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 $I$.

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    $\begingroup$ Regarding your last comment: There exists a non-noetherian ring $A$ and an injective $A$-module $E$, such that $\widetilde{E}$ is not a flasque $\mathcal{O}_{\mathrm{Spec}(A)}$-module (cf. SGA VI, Expose 2, Appendice 1, 0.1 or in the Stack Project stacks.math.columbia.edu/tag/0273). Nevertheless, the category $\mathrm{QCoh}(X)$ has enough injective objects for any scheme since it is a Grothendieck abelian category - a result due to Gabber resp. Franke. $\endgroup$ – Tobias Sitte Jun 17 '15 at 7:15
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    $\begingroup$ Could you please specify your assumptions? Do you mean the following? (a) For every $\Gamma(X,\mathcal{O}_X)$-module $M$ the canonical map $M \to \Gamma(X,\tilde{M})$ is an isomorphism, (b) for every quasi-coherent $\mathcal{O}_X$-module $F$ the canonical map $\widetilde{\Gamma(X,F)} \to F$ is an isomorphism, and (c) for every two quasi-coherent $\mathcal{O}_X$-modules $F,G$ the canonical map $\Gamma(X,F) \otimes \Gamma(X,G) \to \Gamma(X,F \otimes G)$ is an isomorphism? $\endgroup$ – Martin Brandenburg Aug 16 '15 at 13:02
  • $\begingroup$ The assumptions are (a) & (b). So the unit and co-unit of the adjunction between $\Gamma(X,-)$ and $\widetilde{}$ are natural isomorphisms. Removed the part about the tensor structure. $\endgroup$ – bbnkttp Aug 16 '15 at 17:09
  • $\begingroup$ - = 'tilde'. My apologizes for the bad formatting. $\endgroup$ – bbnkttp Aug 16 '15 at 17:15

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.


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