The Stacks project

Lemma 37.52.6. Let $\varphi : A \to B$ be a local ring map of local rings. Let $V \subset \mathop{\mathrm{Spec}}(B)$ be an open subscheme which contains at least one prime not lying over $\mathfrak m_ A$. Assume $A$ is Noetherian, $\varphi $ essentially of finite type, and $A/\mathfrak m_ A \subset B/\mathfrak m_ B$ is finite. Then there exists a $\mathfrak q \in V$, $\mathfrak m_ A \not= \mathfrak q \cap A$ such that $A \to B/\mathfrak q$ is the localization of a quasi-finite ring map.

Proof. Since $A$ is Noetherian and $A \to B$ is essentially of finite type, we know that $B$ is Noetherian too. By Properties, Lemma 28.6.4 the topological space $\mathop{\mathrm{Spec}}(B) \setminus \{ \mathfrak m_ B\} $ is Jacobson. Hence we can choose a closed point $\mathfrak q$ which is contained in the nonempty open

\[ V \setminus \{ \mathfrak q \subset B \mid \mathfrak m_ A = \mathfrak q \cap A\} . \]

(Nonempty by assumption, open because $\{ \mathfrak m_ A\} $ is a closed subset of $\mathop{\mathrm{Spec}}(A)$.) Then $\mathop{\mathrm{Spec}}(B/\mathfrak q)$ has two points, namely $\mathfrak m_ B$ and $\mathfrak q$ and $\mathfrak q$ does not lie over $\mathfrak m_ A$. Write $B/\mathfrak q = C_{\mathfrak m}$ for some finite type $A$-algebra $C$ and prime ideal $\mathfrak m$. Then $A \to C$ is quasi-finite at $\mathfrak m$ by Algebra, Lemma 10.122.2 (2). Hence by Algebra, Lemma 10.123.13 we see that after replacing $C$ by a principal localization the ring map $A \to C$ is quasi-finite. $\square$

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