In this blog post I will define valuations and their specialization. Later on we will concentrate on continuous valuations.

**1.1. Valuations **

Recall that a **totally ordered group** is a commutative group together with a total order such that for all . A homomorphism between two totally ordered groups is a homomorphism of groups such that for all we have .

Whenever we use , we demand additionally that for all and , for all .

Definition 1Let be a commutative ring. Avaluationon is a map where is a totally ordered group and satisfies

- and
- for all
- for all

The subgroup of generated by the image of is called the **value group** of and denoted by . We call the set the **support** of – this is a prime ideal of . Denote by the field of fractions of . We can then define a valuation

by , where are the images of elements . This works also in the other direction. If is a prime ideal and is a valuation on the residue field , then we can compose with the canonical ring homomorphism to obtain a valuation on .

We write for the fraction field of and let be the valuation ring of . We also define the maximal ideal and the residue field of .

Example 2Let be a ring and be a prime ideal of . Then for and for defines a valuation with the value goup . We call this kind of valuations trivial.

We say that two valuations , on are **equivalent** if the following conditions hold

- There is an isomorphism (neccessarily unique) such that .
- and .
- For all one has if and only if .

Definition 3Let be a commutative ring. Thevaluation spectrumof is the set of equivalence classes of valuations on .

Explicitely, consists of pairs where is a prime ideal and is a valuation subring with full fraction field.

We can make the set into a topological space by endowing it with the topology generated by the sets for . Remark that is empty for .

Remark 4Observe that there is a map which sends a valuation to its support . Here is the set of prime ideals of . We claim that this map is continuous with respect the respective topologies on both sets. Indeed, if is a basic open subset then

is open by definition of topology on .

** 1.2. Specializations **

Let . We say that is a **specialization** of if (we also say that is a **generalization** of ). This is equivalent to saying that all open subsets of containing also contain . Looking at rational domains only this is equivalent to saying that whenever , we have also . Let us describe two constructions of specializations.

Let be a valuation with the value group . We say that is a **convex subgroup** of if satisfying for some implies . For any convex subgroup define by

if and otherwise. We call a **vertical generalization** of .

Observe that . Moreover if is a field then . In Proposition 2.14 of [Wed] it is proved that gives a bijection between convex subgroups and vertical generalizations.

The other type of specializations are defined as follows. Let be a subgroup and let be a valuation. We define

by if and otherwise. We call a **horizontal specialization** of if is convex subgroup of containing – the minimal convex subgroup containing . Those conditions are neccessary to ensure that is a valuation and moreover that it is a specialization of . We refer the reader to Proposition 4.3.6 of Conrad’s seminar for details.

The main interest in these two kinds of specializations stems from the following proposition.

The main interest in these two kinds of specializations stems from the following proposition.

Prop 5Let be a ring and let be a valuation on . Let be a specialization of . Then

- is a horizontal specialization of a vertical specialization of , that is, there exists and convex subgroups with such that and .
- is a vertical specialization of such that either is a horizontal specialization of or and is a trivial valuation whose support contains .

*Proof:* See Proposition 4.21 in [Wed].

Bibliography

[Wed] T. Wedhorn, “`Adic spaces”‘, unpublished notes