Thanks to Peter Xu, Linus Hamann, Jack Sempliner, Irit Huq-Kuruvilla for helpful (if somewhat orthogonal) discussion related to the subject material, and to Sean Reynolds for showing me how to make the (rather complicated, at least to me) commutative diagram that shows up in this post.
Oh, and Anja Schulz gets a shout out as well!
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When we discuss the construction of a meromorphic function on a Riemann surface with a given principal part at a given point , we have to find holomorphic functions on such that
on . One can formulate this in a general setting by introducing the concept of sheaf cohomology. Suppose is a topological space and ia sheaf of abelian groups over . Let
be a covering of by open subsets. For every nonnegative integer , we define as follows. An element of is
where is a continuous section of over and is skew-symmetric in . We call the group of alternating -cochains for the covering with coefficients in the sheaf . An element of is an alternating -cochain for the covering with coefficients in the sheaf or simply a -cochain when there is no confusion. For notational convenience, we define to be the zero group when .
We define a map
as follows. The image of
on and means that the index is omitted. We call the map
the coboundary map. Denote by the kernel of
and denote by the image of
The group (respectively ) is respectively called the group of alternating -cocycles (respectively -coboundaries) for the covering with coefficients in the sheaf .
The composite of
is zero, because if the image of the element
Hence, is contained in . Denote by the quotient . The group is called the cohomology group of dimension for the covering with coefficients in the sheaf . It is clear from the definition that is simply the set of all global continuous sections of the sheaf over and is usually denoted also by .
In our construction of a meromorphic function on a Riemann surface with a given principal part at a given point , the collection is a -cocycle with coefficients in the sheaf of germs of holomorphic functions on , and the existence of is equivalent to being a -cboundary. In this formulation, the obstruction to the solution of the problem is the cohomology group of dimension . When we try to piece together local meromorphic functions to form a global meromorphic function, we want to get rid of the discrepancies of the local meromorphic functions, and it would serve the same purpose if we can get rid of the discrepancies by going to a refinement of the covering. The problem is solved if the cohomology group of dimension vanishes when one goes to a refinement of the covering. This suggests that one should take all coverings of the space and consider the direct limit of the cohomology groups for all the coverings.
In the general case when we have a refinement of the covering , we have
We define as the direct limit of as runs through the directed set of all open coverings of . The group is called the cohomology group of dimension of with coefficients in the sheaf . We denote respectively by , , and the direct limits of , , and . Then we have an induced coboundary map
whose kernel is and whose image is . Moreover, is the quotient of by .
Among all cohomology groups of positive dimension, the most important cohomology group is the one of dimension . Its vanishing enables one to piece together local continuous sections of a sheaf to form a global continuous section. Why are the cohomology groups of higher dimensions introduced? They are introduced the help us compute the cohomology group of dimension , because when we have a short exact sequence of three sheaves, we have a long exact sequece of cohomology groups. We are going to discuss this long exact sequence of cohomology groups.
is an exact sequence of sheaves and sheaf-homomorphisms. For every , we have the short exact sequence
This is clear except the surjectivity of . Suppose we have an element
of . We have to show that its restriction to some refinement of is the image of some element . First, let us make a trivial observation. Suppose we have a continuous section of over an open subset of . We give a metric . For every point in , there exists a maximum positive number
such that the ball of radius centered at is contained in , and for every , the restriction of to is the image of some continuous section over . The function is clearly a lower semi-continuous function of so that if
for all in some open neighborhood of . We can assume, without loss of generality, that the covering is locally finite. We choose relatively compact in so that
still covers . For every point in , we let be the minimum of for all containing . Clearly, every admits an open neighborhood on which the function has a positive lower bound. Now, for every point in , let be an open metric ball centered at of radius contained in for some
so that the function
on . Let
Then is a refinement of . Suppose have a common point . We can choose a number such that
for . Let be the metric ball centered at whose radius is . Then contains for , and is contained in . Moreover, be the restriction of to . We skew-symmetrize with respect ot . Then the restriction of to is the image of the element of .
From the commutative diagram with exact rows
we get a long exact sequence
The only map in the long exact sequence that needs some explanation is the so-called connecting homomorphism . It is defined as follows. Take an element of . We can find an element of whose image under is . Let be the image of under the coboundary map . From the above commutative diagram, it follows that is the image of some element under of the element of defined by . The exactness of the sequence is a consequence of straightforward diagram chasing.