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Added the example of smooth manifolds, which have a canonical fully faithful embedding into locally ringed spaces, citing Lucas Braune’s nice proof on stackexchange.
Thanks for the additions.
By the way, your code for the links is correct. For them to come out as links you may have to toggle the option “Format comments as” below the edit pane from “Text” to “Markdown+Itex”.
Thanks! By the way, now that I look at the Lucas’ proof again, he shows that if is a morphism of locally ringed spaces between smooth manifolds where the comorphism is a morphism of sheaves of -algebras (which is local on stalks) then is smooth and is the precomposition operation by . But what if the comorphism is only assumed to be a morphism of sheaves of rings (which is local on stalks)? Does the conclusion still hold?
Ah nevermind, I didn’t realize that the only ring endomorphism is the identity, so looks like the proof still works. My algebra is weak.
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