Example 41.8.2. Property (3) in Theorem 41.4.1 gives us a canonical source of examples for unramified morphisms. Fix a ring R and an integer n. Let I = (g_1, \ldots , g_ m) be an ideal in R[x_1, \ldots , x_ n]. Let \mathfrak q \subset R[x_1, \ldots , x_ n] be a prime. Assume I \subset \mathfrak q and that the matrix
has rank n. Then the morphism f : Z = \mathop{\mathrm{Spec}}(R[x_1, \ldots , x_ n]/I) \to \mathop{\mathrm{Spec}}(R) is unramified at the point x \in Z \subset \mathbf{A}^ n_ R corresponding to \mathfrak q. Clearly we must have m \geq n. In the extreme case m = n, i.e., the differential of the map \mathbf{A}^ n_ R \to \mathbf{A}^ n_ R defined by the g_ i's is an isomorphism of the tangent spaces, then f is also flat x and, hence, is an étale map (see Algebra, Definition 10.137.6, Lemma 10.137.7 and Example 10.137.8).
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