# Fine sheaf

A sheaf of Abelian groups $\mathcal F$ over a paracompact space $X$ with a soft sheaf as sheaf of endomorphisms. A sheaf $\mathcal F$ is fine if and only if for any closed subsets $A,B\subset X$ with $A\cap B=\emptyset$ there is an endomorphism $h\colon\mathcal F\to\mathcal F$ that is the identity on $A$ and zero on $B$, or equivalently if for every open covering $(U_i)_{i\in I}$ of $X$ there is a locally finite collection $(h_i)_{i\in I}$ of endomorphisms of $\mathcal F$ such that $\supp h_i\subset U_i$ $(i\in I)$ and $\sum_{i\in I}h_i$ is the identity endomorphism. Every fine sheaf is soft, and if $\mathcal F$ is a sheaf of rings with an identity, the converse also holds. If $\mathcal F$ is a fine sheaf and $\mathcal L$ is an arbitrary sheaf of Abelian groups, then $\mathcal F\otimes_\mathbf Z\mathcal L$ is also a fine sheaf. An example of a fine sheaf is the sheaf of germs of continuous (or differentiable of class $C^k$) sections of a vector bundle over a paracompact space (respectively, over a paracompact differentiable manifold).