Complex Analytic and Differential Geometry 2024, 50: Sheaves

A "presheaf" A on a topological space X consists of 1.) a collection of nonempty sets A(U) associated with all open sets U in X, 2.) A collection of maps ρ: A(V) → A(U) defined whenever U is a subset of V and satisfies transitivity, 3.) ρ(U, V) ⚬ ρ(V, W) = ρ(U, W). It's a "presheaf of abelian groups" if all sets A(U) are abelian groups, and if the maps are morphisms of these structures. A presheaf is a sheaf, if it satisfies the gluing axioms. If A is a preshaf, the set of germs of A at point x is the abstract inductive limit. It's the set of equivalence of classes of elements in the disjoint union of A(U) taken over all open nighborhoods U, with elements of A(U) and A(V) being equivalent iff their maps out of some neighborhood of both are the same.

The union of inductive limits can be equipped with a natural topology

For topological spaces X, S with a local homeomorphism π mapping S onto X, the S is "sheaf-space" on X and π is the projection from S on X. The space created by the inverse of π through the entirety of some x out of X is the "stalk" of S at x. Topological spaces X, Y with a continuous map f between them, a presheaf A on X defines the direct image, which is a preheaf on Y

A ringed space (X, R) consists of a topological space X and a sheaf of rings Rx on X called the "structure sheaf", with a morphism F: (X, Rx) → (Y, Ry) of ringed spaces in a pair (f, F*) with a continuous f: X → Y and F* a homomorphism of sheaves of rings on X. F*: f ⁻¹ Rx → Ry. A locally ringed space has stalks that are all local rings. A morphism of locally ringed spaces is a morphism of ringed spaces such that their homomorphism are subsets of their maximal ideal. Over some field k, all rings are local k-algebras with a residual field equivalent to k. If k is an algebraically closed field of any characteristic, an affine algebraic set is a subset X of the affine space kⁿ defined by an arbitrary collection S of polynomials.

If S is an ideal, then I(V(S)) = √(S).