- Source: Discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain R that satisfies any one of the following equivalent conditions:
R is a local principal ideal domain, and not a field.
R is a valuation ring with a value group isomorphic to the integers under addition.
R is a local Dedekind domain and not a field.
R is a Noetherian local domain whose maximal ideal is principal, and not a field.
R is an integrally closed Noetherian local ring with Krull dimension one.
R is a principal ideal domain with a unique non-zero prime ideal.
R is a principal ideal domain with a unique irreducible element (up to multiplication by units).
R is a unique factorization domain with a unique irreducible element (up to multiplication by units).
R is Noetherian, not a field, and every nonzero fractional ideal of R is irreducible in the sense that it cannot be written as a finite intersection of fractional ideals properly containing it.
There is some discrete valuation ν on the field of fractions K of R such that R = {0}
∪
{\displaystyle \cup }
{x
∈
{\displaystyle \in }
K : ν(x) ≥ 0}.
Examples
= Algebraic
=Localization of Dedekind rings
Let
Z
(
2
)
:=
{
z
/
n
∣
z
,
n
∈
Z
,
n
is odd
}
{\displaystyle \mathbb {Z} _{(2)}:=\{z/n\mid z,n\in \mathbb {Z} ,\,\,n{\text{ is odd}}\}}
. Then, the field of fractions of
Z
(
2
)
{\displaystyle \mathbb {Z} _{(2)}}
is
Q
{\displaystyle \mathbb {Q} }
. For any nonzero element
r
{\displaystyle r}
of
Q
{\displaystyle \mathbb {Q} }
, we can apply unique factorization to the numerator and denominator of r to write r as 2k z/n where z, n, and k are integers with z and n odd. In this case, we define ν(r)=k.
Then
Z
(
2
)
{\displaystyle \mathbb {Z} _{(2)}}
is the discrete valuation ring corresponding to ν. The maximal ideal of
Z
(
2
)
{\displaystyle \mathbb {Z} _{(2)}}
is the principal ideal generated by 2, i.e.
2
Z
(
2
)
{\displaystyle 2\mathbb {Z} _{(2)}}
, and the "unique" irreducible element (up to units) is 2 (this is also known as a uniformizing parameter). Note that
Z
(
2
)
{\displaystyle \mathbb {Z} _{(2)}}
is the localization of the Dedekind domain
Z
{\displaystyle \mathbb {Z} }
at the prime ideal generated by 2.
More generally, any localization of a Dedekind domain at a non-zero prime ideal is a discrete valuation ring; in practice, this is frequently how discrete valuation rings arise. In particular, we can define rings
Z
(
p
)
:=
{
z
n
|
z
,
n
∈
Z
,
p
∤
n
}
{\displaystyle \mathbb {Z} _{(p)}:=\left.\left\{{\frac {z}{n}}\,\right|z,n\in \mathbb {Z} ,p\nmid n\right\}}
for any prime p in complete analogy.
p-adic integers
The ring
Z
p
{\displaystyle \mathbb {Z} _{p}}
of p-adic integers is a DVR, for any prime
p
{\displaystyle p}
. Here
p
{\displaystyle p}
is an irreducible element; the valuation assigns to each
p
{\displaystyle p}
-adic integer
x
{\displaystyle x}
the largest integer
k
{\displaystyle k}
such that
p
k
{\displaystyle p^{k}}
divides
x
{\displaystyle x}
.
Formal power series
Another important example of a DVR is the ring of formal power series
R
=
k
[
[
T
]
]
{\displaystyle R=k[[T]]}
in one variable
T
{\displaystyle T}
over some field
k
{\displaystyle k}
. The "unique" irreducible element is
T
{\displaystyle T}
, the maximal ideal of
R
{\displaystyle R}
is the principal ideal generated by
T
{\displaystyle T}
, and the valuation
ν
{\displaystyle \nu }
assigns to each power series the index (i.e. degree) of the first non-zero coefficient.
If we restrict ourselves to real or complex coefficients, we can consider the ring of power series in one variable that converge in a neighborhood of 0 (with the neighborhood depending on the power series). This is a discrete valuation ring. This is useful for building intuition with the Valuative criterion of properness.
Ring in function field
For an example more geometrical in nature, take the ring R = {f/g : f, g polynomials in R[X] and g(0) ≠ 0}, considered as a subring of the field of rational functions R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined (i.e. finite) in a neighborhood of 0 on the real axis (with the neighborhood depending on the function). It is a discrete valuation ring; the "unique" irreducible element is X and the valuation assigns to each function f the order (possibly 0) of the zero of f at 0. This example provides the template for studying general algebraic curves near non-singular points, the algebraic curve in this case being the real line.
