I.1: Valuations and their specializations

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 {\Gamma} is a commutative group together with a total order {\leq} such that {\gamma \leq \gamma ' \Rightarrow \gamma \delta \leq \gamma ' \delta} for all {\gamma, \gamma ', \delta \in \Gamma}. A homomorphism between two totally ordered groups {f: \Gamma \rightarrow \Gamma '} is a homomorphism of groups such that for all {\gamma, \gamma ' \in \Gamma} we have {\gamma \leq \gamma ' \Rightarrow f(\gamma) \leq f(\gamma ')}.

Whenever we use {\Gamma \cup \{0\}}, we demand additionally that {0 \leq \gamma} for all {\gamma \in \Gamma} and {0 \cdot 0 = 0}, {\gamma \cdot 0 = 0} for all {\gamma \in \Gamma}.

Definition 1 Let {A} be a commutative ring. A valuation on {A} is a map {|\cdot|: A \rightarrow \Gamma \cup \{0\}} where {\Gamma} is a totally ordered group and {|\cdot|} satisfies

  1. {|0| = 0} and {|1|=1}
  2. {|ab| = |a||b|} for all {a,b \in A}
  3. {|a+b| \leq \max (|a|,|b|)} for all {a,b \in A}

The subgroup of {\Gamma} generated by the image of {|\cdot| \{0\}} is called the value group of {|\cdot |} and denoted by {\Gamma _{|\cdot|}}. We call the set {\mathrm{supp}(|\cdot|) := |\cdot|^{-1}(0)} the support of {|\cdot|} – this is a prime ideal of {A}. Denote by {K} the field of fractions of {A / \mathrm{supp}(|\cdot|)}. We can then define a valuation

\displaystyle \widetilde{|\cdot|}: K \rightarrow \Gamma \cup \{0\}

by {\bar{a}/\bar{b} \mapsto |a||b|^{-1}}, where {\bar{a},\bar{b} \in A / \mathrm{supp}(|\cdot|)} are the images of elements {a,b \in A}. This works also in the other direction. If {\mathfrak{p} \subset A} is a prime ideal and {\widetilde{|\cdot|}} is a valuation on the residue field {\kappa(\mathfrak{p})}, then we can compose {\widetilde{|\cdot|}} with the canonical ring homomorphism {A \rightarrow \kappa(\mathfrak{p})} to obtain a valuation {|\cdot|} on {A}.

We write {K(|\cdot|)} for the fraction field of {A / \mathrm{supp}(|\cdot|)} and let {A(|\cdot|) = \{x\in K(|\cdot|): \widetilde{|x|} \leq 1 \}} be the valuation ring of {|\cdot|}. We also define the maximal ideal {\mathfrak{m}(|\cdot|) = \{x\in K(|\cdot|): \widetilde{|x|} < 1\}} and the residue field {\kappa(|\cdot|) = A(|\cdot|) / \mathfrak{m}(|\cdot|)} of {|\cdot|}.

Example 2 Let {A} be a ring and {\mathfrak{p}} be a prime ideal of {A}. Then {a \mapsto 1} for {a \notin \mathfrak{p}} and {a\mapsto 0} for {a\in \mathfrak{p}} defines a valuation with the value goup {1}. We call this kind of valuations trivial.

We say that two valuations {|\cdot|}, {|\cdot |'} on {A} are equivalent if the following conditions hold

  1. There is an isomorphism {f: \Gamma _{|\cdot|} \simeq \Gamma _{|\cdot|'}} (neccessarily unique) such that {f \circ |\cdot| = |\cdot |'}.
  2. {\mathrm{supp}(|\cdot|) = \mathrm{supp}(|\cdot|')} and {A(|\cdot|) = A(|\cdot|')}.
  3. For all {a,b\in A} one has {|a|\leq |b|} if and only if {|a|' \leq |b|'}.

Definition 3 Let {A} be a commutative ring. The valuation spectrum {\mathrm{Spv}(A)} of {A} is the set of equivalence classes of valuations on {A}.

Explicitely, {\mathrm{Spv}(A)} consists of pairs {(\mathfrak{p},R)} where {\mathfrak{p} \subset A} is a prime ideal and {R \subset \kappa(\mathfrak{p})} is a valuation subring with full fraction field.

