subsheaf of abelian groups

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

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

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

Example.

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

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

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