nLab
sheaf of meromorphic functions

Definition

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

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

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

Properties

Proposition

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

Proposition

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

References

• Alexander Grothendieck, Jean Dieudonné, EGA (IV, 20.1).

category: sheaf theory, algebraic geometry