Fine sheaf - Imported Knowlpedia Content

Definition

Let $\mathcal F$ be a sheaf of abelian groups on a topological space $X$ . It is fine if, for every locally finite open cover $(U_i)_{i\in I}$ , there are sheaf endomorphisms $h_i:\mathcal F\to\mathcal F$ such that the family $(h_i)$ is locally finite,

\sum_{i\in I}h_i=\operatorname{id}_{\mathcal F},

and the support of $h_i$ is contained in $U_i$ . Here the support is the closure of the set of points where the induced map on the stalk of $\mathcal F$ is nonzero. Thus fineness internalizes a partition of unity at the level of sheaf sections.

Module-sheaf criterion

Let $\mathcal A$ be a sheaf of rings that admits partitions of unity subordinate to locally finite covers. Every sheaf of $\mathcal A$ -modules is fine: if $(\varphi_i)$ is such a partition, multiplication by $\varphi_i$ defines the required endomorphism $h_i$ .

In particular, the sheaf of smooth functions is fine, and every sheaf of modules over it is fine. The endomorphisms arise from multiplication by a smooth partition of unity.

Cohomological consequence

On a paracompact Hausdorff space, every fine sheaf is acyclic for the global-section functor: its higher sheaf cohomology groups vanish. This theorem is the sheaf-theoretic reason partitions of unity make many smooth resolutions compute cohomology. Bott and Tu use this mechanism in their treatment of de Rham theory and fine resolutions Bott and Tu, Chapter I.

Conventions and scope

References

Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer, 1982. DOI record. Relevant: Chapter I, partitions of unity, fine sheaves, and the de Rham resolution.