Lemma 61.5.5 (Universal property of the construction). Let A be a ring. Let A \to A_ w be the ring map constructed in Lemma 61.5.3. For any ring map A \to B such that \mathop{\mathrm{Spec}}(B) is w-local, there is a unique factorization A \to A_ w \to B such that \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A_ w) is w-local.
Proof. Denote Y = \mathop{\mathrm{Spec}}(B) and Y_0 \subset Y the set of closed points. Denote f : Y \to X the given morphism. Recall that Y_0 is profinite, in particular every constructible subset of Y_0 is open and closed. Let E \subset A be a finite subset. Recall that A_ w = \mathop{\mathrm{colim}}\nolimits A_ E and that the set of closed points of \mathop{\mathrm{Spec}}(A_ w) is the limit of the closed subsets Z_ E \subset X_ E = \mathop{\mathrm{Spec}}(A_ E). Thus it suffices to show there is a unique factorization A \to A_ E \to B such that Y \to X_ E maps Y_0 into Z_ E. Since Z_ E \to X = \mathop{\mathrm{Spec}}(A) is bijective, and since the strata Z(E', E'') are constructible we see that
is a disjoint union decomposition into open and closed subsets. As Y_0 = \pi _0(Y) we obtain a corresponding decomposition of Y into open and closed pieces. Thus it suffices to construct the factorization in case f(Y_0) \subset Z(E', E'') for some decomposition E = E' \amalg E''. In this case f(Y) is contained in the set of points of X specializing to Z(E', E'') which is homeomorphic to X_{E', E''}. Thus we obtain a unique continuous map Y \to X_{E', E''} over X. By Lemma 61.3.7 this corresponds to a unique morphism of schemes Y \to X_{E', E''} over X. This finishes the proof. \square
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