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.

An open embedding is a morphism that factors as

where the first map is an isomorphism and the second is the inclusion of the open set into (as a ringed space). In this situation, we often write .

A morphism of schemes is quasicompact if for every affine open subset , the preimage is quaiscompact.

A scheme is quasicompact as a topological space if its structure morphism is quasicompact.

A morphisms of schemes is quasiseperated if for every open subset , the preimage is a quasiseparated scheme, i.e the intersection of any two quasicompact open sets is quasicompact.

A scheme is quasiseparated if its structure morphism is quasiseparareted.

We call morphisms that are quasicompact and quasiseparated qcqs.

Most morphisms we encounter in nature are qcqs, since TODO

A morphism of schemes is called affine if for every affine open subset , the open subscheme is an affine scheme.

A morphism of schemes is integral if it is affine and for every affine open set with , the ring is an inegral extension of via

A morphism is locally of finite type if for every affine open subset and every affine open subset , the induces morphism (induced by makes a finitely generated -algebra. A morphism of locally finite type is of finite type if it also quasicompact.

A morphism of schemes is finite if it is affine and for every affine open , the open subscheme is the afine scheme of a finite -algebra .

Finite means integral and finite type

Let ,

The map between topological spaces is the constant map (to the zero ideal). It is trivially continous. The structure sheaf of at is and at empty sets its .

For the morphism of sheaves , give on by restricted to or if is empty.

TODO show its local

… or a base change version.

• polynomial map of affine line
• noether normalization

every open embedding (finite type only if qc)

• (from earlier)
• open embeddings that are qc