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sheaf of meromorphic functions

Context

Topos Theory

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Contents

Definition

Let (X,๐’ช X)(X, \mathcal{O}_X) be a ringed space. Consider the subsheaf of sets ๐’ฎโŠ‚๐’ช X\mathcal{S} \subset \mathcal{O}_X of the structure sheaf such that for each open subset UโŠ‚XU \subset X, ฮ“(U,๐’ฎ)\Gamma(U, \mathcal{S}) consists of only the regular sections of ๐’ช X\mathcal{O}_X over UU, i.e. those elements of ฮ“(U,๐’ช X)\Gamma(U, \mathcal{O}_X) which are not zero divisors. Consider the presheaf of rings on XX

(1)Uโ†ฆฮ“(U,๐’ช X)[ฮ“(U,๐’ฎ) โˆ’1] U \quad\mapsto\quad \Gamma(U, \mathcal{O}_X)[\Gamma(U, \mathcal{S})^{-1}]

which assigns to UU the ring of fractions? of ฮ“(U,๐’ช X)\Gamma(U, \mathcal{O}_X) with denominators in ฮ“(U,๐’ฎ)\Gamma(U, \mathcal{S}); its sheafification โ„ณ X\mathcal{M}_X is called the sheaf of (germs of) meromorphic functions on XX. The sections of โ„ณ X\mathcal{M}_X over XX are called the meromorphic functions on X and we denote this ring M(X)=ฮ“(X,โ„ณ X)M(X) = \Gamma(X, \mathcal{M}_X).

Properties

Proposition

For every open subset UโŠ‚XU \subset X there is a canonical isomorphism between โ„ณ U\mathcal{M}_U and the restriction of โ„ณ X\mathcal{M}_X to UU.

Proposition

For every point xโˆˆXx \in X there is a canonical isomorphism between the stalk โ„ณ X,x\mathcal{M}_{X,x} and ๐’ช X,x[๐’ฎ x โˆ’1]\mathcal{O}_{X,x}[\mathcal{S}_x^{-1}].

References

Revised on July 7, 2014 02:42:43 by David Roberts (82.245.53.167)