# subsheaf of abelian groups

Let $\mathrm{\beta \x84\pm}$ be a sheaf of abelian groups over a topological space^{} $X$. Let $\mathrm{\pi \x9d\x92\u2019}$ be a sheaf
over $X$, such that for every open set $U\beta \x8a\x82X$, $\mathrm{\pi \x9d\x92\u2019}\beta \x81\u2019(U)$ is a subgroup^{} of
$\mathrm{\beta \x84\pm}\beta \x81\u2019(U)$. And further let the on $\mathrm{\pi \x9d\x92\u2019}$ be by those on $\mathrm{\beta \x84\pm}$.
Then $\mathrm{\pi \x9d\x92\u2019}$ is a *subsheaf* of $\mathrm{\beta \x84\pm}$.

Suppose a sheaf of abelian groups $\mathrm{\beta \x84\pm}$ is defined as a disjoint union of stalks ${\mathrm{\beta \x84\pm}}_{x}$ over points $x\beta \x88\x88X$, and $\mathrm{\beta \x84\pm}$ is topologized in the appropriate manner.
In particular, each stalk is an abelian group^{} and the group operations^{} are continuous.
Then a subsheaf $\mathrm{\pi \x9d\x92\u2019}$ is an open subset of $\mathrm{\beta \x84\pm}$ such that ${\mathrm{\pi \x9d\x92\u2019}}_{x}=\mathrm{\pi \x9d\x92\u2019}\beta \x88\copyright {\mathrm{\beta \x84\pm}}_{x}$ is a subgroup of ${\mathrm{\beta \x84\pm}}_{x}$.

When $\mathrm{\pi \x9d\x92\u2019}$ is a subsheaf of $\mathrm{\beta \x84\pm}$, then ${\mathrm{\beta \x84\pm}}_{x}/{\mathrm{\pi \x9d\x92\u2019}}_{x}$ is an abelian group. Considering this to be the stalk over $x$ we have a sheaf which is denoted by $\mathrm{\beta \x84\pm}/\mathrm{\pi \x9d\x92\u2019}$, 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 $z\mathrm{\beta \x88\x88}M\mathrm{,}$ 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 ${\mathrm{M}}^{\mathrm{*}}$.

## References

- 1 Glen E. Bredon. , Springer, 1997.
- 2 Robin Hartshorne. , Springer, 1977.
- 3 Lars HΓΆrmander. , 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 |