Lemma 61.3.3. Let $A \to B$ and $B \to C$ be ring maps.

If $A \to B$ and $B \to C$ are local isomorphisms, then $A \to C$ is a local isomorphism.

If $A \to B$ and $B \to C$ identify local rings, then $A \to C$ identifies local rings.

Lemma 61.3.3. Let $A \to B$ and $B \to C$ be ring maps.

If $A \to B$ and $B \to C$ are local isomorphisms, then $A \to C$ is a local isomorphism.

If $A \to B$ and $B \to C$ identify local rings, then $A \to C$ identifies local rings.

**Proof.**
Omitted.
$\square$

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