Lemma 26.2.2. Let X, Y be locally ringed spaces. If f : X \to Y is an isomorphism of ringed spaces, then f is an isomorphism of locally ringed spaces.
An isomorphism of ringed spaces between locally ringed spaces is an isomorphism of locally ringed spaces.
Proof. This follows trivially from the corresponding fact in algebra: Suppose A, B are local rings. Any isomorphism of rings A \to B is a local ring isomorphism. \square
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