Determining a scheme $X$ is affine from $Qcoh(X)$ My question is a subquestion of this question. And a repost from this MSE question.
The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the stacks project is actually an adjoint equivalence. My goal is to proof that $X$ is affine. 
This is a special case of the general question: can we reconstruct a scheme from its category of quasi-coherent sheafs? I am aware of the available reconstruction theorems, but I feel that this could be done with elementary methods. Feel free to critique this feeling.
For starters, quasi-separatedness of $X$ should be established before you can invoke Gabriel-Rosenberg and quasi-compactness before you can use Serre's vanishing theorem on $X$. Any hints on how to go about this?
What I tried thus far is trying to use Serre's vanishing theorem. The problem: how to prove that there exist 'enough' acyclic resolutions inside $Qcoh(X)$ as is the case for $Qcoh(\mbox{Spec } \Gamma(X,O_X))$, since $\widetilde{I}$ is flasque for injective $\Gamma(X,O_X)$-modules $I$.
 A: First prove that the map $X \to \operatorname{Spec} \Gamma( X, \mathcal O_X)$ is a bijection on points, then that it is a homeomorphism, then that the structure sheaf is equivalent.
Given two points $p_1, p_2$ in the same fiber, let $k$ be a field which contains both $\mathcal O_X/p_1$ and $\mathcal O_x/p_2$. Then unless $p_1=p_2$ you can give it two different structures of an $\mathcal O_X$-module and hence a coherent sheaf, but the pushforward of the two is the same. So $p_1$, $p_2$ are actually the same.
Given a closed set $Z \subseteq X$, we must show it arises by pullback from a closed subset of $X \to \operatorname{Spec} \Gamma( X, \mathcal O_X)$. Well $\mathcal O_Z$ arises by pullback from some sheaf on $\operatorname{Spec} \Gamma( X, \mathcal O_X)$. Moreover as $\mathcal O_Z$ is a quotient of $\mathcal O_X$, so that sheaf must be a quotient of the structure sheaf by an ideal. A point is contained in $Z$ if and only if the map to the structure sheaf of that point factors through $\mathcal O_Z$ if and only if the point is in the vanishing set of this ideal. So $Z$ is the pullback of the vanishing set of this ideal, which is closed.
Finally, the fact that the pushforward of $\mathcal O_X$ is the structure sheaf of $\operatorname{Spec} \Gamma( X, \mathcal O_X)$ implies that the map is an isomorphism on structure sheaves. 
