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I'm starting studying Hartshorne Chapter II and is the first time that I'm studying Schemes. I'm looking for some intuition viewing some examples.

I'm looking for an example of a locally ringed space $(X,\mathcal{O}_X)$ such that is isomorphic as Ringed space to the Spectrum of some ring $A$ but not isomorphic as Locally ringed spaces.

I think that some examples must exist but my intuition on ringed spaces is (at least right now) to vague. Thanks!

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There is no such example: Any isomorphism of ringed spaces between two locally ringed spaces is already an isomorphism of locally ringed spaces.

This follows because a ring isomorphism $A\to B$ of local rings is a local isomorphism.

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