Flat abelian sheaves

Let $X$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). A sheaf $\mathcal{F}$ of abelian groups on $X$ is flat when its stalk ${\mathcal{F}}_x$ is a flat $\mathbb{Z}$-module for every $x\in X$ (Left and right flat modules over an arbitrary ring with $R=\mathbb{Z}$ and additive notation, so that $-{\otimes }_{\mathbb{Z}}{\mathcal{F}}_x$ is exact).

Equivalently, by Stalks, coproducts and right exactness of the abelian sheaf tensor product and Tensor product of abelian sheaves and its total complex, flatness of $\mathcal{F}$ means that the functor $-{\otimes }_{\mathbb{Z}}\mathcal{F}$ on abelian sheaves is exact; the equivalence with exactness of $\mathcal{F}{\otimes }_{\mathbb{Z}}-$ is proved in Flatness criteria and canonical epimorphisms from flat abelian sheaves.

A bounded-above complex ${\mathcal{F}}^{\bullet }$ of abelian sheaves (Bounded, bounded below, and bounded above complexes) is a complex of flat sheaves when each term ${\mathcal{F}}^n$ is flat; the bounded-above complexes of flat sheaves are exactly the source of the flat resolutions constructed on this page. Since a stalk is a filtered colimit (The stalk of a presheaf at a point), flatness is a stalk-local condition, and the zero sheaf is flat because its stalks are the zero modules.