Scheme-theoretic image
= Scheme-theoretic image
The scheme-theoretic image of $f:X\to Y$ is the smallest <closed subscheme> of $Y$ through which $f$ factors. For an affine morphism $\operatorname{Spec}B\to\operatorname{Spec}A$ induced by $\varphi:A\to B$, it is $\operatorname{Spec}(A/\ker\varphi)$, whose underlying set is the closure of the set-theoretic image.
The construction retains the scheme structure on the <closed subscheme>, not merely the set of points in an image.