PlanetMath: subsheaf of abelian groups

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subsheaf of abelian groups

(Definition)

Let $\mathcal{F}$ be a sheaf of abelian groups over a topological space . Let $\mathcal{G}$ be a sheaf over , such that for every open set $U \subset X$ , $\mathcal{G}(U)$ is a subgroup of $\mathcal{F}(U)$ . And further let the restriction morphisms on $\mathcal{G}$ be induced 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 we have a sheaf which is denoted by $\mathcal{F}/\mathcal{G}$ , with the topology being the quotient topology.

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

Bibliography

1: Glen E. Bredon. Sheaf Theory, Springer, 1997.
2: Robin Hartshorne. Algebraic Geometry, Springer, 1977.
3: Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.

"subsheaf of abelian groups" is owned by jirka.

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Other names:

subsheaf, subsheaves

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Cross-references: finite support, integers, dimension, complex, divisors, holomorphic functions, multiplication, functions, meromorphic, germs, complex manifold, quotient topology, continuous, group operations, points, stalks, disjoint union, subgroup, open set, topological space, abelian groups, sheaf
There are 2 references to this entry.

This is version 2 of subsheaf of abelian groups, born on 2007-12-03, modified 2007-12-04.
Object id is 10089, canonical name is Subsheaf.
Accessed 228 times total.

Classification:

AMS MSC:	18F20 (Category theory; homological algebra :: Categories and geometry :: Presheaves and sheaves)
	54B40 (General topology :: Basic constructions :: Presheaves and sheaves)
	14F05 (Algebraic geometry :: homology theory :: Vector bundles, sheaves, related constructions)

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