sheaf of meromorphic functions
Let be a ringed space. By definition, for every we have a ring . If is the set of elements that are not zero divisors, we can construct the localization . If is actually an integral domain, then will be its field of fractions. It is easy to verify that the restriction maps of the sheaf yield restriction maps on the rings , so that we can define a presheaf . Let be the sheafification of this presheaf. Then is called the sheaf of meromorphic functions.
If is a connected complex manifold, then has a sheaf of holomorphic functions making it into a ringed space. These rings are always integral domains, and their quotients are all the same, so is a constant sheaf; in fact it always takes the same value, a field . We recognize as precisely the field of meromorphic functions on .
If is a scheme, then has an associated sheaf making it into a ringed space. If is arbitrary, then will simply be a sheaf of rings. If, however, is integral and quasicompact, then the situation is very similar to the situation of complex manifolds; the ring of regular functions on every Zariski open set is an integral domain, and all the restriction maps are injective. As a result, the sheaf of meromorphic functions is again a constant sheaf that always yields the same value, and this value is called the function field of . This function field is an essential object of study in birational geometry.
If is a differential manifold, the differentiable functions on it form a sheaf of rings making into a ringed space. Here the structure of is much more complicated; a function on has an inverse if and only if its support on has empty interior. So globally, this amounts to allowing functions to have poles provided the support of these poles has empty interior. This complicated structure makes the sheaf of meromorphic functions much less useful in the differentiable category than it is for schemes or complex manifolds.
|Title||sheaf of meromorphic functions|
|Date of creation||2013-03-22 13:52:17|
|Last modified on||2013-03-22 13:52:17|
|Last modified by||archibal (4430)|