# 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 SubsheafOfAbelianGroups 2013-03-22 17:39:21 2013-03-22 17:39:21 jirka (4157) jirka (4157) 5 jirka (4157) Definition msc 14F05 msc 54B40 msc 18F20 subsheaf subsheaves