Discretely-normed ring

discrete valuation ring, discrete valuation domain

A ring with a discrete valuation, i.e. an integral domain with a unit element in which there exists an element $ \pi $ such that any non-zero ideal is generated by some power of the element $ \pi $; such an element is called a uniformizing parameter, and is defined up to multiplication by an invertible element. Each non-zero element of a discretely-normed ring can be uniquely written in the form $ u \pi ^ {n} $, where $ u $ is an invertible element and $ n \geq 0 $ is an integer. Examples of discretely-normed rings include the ring $ \mathbf Z _ {p} $ of $ p $- adic integers, the ring $ k [[ T ]] $ of formal power series in one variable $ T $ over a field $ k $, and the ring of Witt vectors (cf. Witt vector) $ W ( k) $ for a perfect field $ k $.

A discretely-normed ring may also be defined as a local principal ideal ring; as a local one-dimensional Krull ring; as a local Noetherian ring with a principal maximal ideal; as a Noetherian valuation ring; or as a valuation ring with group of values $ \mathbf Z $.

The completion (in the topology of a local ring) of a discretely-normed ring is also a discretely-normed ring. A discretely-normed ring is compact if and only if it is complete and its residue field is finite; any such ring is either isomorphic to $ k [[ T ]] $, where $ k $ is a finite field, or else is a finite extension of $ \mathbf Z _ {p} $.

If $ A \subset B $ is a local homomorphism of discretely-normed rings with uniformizing elements $ \pi $ and $ \Pi $, then $ \pi = u \Pi ^ {e} $, where $ u $ is an invertible element in $ B $. The integer $ e = e ( B / A ) $ is the ramification index of the extension $ A \subset B $, and

$$ [ B / \Pi B : A / \pi A ] = f ( B / A ) $$

is called the residue degree. This situation arises when one considers the integral closure $ B $ of a discretely-normed ring $ A $ with a field of fractions $ K $ in a finite extension $ L $ of $ K $. In such a case $ B $ is a semi-local principal ideal ring; if $ \mathfrak n _ {1} \dots \mathfrak n _ {s} $ are its maximal ideals, then the localizations $ B _ {i} = B _ {\mathfrak n _ {i} } $ are discretely-normed rings. If $ L $ is a separable extension of $ K $ of degree $ n $, the formula

$$ \sum _ {i = 1 } ^ { s } e ( B _ {i} / A ) f ( B _ {i} / A ) = n $$

is valid. If $ L / K $ is a Galois extension, then all $ e ( B _ {i} / A ) $ and all $ f ( B _ {i} / A ) $ are equal, and $ n = sef $. If $ A $ is a complete discretely-normed ring, $ B $ itself will be a discretely-normed ring and $ e ( B / A ) f ( B / A ) = n $. On these assumptions the extension $ A \subset B $( and also $ L $ over $ K $) is known as an unramified extension if $ e ( B / A ) = 1 $ and the field $ B / \mathfrak n $ is separable over $ A / \mathfrak m $; it is weakly ramified if $ e ( B / A ) $ is relatively prime with the characteristic of the field $ A / \mathfrak m $ while $ B / \mathfrak n $ is separable over $ A / \mathfrak m $; it is totally ramified if $ f ( B / A ) = 1 $.

The theory of modules over a discretely-normed ring is very similar to the theory of Abelian groups [3]. Any module of finite type is a direct sum of cyclic modules; a torsion-free module is a flat module; any projective module or submodule of a free module is free. However, the direct product of an infinite number of free modules is not free. A torsion-free module of countable rank over a complete discretely-normed ring is a direct sum of modules of rank one.

References

[1]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
[2]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967)
[3]	J. Kaplansky, "Modules over Dedekind rings and valuation rings" Trans. Amer. Math. Soc. , 72 (1952) pp. 327–340

Comments

Let $ A $ be a discretely-normed ring with uniformizing parameter $ \pi $. The associated valuation is then defined by $ \nu ( a) = n $ if $ a = u \pi ^ {n} $, $ u $ a unit of $ A $. A corresponding norm on $ A $ is defined by $ | a | = c ^ {\nu ( a ) } $, $ | 0 | = 0 $, where $ c $ is a real number between $ 0 $ and $ 1 $. This makes $ A $ a normal ring. If the residue field $ k = A ( \pi ) $ of $ A $ is finite it is customary to take $ c = q ^ {-} 1 $ where $ q $ is the number of elements of $ k $.