Over an algebraically closed field, smooth projective maps are closed immersions exactly when they are injective on closed points and tangent spaces.Suggest a slogan
Let \(k\) be an algebraically closed field and \(f\colon X\to Y\) a morphism of smooth projective varieties over \(k\). Then \(f\) is a closed immersion if and only if \(f\) is injective on closed points and the differential \(df_p\colon T_pX\to T_{f(p)}Y\) is injective for every closed point \(p\in X\) [@Har10a].
The application of Nakayama’s lemma is in the local step: a local homomorphism \(\mathcal O_{Y,f(p)}\to\mathcal O_{X,p}\) of Noetherian local rings that is an isomorphism on residue fields, surjective on cotangent spaces and makes \(\mathcal O_{X,p}\) a finitely generated module is surjective [@Har10a].
Give a geometric application of Nakayama's lemma.