The Stacks project

Lemma 61.5.8. 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 $Y$.