The Stacks project

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

\[ \left(\frac{\partial g_ i}{\partial x_ j}\right) \bmod \mathfrak q \quad \in \quad \text{Mat}(n \times m, \kappa (\mathfrak q)) \]

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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