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sheaf of meromorphic functions

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Definition

Let (X,𝒪 X)(X, \mathcal{O}_X) be a ringed space. Consider the subsheaf of sets 𝒮𝒪 X\mathcal{S} \subset \mathcal{O}_X of the structure sheaf such that for each open subset UXU \subset X, Γ(U,𝒮)\Gamma(U, \mathcal{S}) consists of only the regular sections of 𝒪 X\mathcal{O}_X over UU, i.e. those elements of Γ(U,𝒪 X)\Gamma(U, \mathcal{O}_X) which are not zero divisors. Consider the presheaf of rings on XX

(1)UΓ(U,𝒪 X)[Γ(U,𝒮) 1] U \quad\mapsto\quad \Gamma(U, \mathcal{O}_X)[\Gamma(U, \mathcal{S})^{-1}]

which assigns to UU the ring of fractions? of Γ(U,𝒪 X)\Gamma(U, \mathcal{O}_X) with denominators in Γ(U,𝒮)\Gamma(U, \mathcal{S}); its sheafification X\mathcal{M}_X is called the sheaf of (germs of) meromorphic functions on XX. The sections of X\mathcal{M}_X over XX are called the meromorphic functions on X and we denote this ring M(X)=Γ(X, X)M(X) = \Gamma(X, \mathcal{M}_X).

Properties

Proposition

For every open subset UXU \subset X there is a canonical isomorphism between U\mathcal{M}_U and the restriction of X\mathcal{M}_X to UU.

Proposition

For every point xXx \in X there is a canonical isomorphism between the stalk X,x\mathcal{M}_{X,x} and 𝒪 X,x[𝒮 x 1]\mathcal{O}_{X,x}[\mathcal{S}_x^{-1}].

References

Revised on March 6, 2013 19:46:49 by Zoran Škoda (161.53.130.104)