Lemma 29.6.7. Let $f : X \to Y$ be a morphism of schemes. If $X$ is reduced, then the scheme theoretic image of $f$ is the reduced induced scheme structure on $\overline{f(X)}$.

Proof. This is true because the reduced induced scheme structure on $\overline{f(X)}$ is clearly the smallest closed subscheme of $Y$ through which $f$ factors, see Schemes, Lemma 26.12.7. $\square$

Comment #6820 by Will Chen on

Do we need a quasi-compact hypothesis on $f$ here? (or perhaps replace $\overline{f(X)}$ with the topological space of the scheme theoretic image?)

Comment #6962 by on

No, the lemma and its proof are correct as is. The notation $\overline{f(X)}$ means the closure of $f(X)$ in the topological space underlying the scheme $Y$. Then one has a reduced induced scheme structure on that thing. Call it $Z$. Then $f : X \to Y$ factors through $Z$ by the lemma given in the proof. But clearly you cannot get a smaller closed subscheme of $Y$ that $f$ factors through (and this is exactly what the proof says).

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