subsheaf of abelian groups

Let $\mathcal{F}$ be a sheaf of abelian groups over a topological space $X$ . Let $\mathcal{G}$ be a sheaf over $X$ , such that for every open set $U\subset X$ , $\mathcal{G}(U)$ is a subgroup of $\mathcal{F}(U)$ . And further let the on $\mathcal{G}$ be by those on $\mathcal{F}$ . Then $\mathcal{G}$ is a subsheaf of $\mathcal{F}$ .

Suppose a sheaf of abelian groups $\mathcal{F}$ is defined as a disjoint union of stalks $\mathcal{F}_{x}$ over points $x\in X$ , and $\mathcal{F}$ is topologized in the appropriate manner. In particular, each stalk is an abelian group and the group operations are continuous. Then a subsheaf $\mathcal{G}$ is an open subset of $\mathcal{F}$ such that $\mathcal{G}_{x}=\mathcal{G}\cap\mathcal{F}_{x}$ is a subgroup of $\mathcal{F}_{x}$ .

When $\mathcal{G}$ is a subsheaf of $\mathcal{F}$ , then $\mathcal{F}_{x}/\mathcal{G}_{x}$ is an abelian group. Considering this to be the stalk over $x$ we have a sheaf which is denoted by $\mathcal{F}/\mathcal{G}$ , with the topology being the quotient topology.

Example.

Suppose $M$ is a complex manifold. Let $\mathcal{M}^{*}$ be the sheaf of germs of meromorphic functions which are not identically zero. That is, for $z\in M,$ the stalk $\mathcal{M}^{*}_{z}$ is the abelian group of germs of meromorphic functions at $z$ with the group operation being multiplication. Then $\mathcal{O}^{*}$ , the sheaf of germs of holomorphic functions which are not identically 0 is a subsheaf of $\mathcal{M}^{*}$ .

The sheaf $\mathcal{M}^{*}/\mathcal{O}^{*}$ is then the sheaf of divisors. If $M$ is of (complex) dimension 1, then $\mathcal{M}^{*}/\mathcal{O}^{*}$ is just the sheaf of functions into the integers with finite support.

References

1 Glen E. Bredon. , Springer, 1997.
2 Robin Hartshorne. , Springer, 1977.
3 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.

Title	subsheaf of abelian groups
Canonical name	SubsheafOfAbelianGroups
Date of creation	2013-03-22 17:39:21
Last modified on	2013-03-22 17:39:21
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	5
Author	jirka (4157)
Entry type	Definition
Classification	msc 14F05
Classification	msc 54B40
Classification	msc 18F20
Synonym	subsheaf
Synonym	subsheaves