# Basic property of schemes?

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

$$X\to\mathrm{Spec}\;\Gamma(X,\mathcal{O}_X)$$

is an isomorphism?.

• I think that's what being an affine scheme is all about. – Angina Seng May 29 '19 at 17:22
• 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 $A\cong\mathrm{Spec}\;\Gamma(X,\mathcal{O}_X)$) ? – Dat234 May 29 '19 at 17:26

## 1 Answer

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

• 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. – Dat234 May 30 '19 at 1:55