Lemma 10.151.7. Let R \to S be a ring map. Let \mathfrak q be a prime of S lying over a prime \mathfrak p of R. If
R \to S is of finite type,
\mathfrak p S_{\mathfrak q} is the maximal ideal of the local ring S_{\mathfrak q}, and
the field extension \kappa (\mathfrak q)/\kappa (\mathfrak p) is finite separable,
then R \to S is unramified at \mathfrak q.
Proof. By Lemma 10.151.3 (8) it suffices to show that \Omega _{S \otimes _ R \kappa (\mathfrak p) / \kappa (\mathfrak p)} is zero when localized at \mathfrak q. Hence we may replace S by S \otimes _ R \kappa (\mathfrak p) and R by \kappa (\mathfrak p). In other words, we may assume that R = k is a field and S is a finite type k-algebra. In this case the hypotheses imply that S_{\mathfrak q} \cong \kappa (\mathfrak q). Thus (\Omega _{S/k})_{\mathfrak q} = \Omega _{S_\mathfrak q/k} = \Omega _{\kappa (\mathfrak q)/k} is zero as desired (the first equality is Lemma 10.131.8). \square
Comments (2)
Comment #4876 by HAO on
Comment #5158 by Johan on