Lemma 10.137.17. Let R \to S be a ring map. Let \mathfrak q \subset S be a prime lying over the prime \mathfrak p of R. Assume
there exists a g \in S, g \not\in \mathfrak q such that R \to S_ g is of finite presentation,
the local ring homomorphism R_{\mathfrak p} \to S_{\mathfrak q} is flat,
the fibre S \otimes _ R \kappa (\mathfrak p) is smooth over \kappa (\mathfrak p) at the prime corresponding to \mathfrak q.
Then R \to S is smooth at \mathfrak q.
Proof. By Lemmas 10.136.15 and 10.137.5 we see that there exists a g \in S such that S_ g is a relative global complete intersection. Replacing S by S_ g we may assume S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c) is a relative global complete intersection. For any subset I \subset \{ 1, \ldots , n\} of cardinality c consider the polynomial g_ I = \det (\partial f_ j/\partial x_ i)_{j = 1, \ldots , c, i \in I} of Lemma 10.137.16. Note that the image \overline{g}_ I of g_ I in the polynomial ring \kappa (\mathfrak p)[x_1, \ldots , x_ n] is the determinant of the partial derivatives of the images \overline{f}_ j of the f_ j in the ring \kappa (\mathfrak p)[x_1, \ldots , x_ n]. Thus the lemma follows by applying Lemma 10.137.16 both to R \to S and to \kappa (\mathfrak p) \to S \otimes _ R \kappa (\mathfrak p). \square
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