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The Stacks project

An isomorphism of ringed spaces between locally ringed spaces is an isomorphism of locally ringed spaces.

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.

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


Comments (3)

Comment #1277 by on

Suggested slogan: An isomorphism of ringed spaces between locally ringed spaces is an isomorphism of locally ringed spaces.

Comment #8546 by Shizhang on

The last bit of the proof: ``is a local ring \emph{isomorphism}''.


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