Valuation

logarithmic norm, norm on a field

A mapping $ v : K \rightarrow \Gamma _ \infty $ from a field $ K $ into $ \Gamma _ \infty = \Gamma \cup \{ \infty \} $, where $ \Gamma $ is a totally ordered Abelian group, the adjoined element $ \infty $ is assumed to be larger than any element of $ \Gamma $, and $ x + \infty = \infty + x = \infty $ for all $ x \in \Gamma $. Here the valuation must satisfy the following conditions:

1) $ v ( 0) = \infty $, $ v ( x) < \infty $ for $ x \neq 0 $;

2) $ v ( a \cdot b ) = v ( a) + v ( b) $;

3) $ v ( a - b ) \geq \min ( v ( a) , v ( b) ) $.

The image of $ K ^ {*} = K \setminus \{ 0 \} $ under $ v $ is a subgroup of $ \Gamma $, called the value group of the valuation $ v $. Throughout what follows it is assumed that $ v ( K ^ {*} ) = \Gamma $.

By the same axioms one defines logarithmic valuations of rings. Every ring with a non-Archimedean norm (cf. Norm on a field) can be made into a logarithmically-valued ring if one passes in the groupoid of values from the multiplicative to the additive notation and reverses the order. The element 0 is then naturally denoted by the symbol $ \infty $. The reverse transition from a ring with a logarithmic valuation to one with a non-Archimedean norm is also possible. If in a ring a non-Archimedean real norm is given, then the corresponding transition can be obtained by replacing each positive real number $ \alpha $ by $ - \mathop{\rm log} \alpha $. The resulting logarithmic valuation is also called real.

Two valuations $ v _ {1} : K \rightarrow \Gamma _ \infty ^ {1} $ and $ v _ {2} : K \rightarrow \Gamma _ \infty ^ {2} $ are said to be equivalent if there is an isomorphism $ \phi : \Gamma ^ {1} \rightarrow \Gamma ^ {2} $ of ordered groups such that for all non-zero elements $ x \in K $,

$$ v _ {2} ( x) = \phi ( v _ {1} ( x) ) . $$

The set of those elements $ x $ of $ K $ for which $ v ( x) \geq 0 $ is a subring $ A $ of $ K $, called the valuation ring of $ v $ in $ K $. It is always a local ring. The elements $ x $ of $ K $ for which $ v ( x) > 0 $ form a maximal ideal $ m _ {v} $ of $ A $, called the valuation ideal of $ v $. The quotient ring $ A / m _ {v} $, which is a field, is called the residue field of the valuation $ v $.

Let $ v _ {1} $ and $ v _ {2} $ be two valuations on a field $ K $. The rings of these valuations, regarded as subrings of $ K $, are the same if and only if these valuations are equivalent. Thus, knowing all valuations of a field $ K $( up to equivalence) is the same as knowing all subrings that occur as valuation rings for this field. A subring $ A $ of $ K $ is a valuation ring for $ K $ if and only if for every non-zero element $ x \in K $ at least one of $ x $ and $ x ^ {-} 1 $ belongs to $ A $. Thus, a valuation ring can be defined abstractly as an integral ring (integral domain) that satisfies this condition relative to its field of fractions. Every such ring is the ring of the so-called canonical valuation for its field of fractions, for which the value group is $ K ^ {*} / U $, where $ U $ is the multiplicative group of invertible elements of $ A $, and $ K ^ {*} / U $ is ordered by divisibility.

A valuation ring can be defined in yet another way. If $ A \subseteq B \subseteq K $ are two local rings with maximal ideals $ m $ and $ n $, respectively, then one says that $ B $ dominates $ A $ if $ m \subset n $. Dominance is a partial order relation on the set of subrings of $ K $. The maximal elements of this set are exactly the valuation rings of $ K $. If $ A $ is a valuation ring and $ B \supset A $ is a ring with the same field of fractions as $ A $, then $ B $ is also a valuation ring and is the localization of $ A $ with respect to some prime ideal.

Examples of valuations.