= Scheme-theoretic
=Henselian trait
For a DVR
R
{\displaystyle R}
it is common to write the fraction field as
K
=
Frac
(
R
)
{\displaystyle K={\text{Frac}}(R)}
and
κ
=
R
/
m
{\displaystyle \kappa =R/{\mathfrak {m}}}
the residue field. These correspond to the generic and closed points of
S
=
Spec
(
R
)
.
{\displaystyle S={\text{Spec}}(R).}
For example, the closed point of
Spec
(
Z
p
)
{\displaystyle {\text{Spec}}(\mathbb {Z} _{p})}
is
F
p
{\displaystyle \mathbb {F} _{p}}
and the generic point is
Q
p
{\displaystyle \mathbb {Q} _{p}}
. Sometimes this is denoted as
η
→
S
←
s
{\displaystyle \eta \to S\leftarrow s}
where
η
{\displaystyle \eta }
is the generic point and
s
{\displaystyle s}
is the closed point .
Localization of a point on a curve
Given an algebraic curve
(
X
,
O
X
)
{\displaystyle (X,{\mathcal {O}}_{X})}
, the local ring
O
X
,
p
{\displaystyle {\mathcal {O}}_{X,{\mathfrak {p}}}}
at a smooth point
p
{\displaystyle {\mathfrak {p}}}
is a discrete valuation ring, because it is a principal valuation ring. Note because the point
p
{\displaystyle {\mathfrak {p}}}
is smooth, the completion of the local ring is isomorphic to the completion of the localization of
A
1
{\displaystyle \mathbb {A} ^{1}}
at some point
q
{\displaystyle {\mathfrak {q}}}
.
Uniformizing parameter
Given a DVR R, any irreducible element of R is a generator for the unique maximal ideal of R and vice versa. Such an element is also called a uniformizing parameter of R (or a uniformizing element, a uniformizer, or a prime element).
If we fix a uniformizing parameter t, then M=(t) is the unique maximal ideal of R, and every other non-zero ideal is a power of M, i.e. has the form (t k) for some k≥0. All the powers of t are distinct, and so are the powers of M. Every non-zero element x of R can be written in the form αt k with α a unit in R and k≥0, both uniquely determined by x. The valuation is given by ν(x) = kv(t). So to understand the ring completely, one needs to know the group of units of R and how the units interact additively with the powers of t.
The function v also makes any discrete valuation ring into a Euclidean domain.
Topology
Every discrete valuation ring, being a local ring, carries a natural topology and is a topological ring. We can also give it a metric space structure where the distance between two elements x and y can be measured as follows:
|
x
−
y
|
=
2
−
ν
(
x
−
y
)
{\displaystyle |x-y|=2^{-\nu (x-y)}}
(or with any other fixed real number > 1 in place of 2). Intuitively: an element z is "small" and "close to 0" iff its valuation ν(z) is large. The function |x-y|, supplemented by |0|=0, is the restriction of an absolute value defined on the field of fractions of the discrete valuation ring.
A DVR is compact if and only if it is complete and its residue field R/M is a finite field.
Examples of complete DVRs include
the ring of p-adic integers and
the ring of formal power series over any field
For a given DVR, one often passes to its completion, a complete DVR containing the given ring that is often easier to study. This completion procedure can be thought of in a geometrical way as passing from rational functions to power series, or from rational numbers to the reals.
Returning to our examples: the ring of all formal power series in one variable with real coefficients is the completion of the ring of rational functions defined (i.e. finite) in a neighborhood of 0 on the real line; it is also the completion of the ring of all real power series that converge near 0. The completion of
Z
(
p
)
=
Q
∩
Z
p
{\displaystyle \mathbb {Z} _{(p)}=\mathbb {Q} \cap \mathbb {Z} _{p}}
(which can be seen as the set of all rational numbers that are p-adic integers) is the ring of all p-adic integers Zp.
See also
Category:Localization (mathematics)
Local ring
Ramification of local fields
Cohen ring
Valuation ring
References
Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
Dummit, David S.; Foote, Richard M. (2004), Abstract algebra (3rd ed.), New York: John Wiley & Sons, ISBN 978-0-471-43334-7, MR 2286236
Discrete valuation ring, The Encyclopaedia of Mathematics.
Kata Kunci Pencarian:
- Discrete valuation ring
- Valuation ring
- Discrete valuation
- Henselian ring
- Regular local ring
- Local field
- List of commutative algebra topics
- Valuation (algebra)
- Local ring
- Commutative ring
