set theoretically). In particular, any morphism $Y \to X$ factors as $Y \to X_{red} \to X$.

Proof.Assume $f(Y) \subset Z$ (set theoretically). Let $I \subset \mathcal{O}_X$ be the ideal sheaf of $Z$. For any affine opens $U \subset X$, $\mathop{\rm Spec}(B) = V \subset Y$ with $f(V) \subset U$ and any $g \in \mathcal{I}(U)$ the pullback $b = f^\sharp(g) \in \Gamma(V, \mathcal{O}_Y) = B$ maps to zero in the residue field of any $y \in V$. In other words $b \in \bigcap_{\mathfrak p \subset B} \mathfrak p$. This implies $b = 0$ as $B$ is reduced (Lemma 25.12.2, and Algebra, Lemma 10.16.2). Hence $f$ factors through $Z$ by Lemma 25.4.6. $\square$

The code snippet corresponding to this tag is a part of the file `schemes.tex` and is located in lines 2112–2120 (see updates for more information).

```
\begin{lemma}
\label{lemma-map-into-reduction}
Let $X$ be a scheme.
Let $Z \subset X$ be a closed subscheme.
Let $Y$ be a reduced scheme.
A morphism $f : Y \to X$ factors through $Z$ if and only if
$f(Y) \subset Z$ (set theoretically). In particular, any
morphism $Y \to X$ factors as $Y \to X_{red} \to X$.
\end{lemma}
\begin{proof}
Assume $f(Y) \subset Z$ (set theoretically).
Let $I \subset \mathcal{O}_X$ be the ideal sheaf of $Z$.
For any affine opens $U \subset X$, $\Spec(B) = V \subset Y$
with $f(V) \subset U$ and any $g \in \mathcal{I}(U)$
the pullback $b = f^\sharp(g) \in \Gamma(V, \mathcal{O}_Y) = B$
maps to zero in the residue field of any $y \in V$.
In other words $b \in \bigcap_{\mathfrak p \subset B} \mathfrak p$.
This implies $b = 0$ as $B$ is reduced (Lemma \ref{lemma-reduced}, and
Algebra, Lemma \ref{algebra-lemma-Zariski-topology}).
Hence $f$ factors through
$Z$ by Lemma \ref{lemma-characterize-closed-subspace}.
\end{proof}
```

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