1) The valuation defined by

$$ v ( x) = \ \left \{ is called improper or trivial. Any valuation of a finite field is trivial. 2) Let $ k $ be a field and let $ K = k ( ( t ) ) $ be the field of Laurent series over $ k $. Associating to a series $ a _ {n} t ^ {n} + a _ {n+} 1 t ^ {n+} 1 + \dots $, where $ a _ {n} \neq 0 $, its order $ n $( and $ \infty $ to the null series) is a valuation with value group $ \mathbf Z $( the additive group of integers) and valuation ring $ k [ [ t ] ] $. A valuation with values in $ \mathbf Z $ is called discrete; about their valuation rings see [[Discretely-normed ring|Discretely-normed ring]]. For a description of all valuations of the field of rational numbers, see [[#References|[4]]]. For each totally ordered Abelian group $ \Gamma $ there is a valuation of a certain field with value group $ \Gamma $. =='"`UNIQ--h-1--QINU`"'Ideals in valuation rings.== The set of ideals in a valuation ring is totally ordered by inclusion; every ideal of finite type is principal, that is, a valuation ring is a [[Bezout ring|Bezout ring]]. A more complete description of the structure of ideals in a valuation ring can be given in terms of the value group of the valuation. A subset $ M $ of a totally ordered set is said to be major if $ x \in M $ and $ y > x $ imply $ y \in M $. Let $ A $ be the ring of a valuation $ v $ of a field $ k $ with value group $ \Gamma $, let $ \Gamma ^ {+} $ be the sub-semi-group of positive elements of $ \Gamma $, and let $ M $ be a major set in $ \Gamma ^ {+} $. The mapping $ M \mapsto \alpha ( M) = \{ {x } : {x \in K, v ( x) ; M \cup \{ \infty \} } \} $ is a bijection between the set of major subsets of $ \Gamma ^ {+} $ and the set of ideals of $ A $. Principal ideals correspond to majors having a minimal element. Prime ideals also correspond to majors of special form, namely: $ M _ {H} = \Gamma ^ {+} \setminus H ^ {+} $, where $ H ^ {+} $ is the positive part of a [[Convex subgroup|convex subgroup]] $ H $ of $ \Gamma $. Thus, there is a one-to-one correspondence between the prime ideals of $ A $ and the convex subgroups of the value group $ \Gamma $. Let $ p $ be the prime ideal corresponding to a convex subgroup $ H $. Then the composite mapping $ K \rightarrow ^ {v} \Gamma \rightarrow \Gamma / H $ is a valuation of $ K $ with valuation ring $ A _ {p} $ and valuation ideal $ p A _ {p} $; moreover, on the field $ A _ {p} / p A _ {p} $ there is an induced valuation with values in $ H $ and valuation ring $ A / p $. In this manner a valuation splits into simpler ones. Let $ A $ be a valuation ring. Then the prime spectrum of $ A $ without the zero ( $ \mathop{\rm Spec} A \setminus ( 0) $) is a totally ordered set, and its type is called the height or rank of the corresponding valuation. If $ \mathop{\rm Spec} A $ is finite, then the height of the valuation is the number of elements in $ \mathop{\rm Spec} A \setminus ( 0) $, and this is the same as the number of proper convex subgroups of $ \Gamma $. A valuation of finite rank can be reduced to valuations of rank 1. The latter are characterized by the fact that their value groups are Archimedean (cf. [[Archimedean group|Archimedean group]]), that is, they are isomorphic to a subgroup of the additive group $ \mathbf R $ of real numbers. In this case the mapping $ x \mapsto \mathop{\rm exp} ( - v ( x) ) $ is an [[ultrametric]] norm on $ K $. An important property of valuation rings is that they are integrally closed. Moreover, for an arbitrary integral ring $ A $ its integral closure is equal to the intersection of all valuation rings containing $ A $. A valuation ring is totally integrally closed if and only if its valuation is real, that is, has rank 1. A valuation ring is Noetherian if and only if the valuation is discrete. =='"`UNIQ--h-2--QINU`"'Valuations and topologies.