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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View Lemma 26.2.2 as pdf