sheaf of meromorphic functions

Let (X,𝒪X) be a ringed space. By definition, for every U we have a ring 𝒪X(U). If SU is the set of elements that are not zero divisorsMathworldPlanetmath, we can construct the localizationMathworldPlanetmath KU=SU-1𝒪X(U). If 𝒪X(U) is actually an integral domainMathworldPlanetmath, then K will be its field of fractionsMathworldPlanetmath. It is easy to verify that the restriction maps of the sheaf 𝒪X(U) yield restriction maps on the rings KU, so that we can define a presheafPlanetmathPlanetmath UKU. Let 𝒦X be the sheafificationPlanetmathPlanetmath of this presheaf. Then 𝒦X is called the sheaf of meromorphic functions.

If X is a connected complex manifoldMathworldPlanetmath, then X has a sheaf 𝒪X of holomorphic functionsMathworldPlanetmath making it into a ringed space. These rings are always integral domains, and their quotients are all the same, so 𝒦X is a constant sheaf; in fact it always takes the same value, a field K. We recognize K as precisely the field of meromorphic functions on X.

If X is a scheme, then X has an associated sheaf 𝒪X making it into a ringed space. If X is arbitrary, then 𝒦X will simply be a sheaf of rings. If, however, X 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 X. This function field is an essential object of study in birational geometry.

If X is not reduced, its structure sheaf contains nilpotentsPlanetmathPlanetmath. Thus 𝒦 is not a sheaf of fields, even locally. Such schemes arise when discussing infinitesimal deformations.

If X is reduced but not irreduciblePlanetmathPlanetmath, then each irreducible componentPlanetmathPlanetmath (if it is quasicompact) has a function field, and 𝒦X(X) is in fact the direct sum of these function fields.

If X is a differential manifold, the differentiable functions on it form a sheaf of rings making X into a ringed space. Here the structure of 𝒦X is much more complicated; a function on U has an inversePlanetmathPlanetmath if and only if its supportPlanetmathPlanetmathPlanetmath on U 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
Canonical name SheafOfMeromorphicFunctions
Date of creation 2013-03-22 13:52:17
Last modified on 2013-03-22 13:52:17
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 6
Author archibal (4430)
Entry type Definition
Classification msc 14A99
Defines meromorphic