== Let $ v : K \rightarrow \Gamma _ \infty $ be a valuation on a field $ K $ and let $ V _ \gamma = \{ {x } : {x \in K, v ( x) > \gamma } \} $, where $ \gamma \in \Gamma $. The collection of all $ V _ \gamma $, $ \gamma \in \Gamma $, is a fundamental system of neighbourhoods of zero for a topology $ \tau _ {v} $ of $ K $, which is said to be the topology determined by the valuation $ v $. It is separable and disconnected. The topology induced by $ \tau _ {v} $ on $ A $ is, as a rule, different from that of a local ring. For a non-trivial valuation of $ K $ the topology $ \tau _ {v} $ is locally compact if and only if $ v $ is discrete, the valuation ring is complete, and the residue field of $ v $ is finite; $ A $ is then compact. The completion $ \widehat{K} $ of $ K $ relative to $ \tau _ {v} $ is a field; $ v $ can be extended by continuity to a valuation $ \widehat{v} : \widehat{K} \rightarrow \Gamma _ \infty $, and the topology of $ \widehat{K} $ is the same as $ \tau _ {\widehat{v} } $. The valuation ring $ \widehat{A} $ of $ \widehat{v} $ is the completion of the valuation ring $ A $ of $ v $. Two valuations $ v _ {1} $ and $ v _ {2} $ of $ K $ are called independent if the topologies $ \tau _ {v _ {1} } $ and $ \tau _ {v _ {2} } $ are distinct; this is equivalent to the fact that the valuation rings $ A _ {v _ {1} } $ and $ A _ {v _ {2} } $ generate $ K $. Inequivalent valuations of height 1 are always independent. There is an approximation theorem for valuations: Let $ v _ {i} : K \rightarrow \Gamma _ \infty ^ {i} $, $ 1 \leq i \leq n $, be independent valuations, let $ a _ {i} \in K $, $ \alpha _ {i} \in \Gamma ^ {i} $. Then there is an element $ x $ in $ K $ such that $ v _ {i} ( x - a _ {i} ) \geq \alpha _ {i} $ for all $ i $. =='"`UNIQ--h-3--QINU`"'Extension of valuations.== If $ v : L \rightarrow \Gamma _ \infty ^ \prime $ is a valuation of $ L $ and $ K $ is a subfield of $ L $, then the restriction $ v = v ^ \prime \mid _ {K} $ of $ v ^ \prime $ to $ K $ is a valuation of $ K $, and its value group $ \Gamma $ is a subgroup of $ \Gamma ^ \prime $; $ v ^ \prime $ is then called an extension of $ v $. Conversely, if $ v $ is a valuation on $ K $ and $ L $ is an extension of $ K $, then there is always a valuation of $ L $ that extends $ v $. The index $ [ \Gamma ^ \prime : \Gamma ] $ of $ \Gamma $ in $ \Gamma ^ \prime $ is called the ramification index of $ v ^ \prime $ with respect to $ v $ and is denoted by $ e ( v ^ \prime / v ) $. The residue field $ k _ {v} $ of $ v $ can be identified with a subfield of the residue field $ k _ {v ^ \prime } $ and the degree $ [ k _ {v ^ \prime } : k _ {v} ] $ of the extension is denoted by $ f ( v ^ \prime / v ) $ and is called the residue degree of $ v ^ \prime $ relative to $ v $. An extension $ v ^ \prime $ of a valuation $ v $ is said to be immediate if $ e ( v ^ \prime / v ) = f ( v ^ \prime / v ) = 1 $. Let $ L $ be an extension of $ K $ and let $ \{ {v _ {i} } : {i \in I } \} $ be the set of all extensions of $ v $ to $ L $. If $ L $ is a finite extension of $ K $ of degree $ n $, then the set of all extensions of $ v $ is finite, and $$ \sum _ {i \in I } e ( v _ {i} / v ) f ( v _ {i} / v ) \leq n . $$ In several cases equality holds, for example when $ v $ is discrete and either $ K $ is complete or $ L $ is separable over $ K $. If $ L $ is a normal extension of $ K $, then the extensions of $ v $ to $ L $ are permuted transitively by the $ K $- automorphisms of $ L $; in particular, if $ L $ is a [[purely inseparable extension]] of $ K $, then $ v $ has only one extension. In the case of an arbitrary extension $ K \subset L $ and an extension $ v ^ \prime $ of a valuation $ v $, the transcendence degree of $ L $ over $ K $ is greater than or equal to the sum $$ S ( {v ^ \prime } / v ) + r ( {v ^ \prime } / v ) , $$

where $ S ( v ^ \prime / v ) $ is the transcendence degree of the extension of the residue field of $ v ^ \prime $ over that of $ v $ and $ r ( v ^ \prime / v ) $ is the dimension of the space $ ( \Gamma _ {v ^ \prime } / \Gamma _ {v} ) \otimes \mathbf Q $.

The concept of a valuation was introduced and studied by W. Krull in [1]. It is also widely used in algebraic geometry. Thus, in terms of "valuation rings" one can construct the abstract Riemann surface of a field (cf. [3]).

References

[1]	W. Krull, "Allgemeine Bewertungstheorie" J. Reine Angew. Math. , 167 (1932) pp. 160–196 Zbl 0004.09802 Zbl 58.0148.02
[2]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) MR0360549 Zbl 0279.13001
[3]	O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975) MR0389876 MR0384768 Zbl 0313.13001
[4]	A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) MR0158000 Zbl 0121.25901

References

[a1]	O. Endler, "Valuation theory" , Springer (1972) MR0357379 Zbl 0257.12111