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