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
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