Since we’ve been discussing sheaf cohomology for the last few weeks of the algebraic geometry seminar, and I’m leaving Waterloo soon, I was thinking about possible topics for what will probably be the last seminar talk. I figured that having drudged through all this machinery, it would be nice to look at a cohomological characterisation of affine schemes: namely, the fact that a scheme is affine if and only if all quasi-coherent sheaves of -modules are acyclic, i.e. for . In this post I’ll go over the treatment in [Hartshorne III.3, "Cohomology of a Noetherian Affine Scheme"]. I will probably explain all this stuff more coherently in a video sometime down the road.
This is called Serre’s affineness criterion, and the key to the proof (or at least one direction of it) lies in the fact that if you start with an injective -module , and consider its associated sheaf of -modules (just defined by ), then in fact this is flasque. We saw before that flasque sheaves are acyclic for the global sections functor , so in particular we can use flasque resolutions to compute cohomology (this will be important later).
We also saw that injective sheaves are flasque, so one might be tempted to claim that the “key” we mentioned above is a mere triviality: indeed, why not just observe that (in view of the equivalence of categories) any injective -module will give rise to an injective sheaf, and then finish? The problem with this argument is that the category of -modules is equivalent to the category of quasicoherent sheaves, and not the full category of -modules. So yes, we will always have an injective of the former category, but we would need an injective of the latter category to conclude flasqueness — and in general this does not happen.
The starting point is a theorem of Krull from commutative algebra. The full statement concerns the -adic topology on an -module, and I don’t really know much (nor do I currently have time to read Atiyah-Macdonald) about completions. However, we only really need one containment:
Krull’s Theorem. Let be a Noetherian ring and be an ideal. If are finitely generated -modules, then for any there is such that .
Now, define the following submodule of :
Before proceeding, let us mention a remark about injectives. We said an object of an abelian category was injective if the functor is exact. This (contravariant) functor is always left exact, so the important thing to take away is the following: “ injective” means that if is a submodule and is a morphism, then extends to a morphism .
Surprisingly, the above turns out to be equivalent to the following seemingly weaker condition (Baer’s criterion), namely: if is an ideal of and is a morphism, then extends to a morphism . This equivalence is a basic result from commutative algebra.
This reminds me of a similar thing that came up when trying to formulate the universal property of the Stone-Cech compactification: in some sense the closed unit interval is a “good enough” representative of the class of all compact Hausdorff spaces (this is formalised in the fascinating notion of an injective cogenerator).
Lemma 1. Let be a Noetherian ring, let be an ideal. Then if is an injective -module, then is also an injective -module.
To prove this, we only need to establish Baer’s criterion for , and this is done by observing one can apply Krull’s theorem to the inclusion , pulling back from to , and finally using the natural map to pull back to as required.
Lemma 2. Let be a Noetherian ring, and an injective -module. Then for any , the natural map to the localisation is surjective.
This lemma isn’t very difficult either. If is defined as the annihilator of , then you get some ascending chain of ideals in , but is Noetherian, so , yada yada. Then, letting be the natural map, you take some , write for some and (you can do this by definition of localisation), and define a map by sending (this turns out to be fine since as -modules, and ). Lift to a map by injectivity of , and then let . Then . Magic.
Proposition. If is an injective -module, then is a flasque sheaf of -modules, where .
To establish this, we use Noetherian induction on the support of the sheaf (call it ). The basic idea is, for some open , to choose some and consider some open of the form . Noting that , we can invoke the lemma above, and then the problem reduces to showing is surjective, where . But this follows by induction (put and note this is an injective -module by the lemma, hence , whose support is strictly contained in , is flasque; at this point we win since for all opens ).
Theorem. Let for some Noetherian ring . Then if , for all quasicoherent sheaves on .
To see this, let . Take an injective resolution in the category of -modules, apply the Serre functor to get a flasque resolution of . Applying the global sections functor, we just get back the original resolution, so we’re done.
Theorem (Serre). Let be a Noetherian scheme. Then TFAE:
- is affine.
- for any quasicoherent sheaf on and .
- for any coherent sheaf of ideals on .
We’ve already shown that (1) => (2), and (2) => (3) is easy. (3) => (1) can be proved using the following characterisation of affineness: is affine if and only if there are such that , each set is affine, and is covered by .