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Let $I$ be a small category, $i\mapsto R_i$ a functor from $I$ to commutative rings with identity and $X$ a scheme. Since for every $i$ we have a morphism $\text{Spec}R_i\to\text{Spec}\lim_i R_i$, we obtain a natural map $$\text{Hom}(\text{Spec}\lim_iR_i,X)\to\lim_i\text{Hom}(\text{Spec}R_i,X)$$ If $X=\text{Spec}A$ is affine, then $$\text{Hom}(\text{Spec}\lim_iR_i,\text{Spec}A)=\text{Hom}(A,\lim_iR_i)=\lim_i(A,R_i)=\lim_i\text{Hom}(\text{Spec}R_i,\text{Spec}A)$$ and hence the map above is bijective. Things start to become interesting when we observe that also the converse is true: if the map above is bijective for every small limit of rings, then $X$ is affine.

In fact, suppose that $X$ is such that the map above is bijective for every small limit of rings, and let $X=\bigcup_i\text{Spec} A_i$ be an open affine covering, and $\text{Spec} A_i\cap\text{Spec} A_j=\bigcup_{k}\text{Spec}A_{ijk}$ an affine covering of the intersections. By a small abuse of notation, this allows us to write $X$ as a colimit $\text{colim}_i\text{Spec} A_i$ of affine schemes. On the other hand, we have that the limit $\lim_i A_i$ is naturally the global sections $\Gamma(X)$. Hence we have natural identifications $$\text{Hom}(\text{Spec}\Gamma(X),X)=\text{Hom}(\text{Spec}\lim_iA_i,X)=\lim_i\text{Hom}(\text{Spec}A_i,X)=$$ $$=\text{Hom}(\text{colim}_i\text{Spec}A_i,X)=\text{Hom}(X,X)$$ In particular, we obtain a natural map $\text{Spec}\Gamma(X)\to X$ which is easily checked to be inverse to $X\to\text{Spec}\Gamma(X)$.

The interesting thing is that this weird criterion for affinity can be imposed to a much larger class of objects than schemes, for example it can be applied to algebraic spaces. Hence, let us call an algebraic space affine if it satisfies the criterion above.

Vague question: What are affine algebraic spaces? Have they ever been studied?

Precise question 1: Does it exist an affine algebraic space which is not a scheme?

Precise question 2: If the answer to question n.1 is yes, can we find a Zariski cover of an algebraic space with affine algebraic spaces?

This question is the result of my nocturnal thoughts about my yesterday's question Extension of local morphism in algebraic spaces

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    $\begingroup$ For an algebraic space $X$, you still have a presentation with some ${\rm Spec}\, A_i$, but with etale rather than Zariski transition maps. But this shouldn't matter for your argument since $\mathcal{O}_X$ is also a sheaf for the etale topology. So your argument should prove that $X={\rm Spec}\Gamma(X, \mathcal{O}_X)$, so $X$ is an affine scheme. Am I missing something? $\endgroup$ – Piotr Achinger Apr 6 '18 at 18:12
  • $\begingroup$ I guess you are missing nothing: you are correct, and I am dumb. Maybe one could modify the definition of affine algebraic space requiring the criterion only for those systems whose transition maps are Zariski open embeddings, but it seems a strange definition. $\endgroup$ – Giulio Bresciani Apr 6 '18 at 20:18

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