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.

 

1. The Morita theorem

Let {\mathcal{A}} be an abelian category. How can we tell if {\mathcal{A}} is equivalent to the category {\mathrm{Mod}(R)} of (right) modules over a ring {R}? The key property of {\mathrm{Mod}(R)}, which distinguishes it among general abelian categories, is the following:

Observation: {\mathrm{Mod}(R)} has a compact, projective generator (namely, {R} itself).

The compactness condition states that

\displaystyle \hom(R, \cdot)

commutes with filtered colimits. Together with projectivity, this can be phrased as saying that {\hom(R, \cdot)} commutes with all colimits from {\mathrm{Mod}(R)} to {\mathbf{Ab}}. Being a generator is the condition that {\hom(R, \cdot)} is a conservative functor. Stated equivalently, in this situation: a module vanishes if and only if {\hom(R, \cdot) = 0}.

The point of Morita theory is that this observation actually characterizes abelian categories which are equivalent to {\mathrm{Mod}(R)} for some {R}.

Theorem 1 Let {\mathcal{A}} be a cocomplete abelian category with a compact projective generator {P}. Then {\mathcal{A}} is equivalent to the category of left modules over the ring {\mathrm{End}_{\mathcal{A}}(P, P)^{}}.

The above result encodes the essential data for determining when two categories {\mathrm{Mod}(R)} and {\mathrm{Mod}(R')} are equivalent: this happens precisely when there is a compact projective generator of {\mathrm{Mod}(R')} with endomorphism ring equal to {R}.

Proof: This can be proved by “bare hands,” but another approach is to use the Barr-Beck theorem (which deserves a post of its own). I’d like to sketch this approach (which I learned from Lurie’s “Higher Algebra”) because it generalizes well to the derived setting. The idea is to show that {\mathcal{A}} is monadic over {\mathbf{Ab}} using the Barr-Beck theorem: that is, {\mathcal{A}} can be identified with the category of “modules” over an appropriate monad {T: \mathbf{Ab} \rightarrow \mathbf{Ab}}.

Now, one way to get a monad {\mathbf{Ab} \rightarrow \mathbf{Ab}} is to take an associative algebra {R}, and take {R \otimes \cdot}. This is a monad, and in fact, one has the following:

Observation: Associative rings are the same thing as colimit-preserving monads on the category of abelian groups.

So the strategy of proof is to exhibit {\mathcal{A}} as monadic over {\mathbf{Ab}}, via a colimit-preserving monad. This will complete the proof (and we can identify the ring, too).

Namely, we have a functor

\displaystyle G: \mathcal{A} \rightarrow \mathbf{Ab}, \quad X \mapsto \hom_{\mathcal{A}}(P, X),

which commutes with all limits. This functor has a left adjoint, given by

\displaystyle F: \mathbf{Ab} \rightarrow \mathcal{A}, \quad G(A) = A \otimes P,

where {A \mapsto A \otimes P} is the unique functor which preserves colimits in {A} and such that {\mathbb{Z} \otimes P = P}. In other words, to compute {A \otimes P} for finitely generated {A}, one takes a presentation of {A} by free {\mathbb{Z}}-modules {\mathbb{Z}^m \rightarrow \mathbb{Z}^n \rightarrow A \rightarrow 0}, and defines

\displaystyle A \otimes P = \mathrm{coker}( A^m \rightarrow A^n).

As before, the goal is to make {\mathcal{A}} monadic over {\mathbf{Ab}}, for a colimit-preserving monad on {\mathbf{Ab}}. We need to appeal to the Barr-Beck theorem to do this. The Barr-Beck theorem states that if we have a right adjoint functor like {G} which is conservative and which preserves split coequalizers, then {\mathcal{A}} is the category of “modules” for a certain monad (namely, {GF}) on {\mathbf{Ab}}. But {G} is conservative because {P} is a generator, and {G} actually commutes with all colimits because {P} is compact and projective.

So we find that the Barr-Beck theorem applies, and we get

\displaystyle \mathcal{A} = \mathrm{Mod}(GF),

where {GF} is a certain monad on {\mathbf{Ab}}. Since {G} preserves all colimits, it is a colimit-preserving monad, and we actually find that {\mathcal{A}} is thus the category of modules over a certain ring.

