nLab fine sheaf

Definition

Let $X$ be a paracompact Hausdorff space. A sheaf $F$ of groups over $X$ is fine if for every two disjoint closed subsets $A,B\subset X$, $A\cap B = \emptyset$, there is an endomorphism of the sheaf of groups $F\to F$ which restricts to the identity in a neighborhood of $A$ and to the $0$ endomorphism in a neighborhood of $B$. Every fine sheaf is soft.

A slightly different definition is given in Voisin 2002 (Vol I, Def. 4.35):

A fine sheaf $\mathcal{F}$ over $X$ is a sheaf of $\mathcal{A}$-modules, where $\mathcal{A}$ is a sheaf of rings such that, for every open cover $U_i$ of $X$, there is a partition of unity $1 = \sum f_i$ (where the sum is locally finite) subordinate to this covering.

Another definition is given by Godement 1958 (the paragraph after Theorem II.3.7.2): a sheaf $F$ is fine if the internal hom $Hom(F,F)$ is a soft sheaf.

Properties

• For any fine sheaf $J$ over a paracompact Hausdorff space $M$, the sheaf cohomology groups of positive degree $j$ are all trivial

$$H^j(M,J) = 0.$$

• Any sheaf $S$ of abelian groups admits a resolution by fine sheaves $\{J_i\}$

$$0\to S\to J_0\to J_1\to \cdots$$

These resolutions can be used to compute the sheaf cohomology groups of $S$ in terms of the induced maps of sections

$$d_i^*: \Gamma(M,J_i)\to\Gamma(M,J_{i+1}).$$

as

$$H^q(M,S) \cong (\text{ker}d^*_{q})/(\text{im}d^*_{q-1})$$

(see Section 3 of Gunning (1966).).

Related concepts

• sheaf

• abelian sheaf cohomology

  • soft sheaf

  • flabby sheaf

References

• Roger Godement, Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]

• Robert C. Gunning, Lectures on Riemann Surfaces, Princeton University Press (1966) [pdf]

• Claire Voisin, Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77 (2002/3) [doi:10.1017/CBO9780511615344]

category: sheaf theory