I'm not sure that this is generally true, but Harthorne p73 seems to suggest it. If it is true could someone give me a hint for the proof?
If $f$ an isomorphism of ringed spaces, is $f$ necessarily an isomorphism of locally ringed spaces?
0
$\begingroup$
algebraic-geometry
commutative-algebra
schemes
-
1@EdwardHughes:I am sorry but I don't quite see what you mean. See if you agree with this: $f_{*,x}:\mathcal O_{Y,f(P)} \to f_*(\mathcal O_X)_{f(P)}$ is an isomorphism. – 2012-10-16
1 Answers
4
Yes. See Stacks project, (21.2.2).