I'm currently trying to understand the sort of ad hoc way of computing sheaf cohomology through acyclic resolutions, and related ideas. I understand that if you want to compute the sheaf cohomology of a sheaf $\mathcal{F}$, you can find a resolution by acyclic objects $\mathcal{C}^{i}$, with an exact sequence
$0 \to \mathcal{F} \to \mathcal{C}^{0} \to \mathcal{C}^{1} \to \ldots$
We can then define the sheaf cohomology of $\mathcal{F}$ to be the cohomology of the complex of global sections of the acyclic objects:
$H^{q}(X, \mathcal{F}) = H^{q}(\Gamma(\mathcal{C}^{*}))$
(I) So I'm curious what the point is in defining more restrictive notions of acyclicity like fine, soft, or flasque sheaves? In particular, I'm curious about fine sheaves. How exactly does the existence of the partition of unity property help us to compute sheaf cohomology? Perhaps the idea is that when you have more restrictive definitions, you have more structure available to exhibit the existence of these objects?
(II) I think I understand the proofs that both flasque and fine sheaves are individually also soft and acyclic. However, how do flasque and fine sheaves compare? Are they just similarly refined, yet unrelated notions which are helpful in different circumstances?
(III) Finally, I was hoping for a bit of intuition about fine sheaves themselves. Is it perhaps a helpful intuitive crutch to think of a fine sheaf as a $C^{\infty}(X)$-module? I think these are examples of fine sheaves, but perhaps the partition of unity definition is most intuitive when you imagine being able to multiply sections by bump functions on the space.
Thanks in advance for any help!
Best Answer
1) The only useful result about fine sheaves in differential geometry is that a sheaf of $C^{\infty}_X$-modules is acyclic.
There are many such $C^{\infty}_X$-modules, the best known are the locally free sheaves, and in particular the sheaves $\Omega^k_X $ of differential forms on a differential manifold $X$.
The fact that the complex $\Omega^\ast _X$ is an acyclic resolution of the constant sheaf $\underline {\mathbb R}$ immediately yields De Rham's theorem, as brilliantly observed by André Weil.
That said, I'm trying to forget the boring definition(s) of fine sheaf, which I have never seen used in practice: only its consequence mentioned above, the acyclicity of $C^{\infty}_X$-modules, is ever applied.
2) a) Flasque (=flabby) sheaves are acyclic and are a powerful theoretical tool in the abstract cohomology theory of sheaves, but the only natural flasque sheaves arising in differential geometry seem to be sky-scraper sheaves: they have thus two reasons to be acyclic, since they are also $C^{\infty}_X$-modules :-)
Beware that constant sheaves on manifolds are never flasque (except in completely trivial situations).
b) On the other hand flasque sheaves are more common in algebraic geometry: constant sheaves are flasque on an algebraic variety $X$ , a useful example being the constant sheaf of rational functions $\mathcal {Rat}_X=\underline {K(X)}$.