Let \(f,g\colon X\to Y\) be morphisms of \(S\)-schemes with \(X\) reduced and \(Y\) separated over \(S\). If \(f\) and \(g\) agree on a dense open subset of \(X\), then \(f=g\). Separatedness is needed: the two open immersions of \(\AA^1_k\) into the line with doubled origin agree on \(\AA^1_k\setminus\{0\}\). Reducedness is needed: on \(X=\Spec k[x,\varepsilon]/(\varepsilon^2,x\varepsilon)\), the morphisms \(X\to\AA^1_k\) given by \(0\) and \(\varepsilon\) agree on the dense open subset \(D(x)\), where \(\varepsilon=0\).
Separated and proper, the diagonal, the valuative criteria, and the counterexamples for each dropped hypothesis.
Finite and flat, going up and going down, finite flat morphisms, and the Hilbert polynomial in flat families.
Classes of morphism, the immersions, the affine and finite and projective chain, and what survives base change.
Smooth, unramified, and étale, the Jacobian criterion, ramification of curve morphisms, regularity against smoothness over imperfect fields, and the failure of generic smoothness for Frobenius.
Fibres and images, the fibre dimension theorem, Zariski’s main theorem, and Stein factorisation.
Inseparable morphisms and Lüroth, the Frobenius twist, purely inseparable morphisms of curves as composites of Frobenius morphisms, and Lüroth’s theorem.