Can we identify this ring? That corresponds to identifying the monad {GF}: in other words, to computing the law of composition in {GF(\mathbb{Z})}. But this is {G ( P) = \hom_{\mathcal{A}}(P, P)}. \Box

2. Serre’s criterion

So let’s suppose {X} is a quasi-compact, separated scheme. We will study {X} using its category {\mathrm{QCoh}(X)} of quasi-coherent sheaves. This is an abelian category, admitting all colimits. (In fact, it is a presentable category.) Our first goal is to show that if {X} satisfies the hypotheses of Serre’s criterion, then {\mathrm{QCoh}(X)} has a compact projective generator.

Proposition 2 Suppose {X} is a quasi-compact, separated scheme such that {H^i(X, \mathcal{F}) =0} for all {i > 0} and quasi-coherent sheaves {\mathcal{F} \in \mathrm{QCoh}(X)}. Then the sheaf {\mathcal{O}_X} is a compact projective generator of {\mathrm{QCoh}(X)}.

The first observation is that {\hom(\mathcal{O}_X, \cdot)}, or equivalently the global section functor {\Gamma}, commutes with filtered colimits. This is a consequence of quasi-compactness and quasi-separatedness. That is, quasi-compactness and separatedness allows us to write {\Gamma(X, \mathcal{F}) } for any {\mathcal{F} \in \mathrm{QCoh}(X)} as a finite equalizer of {\mathcal{F}} evaluated at affine open sets. For {U \subset X} affine open, the functor {\Gamma(U, \cdot)} commutes with all colimits in {\mathrm{QCoh}(X)}. This allows us to conclude that {\Gamma(X, \cdot)} does too.

The more interesting, and relevant, observation is that {\Gamma()} is actually an exact functor on {\mathrm{QCoh}(X)}. This is a consequence of the long exact sequence in cohomology and the vanishing hypothesis on {X}.

Together, these imply that {\mathcal{O}_X} is a compact projective generator of {\mathrm{QCoh}(X)}, and the Morita principle implies an equivalence of categories

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

given by the global section functor {\Gamma}.

From this, we would like to conclude that the scheme {X} itself is equivalent to {\mathrm{Spec} \Gamma(X, \mathcal{O}_X)}. We are not quite there yet. For instance, there are many rings with the same module category as {\mathrm{Mod}( \Gamma(X, \mathcal{O}_X))}; this is the phenomenon of Morita equivalence. Granted, this turns out not to be a problem because one is only interested here in commutative rings, and I have heard in fact under many conditions one can reconstruct a scheme from {\mathrm{QCoh}(X)}; however, I don’t know what the conditions are.

Edit: Below, John points to the Gabriel-Rosenberg theorem. In this case, it implies that we can conclude the desired direction of Serre’s criterion from here itself.

But anyway, it will be necessary to have a little more. The point is that while {\mathrm{QCoh}(X)}, as an abstract category, does not recover {X}, the symmetric monoidal category {\mathrm{QCoh}(X)} contains enough information to recover {X}. This is the “Tannakian” philosophy. The functor {\Gamma} above is a lax symmetric monoidal functor. In this case, one can check that the natural map

\displaystyle \Gamma(X, \mathcal{F}) \otimes_{\Gamma(X, \mathcal{O}_X)} \Gamma(X, \mathcal{G}) \rightarrow \Gamma(X, \mathcal{F} \otimes_X \mathcal{G})

is actually an equivalencefor all {\mathcal{F}, \mathcal{G}}. In fact, if we fix {\mathcal{F}}, the set of {\mathcal{G}} for which this is true is stable under colimits and contains {\mathcal{O}_X}, so it contains everything.

In other words, we have a symmetric monoidal equivalence between the symmetric monoidal category {\mathrm{QCoh}(X)} of quasi-coherent sheaves on {X}, and the category of modules over {\mathrm{Mod}( \Gamma(X, \mathcal{O}_X))}.

Serre’s theorem is now a consequence of the following (see this paper).

Theorem 3 (Brandenburg and Chirvasitu) Let {X} be quasi-compact and quasi-separated, and let {Y} be a scheme. Then we have an equivalence of categories

\displaystyle \hom(Y, X) \simeq \mathrm{Fun}^{\otimes}(\mathrm{QCoh}(X), \mathrm{QCoh}(Y))

between the category (set) of maps {X \rightarrow Y} and the category of colimit-preserving tensor functors {\mathrm{QCoh}(Y) \rightarrow \mathrm{QCoh}(X)}.

In fact, in view of this, we find that the above symmetric monoidal equivalence between {\mathrm{QCoh}(X)} and {\mathrm{Mod}(\Gamma(X, \mathcal{O}_X)} actually comes from an isomorphism

\displaystyle X \simeq \mathrm{Spec} \Gamma(X, \mathcal{O}_X),

which is what we wanted.

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