The Stacks project

Proof. Set $\bar\Lambda = \Lambda _\mathfrak q/\mathfrak q^ n\Lambda _\mathfrak q$. Note that $\dim H_1(L_{\kappa (\mathfrak q)/k}) < \infty $ by More on Algebra, Proposition 15.35.1. Pick $A \subset \bar\Lambda $ containing $E$ such that $A$ is local Artinian, essentially of finite type over $k$, the map $A \to \bar\Lambda $ is flat, and $\mathfrak m_ A$ generates the maximal ideal of $\bar\Lambda $, see Lemma 16.11.1. Denote $F = A/\mathfrak m_ A$ the residue field so that $k \subset F \subset K$. Pick $\lambda _1, \ldots , \lambda _ t \in \Lambda $ which map to elements of $A$ in $\bar\Lambda $ such that moreover the images of $\text{d}\lambda _1, \ldots , \text{d}\lambda _ t$ form a basis of $\Omega _{F/k}$. Consider the map $\varphi ' : k[y_1, \ldots , y_ t] \to \Lambda $ sending $y_ j$ to $\lambda _ j$. Set $\mathfrak p' = (\varphi ')^{-1}(\mathfrak q)$. By More on Algebra, Lemma 15.35.2 the ring map $k[y_1, \ldots , y_ t]_{\mathfrak p'} \to \Lambda _\mathfrak q$ is flat and $\Lambda _\mathfrak q/\mathfrak p' \Lambda _\mathfrak q$ is regular. Thus we can choose further elements $\lambda _{t + 1}, \ldots , \lambda _ m \in \Lambda $ which map into $A \subset \bar\Lambda $ and which map to a regular system of parameters of $\Lambda _\mathfrak q/\mathfrak p' \Lambda _\mathfrak q$. We obtain $\varphi : k[y_1, \ldots , y_ m] \to \Lambda $ having property (1) such that $k[y_1, \ldots , y_ m]_\mathfrak p/\mathfrak p^ n k[y_1, \ldots , y_ m]_\mathfrak p \to \bar\Lambda $ factors through $A$. Thus $k[y_1, \ldots , y_ m]_\mathfrak p/\mathfrak p^ n k[y_1, \ldots , y_ m]_\mathfrak p \to A$ is flat by Algebra, Lemma 10.39.9. By construction the residue field extension $F/\kappa (\mathfrak p)$ is finitely generated and $\Omega _{F/\kappa (\mathfrak p)} = 0$. Hence it is finite separable by More on Algebra, Lemma 15.34.1. Thus $k[y_1, \ldots , y_ m]_\mathfrak p/\mathfrak p^ n k[y_1, \ldots , y_ m]_\mathfrak p \to A$ is finite by Algebra, Lemma 10.54.4. Finally, we conclude that it is étale by Algebra, Lemma 10.143.7. Since an étale ring map is certainly essentially smooth we win. $\square$