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Reference. Algebraic Geometry Rainer Sinn [00C4]

Lecture
Algebraic Geomemetry
Sommersemester 2026, Goettingen
Rainer Sinn

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Overview. Morphisms of Schemes [00AH]

This page collects some notes I took for a small presentation on classes of morphisms of schemes and notions of finiteness. These notes are almost exclusively based on this lecture.

Preliminaries [00CM]

Definition. Morphism of locally ringed spaces [00AO]

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 .

Definition. Morphism of schemes [00BF]

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

Proposition. [00BG]

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

Proof. See 6.15

Definition. Structure morphism [00BI]

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.

Idea. [00BJ]

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.

Terminology. Reasonable class of morphisms [00BN]

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

All isomorphisms are in the class.
The class is closed under composition.
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.
(The class is local on the target)

Remark [00BU]

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

Meta-Theorem. [00CL]

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

Classes of morphisms [00CN]

Definition. open embedding [00BP]

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 .

Definition. Quasicompact morphism [00BV]

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

Remark. [00C6]

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

Definition. Quasi separated [00C9]

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.

Remark. [00CA]

A scheme is quasiseparated if its structure morphism is quasiseparareted.

Terminology. qcqs [00CC]

We call morphisms that are quasicompact and quasiseparated qcqs.

Remark. [00CD]

Most morphisms we encounter in nature are qcqs, since TODO

Definition. Affine morphism [00BT]

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

Definition. Integral morphism [00BS]

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

Definition. Morpisms of (locally) finite type [00BQ]

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.

Definition. Finite morphism [00BR]

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 .

Theorem. [00CK]

Finite means integral and finite type

Examples [00CO]

Example. Morphism of schemes [00CJ]

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

Example. Affine but not integral [00C3]

… or a base change version.

Example. integral but not finite [00BY]

Example. Finite [00CP]

polynomial map of affine line
noether normalization

Example. Locally finite type but not finite type [00C2]

every open embedding (finite type only if qc)

Example. finite type [00C1]

(from earlier)
open embeddings that are qc

Hierachy of notions of finiteness [00BK]