For \(f: X \to Y\), the image of \(f\) lies in a closed subscheme \(Z \subseteq Y\) with ideal sheaf \(\mci_Z\) if the composite \(\mci_Z \to \OO_Y \to f_* \OO_X\) is zero, equivalently if \(f\) factors through \(Z\). The scheme-theoretic image of \(f\) is the smallest closed subscheme of \(Y\) in which the image of \(f\) lies, the intersection of all such \(Z\). When \(\mci \da \ker(\OO_Y \to f_* \OO_X)\) is quasicoherent — for instance when \(f\) is quasicompact and quasiseparated — this ideal sheaf defines it.
(Constructible set)
In a Noetherian topological space \(X\), the constructible subsets form the smallest family of subsets containing every open set and closed under finite intersections and complements. A subset of \(X\) is constructible if and only if it is a finite disjoint union of locally closed subsets. [@Har10a, Exercise II.3.18]
(Scheme-theoretic closure)
The scheme-theoretic closure of a locally closed immersion \(i \colon X \to Y\) is the scheme-theoretic image of \(i\).
If \(X\) is reduced, the scheme-theoretic image is \(\closure{f(X)}\) with its reduced induced structure. For \(f: \Spec B \to \Spec A\) induced by \(\varphi: A \to B\), the image is \(\Spec(A/\ker \varphi)\).
The set-theoretic image of a morphism need not be closed, open, or locally closed: \(\AA^2 \to \AA^2\), \((x,y) \mapsto (x, xy)\), has image the plane minus the \(y\)-axis, together with the origin. By Chevalley’s theorem, the image of a constructible set under a finite-type morphism of Noetherian schemes is constructible.
The scheme-theoretic image of the morphism \(\Spec k[\eps]/\eps^2 \to \AA^1\) given by \(t\mapsto\eps\) is the double point \(V(t^2)\), because the kernel of \(k[t] \to k[\eps]/\eps^2\) is \((t^2)\). The closure of its set-theoretic image, with the reduced induced structure, is the reduced point \(V(t)\).