If you use sheaves to study differential geometry, one of the basic lemmas you’ll want is the following: Let be a smooth manifold and let be a sheaf of modules over . (For example, might be the sheaf of sections of a vector bundle.) Then all higher sheaf cohomology of vanishes.
The proof of this theorem is basically homological algebra plus the existence of partitions of unity. This gives rise to a slogan “when you have partitions of unity, sheaf cohomology vanishes.” One way to make this definition precise is through the technology of fine sheaves.
As Wikipedia says today, “[f]ine sheaves are usually only used over paracompact Hausdorff spaces”. That means they are not used when working with the Zariski topology on schemes, for example. When I started digging into this, I realized there were good reasons: The technology of fine sheaves (and the closely related technology of soft sheaves) does not include the scheme theory cases which we would want it to.
However, there are theorems of the form “when you have partitions of unity, sheaf cohomology vanishes” on schemes and on complex manifolds. I put up a question at MathOverflow asking whether there were better formulations that included these examples, but I probably didn’t formulate it well. I think spelling out all my issues would be too discursive for MathOverflow, so I’m bringing it over here.
What does it mean to vanish?
Let me start with a technical point that caused me a great deal of confusion. Let be a topological space, an open subset of , and be a point in . Let be a sheaf of abelian groups on and a section in . In what sense could we say that vanishes at ?
In this generality, there is only one reasonable definition: That the image of in the stalk is zero. Unpacking the definition of the stalk, this means that there is an open set , with , such that .
Now, think about the case where is a sheaf of functions on , with restriction meaning honest-to-God-restriction of functions. The above definition is not what we mean when we say vanishes at ! Rather, it is the concept we would express as “ vanishes on a neighborhood of .”
In order to get a concept which generalizes the ordinary meaning of vanishing at a point, we need to restrict to the case where is a locally ringed space, and a sheaf of -modules. In that case, is a module over the local ring . And the image of in is the best analogue to “the value of at “, where is the residue field of .
Therefore, in this blog post, I make the following definitions:
With the above notation, I say that vanishes on a neighborhood of if the image of in is zero or, equivalently, if there is an open set such that .
I say that vanishes at if the image of in is zero.
Let be a closed subset of . We have the following, analogous definitions:
The function vanishes on a neighborhood of if either of the equivalent definitions holds: (1) There is an open set containing , such that is zero or (2) For every , the function vanishes in a neighborhood of .
The function vanishes on if, for every , the function vanishes in at .
If more books had adopted this terminology, I would have spent far less time confused about exactly what they meant when they claimed some space had partitions of unity.
Partitions of unity implies vanishing sheaf cohomology, the standard version
With these definitions out of the way, we can show that the existence of partitions of unity implies vanishing sheaf cohomology.
Theorem 1: Let be a locally ringed space, and assume that is paracompact. (Every cover has a locally finite
subcover refinement.) Suppose that, for every open cover of , there are global functions so that vanishes in a neighborhood of and . Let be any sheaf of modules. Then for all .
Proof Sketch: Our proof is by induction on . Let be an injective sheaf with an injection ; and let be the cokernel of . For , the long exact sequence gives , the right hand side of which is zero by induction. So we simply must establise the base case, that .
We know that is exact, so it is enough to show is surjective. Let be a global section of . Since is surjective, there is an open cover of , and functions such that .
Take as in the hypothesis. For each index , let be an open set containing such that vanishes. Let be the section of which is on and is on . Such a section exists by the gluing axiom, applied to the open cover . Let . (By paracompactness, we may assume that the cover is locally finite, so the sum makes sense.) We claim that .
It is enough to check this claim on stalks. Near any point , we have . By construction, this maps to . QED
I would feel guilty if I never defined a fine sheaf in this post. The idea of fine sheaves is that, rather than starting with the sheaf of rings , we can start with the sheaf and define . The sheaf is called fine if , defined in this manner, has partitions of unity in the above sense. Of course, may not be commutative and the stalks of may not be local, but it turns out that we can still prove Theorem 1 in this setting: A fine sheaf on a paracompact space has no cohomology. Unfortunately, in the examples I discuss below, the extra elements of still don’t create partitions of unity.
The Regularity Trick
The above proof asks for to vanish on a neighborhood of . When is nice enough, we can ask that just vanish on .
