Unramified morphisms

The first condition says that for every \(y\in Y\) and \(x\in X_y\), the local ring \(\OO_{X_y,x}\) is the field \(\kappa(x)\), a finite separable extension of \(\kappa(y)\). Since the diagonal of an unramified morphism is open, two sections \(s,t\colon Y\to X\) of an unramified morphism \(f\) agree on an open subset of \(Y\), namely \((s,t)^{-1}(\Delta_{X/Y})\).

For a nonconstant map of smooth integral curves over an algebraically closed field, the local picture at a closed point is \(f^\sharp(\mfm_{f(p)}) \OO_{X,p} = \mfm_p^{e_p}\), and \(f\) is unramified exactly when every ramification index \(e_p\) equals \(1\). For \(n>1\), \(t \mapsto t^n\) on \(\AA^1\) is ramified at the origin. In characteristic \(p\) dividing \(n\), its differential vanishes everywhere and the induced function-field extension is inseparable. For an unramified finite separable morphism \(f\colon X\to Y\) of smooth projective curves, the ramification divisor is \(0\), and Riemann–Hurwitz gives \(2g_X-2=(\deg f)(2g_Y-2)\).