Valuation
logarithmic norm, norm on a field
A mapping 
from a field 
into 
, where 
is a totally ordered Abelian group, the adjoined element 
is assumed to be larger than any element of 
, and 
for all 
. Here the valuation must satisfy the following conditions:
1) 
, 
for 
;
2) 
;
3) 
.
The image of 
under 
is a subgroup of 
, called the value group of the valuation 
. Throughout what follows it is assumed that 
.
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 
. 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 
by 
. The resulting logarithmic valuation is also called real.
Two valuations 
and 
are said to be equivalent if there is an isomorphism 
of ordered groups such that for all non-zero elements 
,
 |
The set of those elements 
of 
for which 
is a subring 
of 
, called the valuation ring of 
in 
. It is always a local ring. The elements 
of 
for which 
form a maximal ideal 
of 
, called the valuation ideal of 
. The quotient ring 
, which is a field, is called the residue field of the valuation 
.
Let 
and 
be two valuations on a field 
. The rings of these valuations, regarded as subrings of 
, are the same if and only if these valuations are equivalent. Thus, knowing all valuations of a field 
(up to equivalence) is the same as knowing all subrings that occur as valuation rings for this field. A subring 
of 
is a valuation ring for 
if and only if for every non-zero element 
at least one of 
and 
belongs to 
. 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 
, where 
is the multiplicative group of invertible elements of 
, and 
is ordered by divisibility.
A valuation ring can be defined in yet another way. If 
are two local rings with maximal ideals 
and 
, respectively, then one says that 
dominates 
if 
. Dominance is a partial order relation on the set of subrings of 
. The maximal elements of this set are exactly the valuation rings of 
. If 
is a valuation ring and 
is a ring with the same field of fractions as 
, then 
is also a valuation ring and is the localization of 
with respect to some prime ideal.
Examples of valuations.
1) The valuation defined by
 |
is called improper or trivial. Any valuation of a finite field is trivial.
2) Let 
be a field and let 
be the field of Laurent series over 
. Associating to a series 
, where 
, its order 
(and 
to the null series) is a valuation with value group 
(the additive group of integers) and valuation ring 
.
A valuation with values in 
is called discrete; about their valuation rings see Discretely-normed ring. For a description of all valuations of the field of rational numbers, see [4].
For each totally ordered Abelian group 
there is a valuation of a certain field with value group 
.
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. 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 
of a totally ordered set is said to be major if 
and 
imply 
. Let 
be the ring of a valuation 
of a field 
with value group 
, let 
be the sub-semi-group of positive elements of 
, and let 
be a major set in 
. The mapping 
is a bijection between the set of major subsets of 
and the set of ideals of 
. Principal ideals correspond to majors having a minimal element. Prime ideals also correspond to majors of special form, namely: 
, where 
is the positive part of a convex subgroup 
of 
. Thus, there is a one-to-one correspondence between the prime ideals of 
and the convex subgroups of the value group 
.
Let 
be the prime ideal corresponding to a convex subgroup 
. Then the composite mapping 
is a valuation of 
with valuation ring 
and valuation ideal 
; moreover, on the field 
there is an induced valuation with values in 
and valuation ring 
. In this manner a valuation splits into simpler ones. Let 
be a valuation ring. Then the prime spectrum of 
without the zero (
) is a totally ordered set, and its type is called the height or rank of the corresponding valuation. If 
is finite, then the height of the valuation is the number of elements in 
, and this is the same as the number of proper convex subgroups of 
. 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), that is, they are isomorphic to a subgroup of the additive group 
of real numbers. In this case the mapping 
is an ultrametric norm on 
.
An important property of valuation rings is that they are integrally closed. Moreover, for an arbitrary integral ring 
its integral closure is equal to the intersection of all valuation rings containing 
. 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.
Valuations and topologies.
Let 
be a valuation on a field 
and let 
, where 
. The collection of all 
, 
, is a fundamental system of neighbourhoods of zero for a topology 
of 
, which is said to be the topology determined by the valuation 
. It is separable and disconnected. The topology induced by 
on 
is, as a rule, different from that of a local ring. For a non-trivial valuation of 
the topology 
is locally compact if and only if 
is discrete, the valuation ring is complete, and the residue field of 
is finite; 
is then compact. The completion 
of 
relative to 
is a field; 
can be extended by continuity to a valuation 
, and the topology of 
is the same as 
. The valuation ring 
of 
is the completion of the valuation ring 
of 
.
Two valuations 
and 
of 
are called independent if the topologies 
and 
are distinct; this is equivalent to the fact that the valuation rings 
and 
generate 
. Inequivalent valuations of height 1 are always independent. There is an approximation theorem for valuations: Let 
, 
, be independent valuations, let 
, 
. Then there is an element 
in 
such that 
for all 
.
Extension of valuations.
If 
is a valuation of 
and 
is a subfield of 
, then the restriction 
of 
to 
is a valuation of 
, and its value group 
is a subgroup of 
; 
is then called an extension of 
. Conversely, if 
is a valuation on 
and 
is an extension of 
, then there is always a valuation of 
that extends 
. The index 
of 
in 
is called the ramification index of 
with respect to 
and is denoted by 
. The residue field 
of 
can be identified with a subfield of the residue field 
and the degree 
of the extension is denoted by 
and is called the residue degree of 
relative to 
. An extension 
of a valuation 
is said to be immediate if 
.
Let 
be an extension of 
and let 
be the set of all extensions of 
to 
. If 
is a finite extension of 
of degree 
, then the set of all extensions of 
is finite, and
 |
In several cases equality holds, for example when 
is discrete and either 
is complete or 
is separable over 
. If 
is a normal extension of 
, then the extensions of 
to 
are permuted transitively by the 
-automorphisms of 
; in particular, if 
is a purely inseparable extension of 
, then 
has only one extension. In the case of an arbitrary extension 
and an extension 
of a valuation 
, the transcendence degree of 
over 
is greater than or equal to the sum
 |
where 
is the transcendence degree of the extension of the residue field of 
over that of 
and 
is the dimension of the space 
.
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 |
Comments
References
| [a1] | O. Endler, "Valuation theory" , Springer (1972) MR0357379 Zbl 0257.12111 |
Valuation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Valuation&oldid=49103