subsheaf of abelian groups

Let β„± be a sheaf of abelian groups over a topological spaceMathworldPlanetmath X. Let 𝒒 be a sheaf over X, such that for every open set UβŠ‚X, 𝒒⁒(U) is a subgroupMathworldPlanetmathPlanetmath of ℱ⁒(U). And further let the on 𝒒 be by those on β„±. Then 𝒒 is a subsheaf of β„±.

Suppose a sheaf of abelian groups β„± is defined as a disjoint union of stalks β„±x over points x∈X, and β„± is topologized in the appropriate manner. In particular, each stalk is an abelian groupMathworldPlanetmath and the group operationsMathworldPlanetmath are continuous. Then a subsheaf 𝒒 is an open subset of β„± such that 𝒒x=π’’βˆ©β„±x is a subgroup of β„±x.

When 𝒒 is a subsheaf of β„±, then β„±x/𝒒x is an abelian group. Considering this to be the stalk over x we have a sheaf which is denoted by β„±/𝒒, with the topology being the quotient topology.


Suppose M is a complex manifoldMathworldPlanetmath. Let M* be the sheaf of germs of meromorphic functions which are not identically zero. That is, for z∈M, the stalk Mz* is the abelian group of germs of meromorphic functions at z with the group operation being multiplication. Then O*, the sheaf of germs of holomorphic functionsMathworldPlanetmath which are not identically 0 is a subsheaf of M*.

The sheaf M*/O* is then the sheaf of divisorsMathworldPlanetmathPlanetmathPlanetmath. If M is of (complex) dimensionMathworldPlanetmath 1, then M*/O* is just the sheaf of functions into the integers with finite support.


  • 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