Let {X} be a quasi-compact, separated scheme. Then a criterion of Serre asserts that {X} is affine if and only if

\displaystyle H^i(X, \mathcal{F}) =0 , \quad i > 0,

for all quasi-coherent sheaves {\mathcal{F}} on {X}: that is, affine schemes are characterized by the vanishing of the higher cohomology of all quasi-coherent sheaves. The purpose of this post is to explain an interpretation of Serre’s theorem (or rather, the “if” direction) in terms of category theory. Namely, the idea is that if {X} satisfies the cohomological vanishing condition, then the functor

\displaystyle \Gamma: \mathrm{QCoh}(X) \rightarrow \mathrm{Mod}(\Gamma(X,\mathcal{O}_X)),

from the category of quasi-coherent sheaves on {X} to the category of modules over {\Gamma(X, \mathcal{O}_X)}, turns out to be a symmetric monoidal equivalence for formal reasons. A version of Tannakian formalism now shows that {X} is itself isomorphic to {\mathrm{Spec} \Gamma(X, \mathcal{O}_X)}: that is, the category {\mathrm{QCoh}(X)} together with its symmetric monoidal structure recovers {X}.

Edit: I was sure this material was well-known folklore, but didn’t have a reference when I posted this. Now I do; see Knutson’s Algebraic Spaces.

Second edit (11/19): I just realized that the argument below leaves out an important piece: it doesn’t show that the structure sheaf is a generator! I’ll try to fix this soon.

 

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