Subsheaf

Subsheaf is a sheaf G on a topological space X together with a monomorphism of sheaves i: G → F into a fixed sheaf F on X. Concretely, for every open set U ⊆ X, G(U) is a subset of F(U) and the restriction maps of G are the restrictions of F; the inclusions are compatible with all restriction maps. The inclusion induces, for each point x ∈ X, a map on stalks G_x → F_x that is injective, so each stalk G_x is a subobject of F_x.

Examples include the subsheaf of continuous functions vanishing at a fixed point x0 inside the sheaf of continuous functions, and the subsheaf of holomorphic functions vanishing on a divisor inside the sheaf of holomorphic functions on a complex manifold. In general, if F is a sheaf of abelian groups (or modules, or rings), a subsheaf is a subobject in that category, with G(U) ⊆ F(U) and the same restriction maps.

The smallest subsheaf containing a given family of sections is the intersection of all subsheaves that include those sections; equivalently, it is the subsheaf generated by them. Intersections of subsheaves are subsheaves, and the union of an ascending chain of subsheaves is a subsheaf. If G is a subsheaf of F, one can form the quotient sheaf F/G with (F/G)(U) = F(U)/G(U).

Subsheaves are fundamental in sheaf theory and appear in contexts such as cohomology, localization, and the study of torsion, support, and other structural properties of sheaves.