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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