The Stacks project

Lemma 29.55.3. Let $A \to B$ be a ring map inducing a dominant morphism $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ of spectra. There exists an $A$-subalgebra $B' \subset B$ such that

  1. $\mathop{\mathrm{Spec}}(B') \to \mathop{\mathrm{Spec}}(A)$ is a universal homeomorphism inducing isomorphisms on residue fields,

  2. given a factorization $A \to C \to B$ such that $\mathop{\mathrm{Spec}}(C) \to \mathop{\mathrm{Spec}}(A)$ is a universal homeomorphism inducing isomorphisms on residue fields, the image of $C \to B$ is contained in $B'$.

Proof. This proof is exactly the same as the proof of Lemma 29.55.1 except we use Proposition 29.46.7 in stead of Proposition 29.46.8 $\square$

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