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, in Hodge Theory and Complex Algebraic Geometry I (Definition 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 in Topologie algébrique et théorie des faisceaux (the paragraph after Theorem II.3.7.2): a sheaf $F$ is fine if the internal hom $Hom(F,F)$ is a soft sheaf.

Related concepts

• sheaf

• abelian sheaf cohomology

  • soft sheaf

  • flabby sheaf

category: sheaf theory