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
,
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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.
Contents
Examples of valuations.
1) The valuation defined by
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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
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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
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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 |
[2] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |
[3] | O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975) |
[4] | A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) |
Comments
References
[a1] | O. Endler, "Valuation theory" , Springer (1972) |
Valuation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Valuation&oldid=13681