Fixed points of the adjunction between locally ringed spaces and rings

Question

There is an adjunction between locally ringed spaces and rings given by $R\mapsto \text{Spec}(R)$ and $X\mapsto \Gamma(X, \mathcal O_X)$.

Is that adjunction idempotent?

Roughly equivalent: Is it true that the canonical morphism $R\to \Gamma(\text{Spec}_R, \mathcal O_{\text{Spec}(R)})$ is always an isomorphism and that the canonical morphism $X\to \text{Spec}(\Gamma(X, \mathcal O_X))$ is an isomorphism if and only if $X$ is an affine scheme?

Answer 1

Yes.

1. $R$ and $\Gamma(\text{Spec}(R), \mathcal{O}_{\text{Spec}(R)})$ are canonically isomorphic, see here.
2. If $X \to \text{Spec}(\Gamma(X,\mathcal{O}_X))$ is an isomorphism, then this same isomorphism witnesses $X$ as an affine scheme (it's $\text{Spec}$ of something).

More categorically, we know that the category of affine schemes is antiequivalent to the category of rings (see here) and this equivalence is given by $\text{Spec}$ and $\Gamma$. Idempotent adjunctions always give rise to an equivalence of categories between the categories of fixed points, and we see that behavior here.

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I hope this helps ^_^