We can make the set {\mathrm{Spv}(A)} into a topological space by endowing it with the topology generated by the sets {U(f/g) = \{|\cdot|\in \mathrm{Spv}(A): |f|\leq|g| \not = 0\}} for {f,g \in A}. Remark that {U(f/g)} is empty for {g = 0}.

Remark 4 Observe that there is a map {\phi: \mathrm{Spv}(A) \rightarrow \mathrm{Spec}(A)} which sends a valuation {|\cdot|} to its support {\mathrm{supp}(|\cdot|)}. Here {\mathrm{Spec}(A)} is the set of prime ideals of {A}. We claim that this map is continuous with respect the respective topologies on both sets. Indeed, if {D(b) \subset \mathrm{Spec}(A)} is a basic open subset then

\displaystyle \phi ^{-1}(D(b))=\{|\cdot| \in \mathrm{Spv}(A): |b| \not = 0\} = \{|\cdot| \in \mathrm{Spv}(A): |0| \leq |b| \not = 0\} = U(0/b)

is open by definition of topology on {\mathrm{Spv}(A)}.

1.2. Specializations

Let {|\cdot|, |\cdot|' \in \mathrm{Spv}(A)}. We say that {|\cdot|} is a specialization of {|\cdot|'} if {|\cdot| \in \overline{\{|\cdot|'\}}}   (we also say that {|\cdot|'} is a generalization of {|\cdot|}). This is equivalent to saying that all open subsets of {\mathrm{Spv}(A)} containing {|\cdot|} also contain {|\cdot|'}. Looking at rational domains {U(f/g)} only this is equivalent to saying that whenever {|f|\leq |g| \not = 0}, we have also {|f|' \leq |g|' \not = 0}. Let us describe two constructions of specializations.

Let {|\cdot|: A \rightarrow \Gamma \cup \{0\}} be a valuation with the value group {\Gamma _{|\cdot|}}. We say that {H \subset \Gamma} is a convex subgroup of {\Gamma} if {\gamma \in \Gamma } satisfying {h \leq \gamma \leq h'} for some {h,h' \in H} implies {\gamma \in H}. For any convex subgroup {H \subset \Gamma} define {|\cdot|_{/H}: A \rightarrow (\Gamma/H) \cup \{0\}} by

\displaystyle |a|_{/H} := |a| \mod H

if {|a| \not = 0} and {0} otherwise. We call {|\cdot|_{/H}} a vertical generalization of {|\cdot|}.

Observe that {\mathrm{supp}(|\cdot|) = \mathrm{supp}(|\cdot|_{/H})}. Moreover if {A} is a field then {A(|\cdot|_{/H}) \subset A(|\cdot|)}. In Proposition 2.14 of [Wed] it is proved that {H \mapsto |\cdot|_{/H}} gives a bijection between convex subgroups and vertical generalizations.

The other type of specializations are defined as follows. Let {H \subset \Gamma} be a subgroup and let {|\cdot|: A \rightarrow \Gamma \cup \{0\}} be a valuation. We define

\displaystyle |\cdot|_{|H}: A \rightarrow \Gamma \cup \{0\}

by {|a|_{|H} = |a|} if {|a| \in H} and {0} otherwise. We call {|\cdot|_{|H}} a horizontal specialization of {|\cdot|} if {H} is convex subgroup of {\Gamma _{|\cdot|}} containing {\Gamma _{|\cdot|}^c} – the minimal convex subgroup containing {|A|\cap \Gamma _{\geq 1}}. Those conditions are neccessary to ensure that {|\cdot|_{|H}} is a valuation and moreover that it is a specialization of {|\cdot|}. 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 5 Let {A} be a ring and let {v} be a valuation on {A}. Let {w} be a specialization of {v}. Then

  1. {w} is a horizontal specialization of a vertical specialization of {v}, that is, there exists {u \in \mathrm{Spv} A} and convex subgroups {H,G \subset \Gamma _{u}} with {\Gamma ^c _u \subset G} such that {w = u _{|G}} and {v = u / H}.

  2. {w} is a vertical specialization of {u \in \mathrm{Spv} A} such that either {u} is a horizontal specialization of {v} or {\Gamma ^c _u = 1} and {u} is a trivial valuation whose support contains {\mathrm{supp} (v _{|1})}.

Proof: See Proposition 4.21 in [Wed]. \Box

Bibliography

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

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