The Stacks project

Lemma 60.5.8. Let $A$ be a ring such that $X = \mathop{\mathrm{Spec}}(A)$ is w-local. Let $I \subset A$ be the radical ideal cutting out the set $X_0$ of closed points in $X$. Let $A \to B$ be a ring map inducing algebraic extensions on residue fields at primes. Then

  1. every point of $Z = V(IB)$ is a closed point of $\mathop{\mathrm{Spec}}(B)$,

  2. there exists an ind-Zariski ring map $B \to C$ such that

    1. $B/IB \to C/IC$ is an isomorphism,

    2. the space $Y = \mathop{\mathrm{Spec}}(C)$ is w-local,

    3. the induced map $p : Y \to X$ is w-local, and

    4. $p^{-1}(X_0)$ is the set of closed points of $Y$.

Proof. By Lemma 60.5.6 applied to $A/I \to B/IB$ all points of $Z = V(IB) = \mathop{\mathrm{Spec}}(B/IB)$ are closed, in fact $\mathop{\mathrm{Spec}}(B/IB)$ is a profinite space. To finish the proof we apply Lemma 60.5.7 to $IB \subset B$. $\square$


Comments (2)

Comment #2532 by Brian Conrad on

In (2) you should say can also be taken to be flat (used in paragraph 2 of the proof of 52.6.7 in 097B).

Comment #2533 by Brian Conrad on

Oops, disregard the preceding comment (since ind-Zariski maps are always flat); it's been a long day...


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