Ringed Space - Morphisms

Morphisms

A morphism from (X, O_X) to (Y, O_Y) is a pair (f, φ), where f: X → Y is a continuous map between the underlying topological spaces, and φ: O_Y → f_*O_X is a morphism from the structure sheaf of Y to the direct image of the structure sheaf of X. In other words, a morphism from (X, O_X) to (Y, O_Y) is given by the following data:

• a continuous map f : X → Y
• a family of ring homomorphisms φ_V : O_Y(V) → O_X(f -1(V)) for every open set V of Y which commute with the restriction maps. That is, if V₁ ⊂ V₂ are two open subsets of Y, then the following diagram must commute (the vertical maps are the restriction homomorphisms):

There is an additional requirement for morphisms between locally ringed spaces:

• the ring homomorphisms induced by φ between the stalks of Y and the stalks of X must be local homomorphisms, i.e. for every x ∈ X the maximal ideal of the local ring (stalk) at f(x) ∈ Y is mapped to the maximal ideal of the local ring at x ∈ X.

Two morphisms can be composed to form a new morphism, and we obtain the category of ringed spaces and the category of locally ringed spaces. Isomorphisms in these categories are defined as usual.