The Stacks project

Lemma 33.18.3. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Let $V = \mathop{\mathrm{Spec}}(A)$ be an affine open neighbourhood of $f(x)$ in $S$. If $f$ is unramified at $x$, then there exist exists an affine open $U \subset X$ with $x \in U$ and $f(U) \subset V$ such that we have a commutative diagram

\[ \xymatrix{ X \ar[d] & U \ar[l] \ar[rd] \ar[r]^-j & \mathop{\mathrm{Spec}}(A[t]_{g'}/(g)) \ar[d] \ar[r] & \mathop{\mathrm{Spec}}(A[t]) = \mathbf{A}^1_ V \ar[ld] \\ Y & & V \ar[ll] } \]

where $j$ is an immersion, $g \in A[t]$ is a monic polynomial, and $g'$ is the derivative of $g$ with respect to $t$. If $f$ is ├ętale at $x$, then we may choose the diagram such that $j$ is an open immersion.

Proof. The unramified case is a translation of Algebra, Proposition 10.152.1. In the ├ętale case this is a translation of Algebra, Proposition 10.144.4 or equivalently it follows from Morphisms, Lemma 29.36.14 although the statements differ slightly. $\square$


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