To continue, I am now going to have to use the language of sheaves. For it, and for all details I will omit here, I refer the reader to Charles Siegel’s post at Rigorous Trivialties and Hartshorne’s Algebraic Geometry. When I talk about sheaf cohomology, it will always be the derived functor cohomology. I will briefly review some of these ideas.
The basic properties of this are as follows.
First, if is a topological space and , then is a covariant additive functor from sheaves on to the category of abelian groups. We have
that is to say, the global sections. Also, if
is a short exact sequence of sheaves, there is a long exact sequence
Finally, sheaf cohomology (except at 0) vanishes on injectives in the category of sheaves.
In other words, sheaf cohomology consists of the derived functors of the (left-exact) global section functor.
When is flasque (i.e., for any open with , restriction is surjective), we have
(This is basically because of a well-known fact about sheaves: if is exact and is flasque, then the sequence of global sections is also exact.)
Moreover, this gives a way to compute the cohomology of a sheaf . If we have a flasque resolution , then consider the complex of the global sections of the sheaves (and ). The key formula is then:
There are many interesting results about the cohomology of a coherent sheaf over schemes, though they are not really relevant to us now. We are interested in the analytic applications.
Cech cohomology gives a reasonable way to actually compute these cohomology groups. For instance, Hartshorne uses it to calculate the cohomology of line bundles on projective space. In general, Cech cohomology does not equal derived functor cohomology, though it does in certain cases (in algebraic geometry, if the scheme is separated, then it is ok).
In the analytic case, we are working with paracompact spaces, so Cech cohomology will always be acceptable, as we will see.
Let be a covering of , and let be a sheaf on . Consider the chain complex
with boundary maps defined as follows. Let with corresponding
for every -tuple . Define
where the hat denotes omission, as usual. (Also, when .) This can be checked to be a complex. Then we define the Cech cohomology groups
as the cohomology of this (co)chain complex.
There is another way to interpret the Cech cohomology groups that better illustrates their connection with regular cohomology. Given , consider the sheaves defined by
The boundary map is defined the same as before, so we have a complex of sheaves
Proposition 1 The above complex is exact.
Exactness at the first step is basically the definition of a sheaf: cocyles in represents collections of elements of that agree on the common intersections, so piece together to a section of .
Now consider the other more general case. We need to check exactness on the stalks, say at . Choose with . Let be such that ; we can assume . We need to lift on some neighborhood containing to get some that goes to via .
Thus, for all -tuples , define
Then is a section of on . In this way, we get a map from , and it is easy to check that it is a chain homotopy, which implies exactness.
Cech cohomology versus derived functor cohomology
I will now discuss some examples of Cech cohomology. The notation remains the same.
Proposition 2 .
This is really just the sheaf axiom; see the beginning of the proof of exactness of the Cech complex of sheaves.
The next proposition gives us some reason to suspect a relation between tehse two types of cohomology.
Proposition 3There is a natural transformation of -functors
be an injective resolution of . Then by a basic result in homological algebra, there is a unique commutative diagram of resolutions
which induces a corresponding commutative diagram on the global sections. Taking the cohomology of the complex then gives the result.
Next up: the Leray theorem.
Incidentally, I’m not sure how I should categorize this post. I’m using “algebraic geometry” because that subject seems to depend the most heavily on sheaves.