nLab sheaf of meromorphic functions

Definition

Let $(X, \mathcal{O}_X)$ be a ringed space. Consider the subsheaf of sets $\mathcal{S} \subset \mathcal{O}_X$ of the structure sheaf such that for each open subset $U \subset X$, $\Gamma(U, \mathcal{S})$ consists of only the regular sections of $\mathcal{O}_X$ over $U$, i.e. those elements $s\in\Gamma(U, \mathcal{O}_X)$ for which $s|_V f=0$ implies $f=0$ for all $f\in\Gamma(V,\mathcal{O}_X)$ for all opens $V\subseteq X$. Consider the presheaf of rings on $X$

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

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

Properties

Proposition

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

Proposition

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

References

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

• Stacks Project, sheaf of meromorphic functions and sections, tag 01X1.

• Steven L. Kleiman, Misconceptions about $K_X$, L’Enseignement Mathématique 25 (1979), 203–206 (doi:10.5169/seals-50379)

category: sheaf theory, algebraic geometry