Definition 38.6.1. Let $R \to S$ be a ring map. Let $\mathfrak q$ be a prime of $S$ lying over the prime $\mathfrak p$ of $R$. A elementary étale localization of the ring map $R \to S$ at $\mathfrak q$ is given by a commutative diagram of rings and accompanying primes
\[ \xymatrix{ S \ar[r] & S' \\ R \ar[u] \ar[r] & R' \ar[u] } \quad \quad \xymatrix{ \mathfrak q \ar@{-}[r] & \mathfrak q' \\ \mathfrak p \ar@{-}[u] \ar@{-}[r] & \mathfrak p' \ar@{-}[u] } \]
such that $R \to R'$ and $S \to S'$ are étale ring maps and $\kappa (\mathfrak p) = \kappa (\mathfrak p')$ and $\kappa (\mathfrak q) = \kappa (\mathfrak q')$.
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