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    • CommentRowNumber1.
    • CommentAuthorBartek
    • CommentTimeFeb 22nd 2017
    • (edited Feb 22nd 2017)

    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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 22nd 2017
    • (edited Feb 22nd 2017)

    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”.

    • CommentRowNumber3.
    • CommentAuthorBartek
    • CommentTimeFeb 22nd 2017
    • (edited Feb 22nd 2017)

    Thanks! By the way, now that I look at the Lucas’ proof again, he shows that if (f,ψ):MN(f, \psi): M \to N is a morphism of locally ringed spaces between smooth manifolds where the comorphism ψ\psi is a morphism of sheaves of \mathbb{R}-algebras (which is local on stalks) then ff is smooth and ψ\psi is the precomposition operation by ff. But what if the comorphism ψ\psi is only assumed to be a morphism of sheaves of rings (which is local on stalks)? Does the conclusion still hold?

    • CommentRowNumber4.
    • CommentAuthorBartek
    • CommentTimeFeb 22nd 2017

    Ah nevermind, I didn’t realize that the only ring endomorphism \mathbb{R} \to \mathbb{R} is the identity, so looks like the proof still works. My algebra is weak.