Let $X$ be a scheme. If $X$ happens to be affine then is it true that the canonical morphism


is an isomorphism?.

  • 1
    $\begingroup$ I think that's what being an affine scheme is all about. $\endgroup$ – Angina Seng May 29 '19 at 17:22
  • $\begingroup$ I suppose the definition of an affine scheme is that there is an (not necessarily canonical) isomorphism of locally ringed spaces $X\cong \mathcal{O}_{\mathrm{Spec}\;A}$ for a ring $A$ (then of course $A\cong\mathrm{Spec}\;\Gamma(X,\mathcal{O}_X)$) ? $\endgroup$ – Dat234 May 29 '19 at 17:26

Yes. Moreover, this property characterizes affine schemes among all locally ringed spaces. See this MO comment for a proof sketch.

  • $\begingroup$ It seems that the link you provided above just contains the assertions with no proof. Can you clarify it a bit more for me? Thank you. $\endgroup$ – Dat234 May 30 '19 at 1:55

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