Definition 61.3.1. Let $\varphi : A \to B$ be a ring map.

We say $A \to B$ is a

*local isomorphism*if for every prime $\mathfrak q \subset B$ there exists a $g \in B$, $g \not\in \mathfrak q$ such that $A \to B_ g$ induces an open immersion $\mathop{\mathrm{Spec}}(B_ g) \to \mathop{\mathrm{Spec}}(A)$.We say $A \to B$

*identifies local rings*if for every prime $\mathfrak q \subset B$ the canonical map $A_{\varphi ^{-1}(\mathfrak q)} \to B_\mathfrak q$ is an isomorphism.

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