Theorem 2 Let be a paracompact regular topological space and be a locally ringed space. Suppose that, for any open cover , there are global functions such that vanishes on and . Then, for any open cover , there are global functions so that vanishes in a neighborhood of and .
Proof sketch: Take your open cover . For every point in , let contain . Choose disjoint open sets and such that and . Find functions such that and vanishes on . Then will vanish on and hence on a neighborhood of . QED
Because of the above argument, mathematicians who work on metrizable spaces don’t worry very much about the distinction between vanishing on a closed set and in the neighborhood of a closed set. But the Zariski topology is not metrizable…
The Zariski world: Cause for caution!
Let us begin, right away, by pointing out that there are affine schemes and sheaves of modules on them which have nontrivial cohomology.
Let be the scheme . Let be the following sheaf: If then is the local ring , otherwise . The obvious map is a surjection of sheaves (exercise!), yet the map on global sections is not surjective. So, if is the kernel of , then .
So any theorem about sheaf cohomology vanishing on affine spaces needs to be phrased carefully.
The Zariski world: Cause for hope!
Affine schemes, with the Zariski topology, do not have partitions of unity in the sense of Theorem 1. Indeed, there do not exist two polynomials on the affine line adding to , the first of which is zero on a neighborhood of and the other on a neighborhood of . (Since any polynomial which is zero in a neighborhood of a point must be identically zero.)
Nonetheless, we have the following theorem, originally due to Serre:
Theorem 3: (Hartshorne III.3.5, EGA III.1.3.1) Coherent sheaves on an affine scheme have no cohomology.*
Proof Sketch: As before, we reduce to the case of showing that, if is a surjective map of coherent sheaves, then is surjective.
Let be a global section of . Let be a basic open cover of . The adjective basic means that for some global function . Let be a preimage of . There is some such that extends to a section of . (Exercise! This is the point that the coherence hypothesis is used.)
Since the are a cover; the functions have no common zero, and the functions also have none. So, by the Nullstellansatz, there are global functions such that . So we can find
global sections extending and, as in the proof of Theorem 1, is a preimage of . QED
This proof used two important facts. In order to avoid the language of basic opens, I’ll phrase them in terms of ideal sheaves; the reader might enjoy rephrasing the above proof in this language.
Fact 1: Let be an open set in , let be a coherent sheaf and a section in . Let be the ideal of such that extends to . Then .
Fact 2: If , , …, is a collection of coherent ideal sheaves on such that then .
Fact 2, to my mind, is a good generalization of the existence of partitions of unity. It is weaker than asking for partitions of unity in the sense of vanishing on neighborhoods of closed sets, but stronger than just asking for partitions of unity in the sense of vanishing on closed sets. I had hoped that the correct generalization of “partitions of unity implies sheaf cohomology vanishing” would be “Facts 1 and 2 imply vanishing of cohomology for coherent sheaves”. But, when I started reading about complex manifolds, I realized this was not the way to go.
The Stein World: Cause for puzzlement!
A Stein space is a closed complex-submanifold of .
They are the analogues of (smooth) affine schemes for complex analysis. We can talk about the sheaf of holomorphic functions on any Stein space.
Stein spaces have Fact 2; this is a consequence of Rückert’s Nullstellansatz. I am willing to consider this a good generalization of the existence of partitions of unity. Of course, Stein Spaces don’t have partitions of unity in the sense of Theorem 1 for the same reason polynomials don’t: An analytic function that vanishes on a neighborhood of a point must be identically .
Stein spaces also have Theorem 3. This is called Cartan’s Theorem B.
But Stein spaces don’t have Fact 1! Consider , let . Consider the section of . There is no holomorphic function such that is holomorphic! So we can’t use Fact 2 to prove Theorem 3.
I assumed that there was some minor trick which was used to get around this. But I just read through the proof of Cartan’s Theorem B in Grauert and Remmert’s Theory of Stein Spaces and it looks nothing like the proofs of Theorems 1 and 3.
This is where I run out of ideas. But I know we have readers who think about sheaves and homological methods on a much deeper level than I do. So, what is the version of “Partitions of unity imply cohomology vanishing” which works for Stein Spaces?
* Two footnotes for experts: Yes, this also holds for quasi-coherent sheaves. I stated the weaker version because I want to make the analogy to Stein spaces, and I’m not sure if the corresponding result is true for quasi-coherent on Stein spaces. Second, I am implicitly assuming noetherianness, in order to make sure my sums are finite. But the theorem is true without this.