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

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

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

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

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

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

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

## Comments (2)

Comment #2532 by Brian Conrad on

Comment #2533 by Brian Conrad on