Lemma 29.55.4. Let $A \to B$ be a ring map inducing a dominant morphism $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ of spectra. Formation of the $A$-subalgebra $B' \subset B$ in Lemma 29.55.3 commutes with localization (see proof for explanation).

**Proof.**
The proof is the same as the proof of Lemma 29.55.2.
$\square$

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