Sheaves and locally ringed spaces

A presheaf assigns an object to every open set and a restriction map to every inclusion, with identity and composition laws. For a sheaf, sections that agree on every pairwise overlap glue to exactly one section on the union. Existence and uniqueness are separate requirements: agreement prevents contradictory local instructions, while uniqueness prevents hidden global differences invisible on the cover. The empty open has the terminal object, which is the zero ring for sheaves of rings.

The condition concerns an actual open covering and restrictions to its overlaps. It does not say that any arbitrary list of local functions glues, nor that a section on a small open extends to the entire space. A rational expression may be regular on one chart while having a pole elsewhere. For an abelian sheaf, compatibility can be written as the kernel of a difference map; for rings, the more intrinsic formulation is an equalizer of two restriction maps.

The stalk at a point consists of germs of sections defined near that point. Two representatives give the same germ when they agree on some smaller common neighborhood; equality on their original entire domains is unnecessary. Algebraically this is a filtered colimit over neighborhoods. Stalks are effective for checking equality and exactness of sheaf maps, but one cannot replace a sheaf by a bare list of stalks without the information explaining how germs vary and extend locally.

A locally ringed space is a ringed space whose stalk at every point is a local ring. Its unique maximal ideal consists of the germs that are nonunits, and the quotient is the residue field at that point. Residue fields need not equal a fixed ground field: the generic point of an integral affine scheme has its function field. This is why evaluating only at familiar rational points does not recover the full geometric object.

A ringed-space morphism includes a continuous map f and a sheaf map from the target's functions to the direct image of the source's functions. It induces a ring homomorphism from the target stalk at f(x) to the source stalk at x. For a morphism of locally ringed spaces this homomorphism must be local: the inverse image of the source maximal ideal equals the target maximal ideal. This ensures that a germ vanishing at the image pulls back to a germ vanishing at the source point.

A unital homomorphism between local rings need not be local. For example, the inclusion of the local ring of integers at the prime three into the rationals sends the nonunit three to a unit. It cannot be a stalk map at the closed point with the proposed image. The associated scheme morphism from the spectrum of the rationals instead lands at the generic point, where the appropriate target stalk is already a field. Keeping the point and stalk map together resolves the apparent discrepancy.