The Stacks project

Proposition 61.6.6. Let $A \to B$ be a ring map which identifies local rings. Then there exists a faithfully flat, ind-Zariski ring map $B \to B'$ such that $A \to B'$ is ind-Zariski.

Proof. Let $A \to A_ w$, resp. $B \to B_ w$ be the faithfully flat, ind-Zariski ring map constructed in Lemma 61.5.3 for $A$, resp. $B$. Since $\mathop{\mathrm{Spec}}(B_ w)$ is w-local, there exists a unique factorization $A \to A_ w \to B_ w$ such that $\mathop{\mathrm{Spec}}(B_ w) \to \mathop{\mathrm{Spec}}(A_ w)$ is w-local by Lemma 61.5.5. Note that $A_ w \to B_ w$ identifies local rings, see Lemma 61.3.4. By Lemma 61.6.5 this means $A_ w \to B_ w$ is ind-Zariski. Since $B \to B_ w$ is faithfully flat, ind-Zariski (Lemma 61.5.3) and the composition $A \to B \to B_ w$ is ind-Zariski (Lemma 61.4.3) the proposition is proved. $\square$

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