Subsheaves

Subsheaves are a fundamental concept in sheaf theory, describing subobjects of a given sheaf on a topological space. If F is a sheaf of sets, groups, rings, or modules on a space X, a subsheaf G of F consists of a sheaf G equipped with a monomorphism i: G → F such that for every open set U ⊆ X, G(U) is a subset of F(U) and the restriction maps of G are the restrictions coming from F. In other words, G is a subobject of F in the category of sheaves.

Equivalently, G can be viewed as a subpresheaf of F that also satisfies the sheaf condition. For each open U, a section s ∈ F(U) lies in G(U) if and only if its restrictions to a covering {Ui} of U lie in G(Ui) and the local data glue to a section of G. Stalkwise, for every x ∈ X, the germ Gx is a subobject of Fx, so a subsheaf is determined by its stalks subject to the sheaf gluing condition.

Examples illustrate the concept. On a complex manifold X, the sheaf OX of holomorphic functions has subsheaves such as the subsheaf of holomorphic functions vanishing on a fixed closed analytic subset Y. The subsheaf of continuous functions vanishing on a closed subset Z is a subsheaf of the sheaf of continuous functions. More generally, a sheaf of modules F over a ringed space has subsheaves obtained by taking submodules on each open, stable under restriction.

Properties and constructions include: intersections of arbitrary families of subsheaves yield subsheaves; kernels of morphisms of sheaves are subsheaves of the domain, and images are subsheaves of the codomain. If a subpresheaf is not a sheaf, one can sheafify it to obtain the smallest subsheaf containing it. Unions of an ascending chain of subsheaves remain subsheaves, while arbitrary unions need directedness to preserve the subsheaf property.