Lemma 29.58.1. Let $f : Y \to X$ be a morphism of schemes. Let $x \in X$ be a point. Assume that $Y$ is reduced and $f(Y)$ is set-theoretically contained in $\{ x\} $. Then $f$ factors through the canonical morphism $x = \mathop{\mathrm{Spec}}(\kappa (x)) \to X$.
Proof. Omitted. Hints: working affine locally one reduces to a commutative algebra lemma. Given a ring map $A \to B$ with $B$ reduced such that there exists a unique prime ideal $\mathfrak p \subset A$ in the image of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$, then $A \to B$ factors through $\kappa (\mathfrak p)$. This is a nice exercise. $\square$
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