A morphism of locally ringed spaces is a morphism of ringed spaces , such that the induces map on stalks is a local homomorphism of local rings. I.e the unique maximal ideal is sent to the unique maximal ideal.

Recall that is a continous map, and is a morphism of sheaves, where is a sheaf on by .

A morphism of schemes is a morphism of locally ringed spaces between schemes.

Let be a ring, let be a scheme. Then the morphisms are in bijection with ring homomorphisms to global sections .

Proof. See 6.15

The structure morphism of a scheme is the unique morphism of schemes . This is well-defined since is the final object in the category of schemes.

A scheme should have property if its structure morphism has property . This is part of the philosophy, that we should understand objects of a category by their interaction with other objects via morphisms. This principle will not hold for all properties of schemes that we define.

We say a class of morphisms of schemes is reasonable if it fullfills the following properties.

0. All isomorphisms are in the class.
1. The class is closed under composition.
2. The class is closed under base change: if is in the class, and is in the class, then the induced map is also in the class.
3. (The class is local on the target)

Property 0. and 1. ensure that the schemes with only a particular class of morphisms still forms a category.

All classes we will see today are reasonable. Proof. Almost the entire chapter on classes of morphisms.