Completion (ring theory)
Completion or completion of a ring or of a module is a technique in the commutative algebra in which a ring or a module is completed with respect to a given metric, which is usually induced by an ideal. The term is geometrically related to the localization of a ring: Both ring extensions examine the neighborhood of a point in the spectrum of a ring, but with even more reflects the completion of the local appearance.
This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. Ring homomorphisms form elements from one to one elements. For more details see Commutative Algebra.
- 5.1 generalizations to modules by filtrations
- 5.2 metric Rings as (pseudo ) spaces
- 5.3 Completion as an inverse limit
Complete a ring with respect to an ideal
Be a ring and an ideal.
In the ring
Is a sequence
Zero sequence called when it is a once and for all, so that:
Is the ideal of all null sequences.
One consequence of
Is called when there is and for all, so that applies Cauchy sequence:
Be the subring of all Cauchy sequences
The ring
Is referred to as the completion of regard.
For is
A null sequence.
The figure
Is injective if:
The ring is called complete (complete) (with respect to ) if is an isomorphism.
Examples
Formal power series
If the corresponding polynomial ring over a field and the ideal, so Cauchy sequences of polynomials infinite polynomials
The completion is isomorphic to the ring of formal power series
P- adic numbers
The p- adic numbers are described as the completion of with respect to the -adic metric: Are and rational numbers with
With and and does not share, then
A Cauchy sequence A sequence of integers is exactly then with respect to the -adic metric if it is a Cauchy sequence with respect to the ideal. Therefore gives an embedding:
Here denotes the left side, the completion of respect. This embedding even an isomorphism of the ring of p- adic numbers. Due to the hensel between lemma exist in many non-rational algebraic numbers, such as the roots of unity.
Geometric example
Is the two-dimensional plane algebraic curve in affine space by the equation
Is defined. In the zero point, the curve intersects. It is called the Newtonian nodes and looks around the zero point (clearly ) is locally the same as the curve defined by the equation:
This geometrical situation corresponds to the isomorphism:
With
And
The local rings of the points are not isomorphic, but their well completions with respect to their maximal ideals.
The ring on the left side of the " isomorphism equation " is also an example that the completion of an integral domain need not be an integral domain.
Analytically, the Newtonian node is a subset of the complex plane as a whole, although irreducible, but decomposes locally around the zero into two branches. Because for is the root of holomorphic, one can write:
With two holomorphic functions.
Algebraic- geometric interpretation
The importance of the completion of the algebraic geometry is that one can study the local appearance of the variety in the completed ring. If two points of two irreducible varieties isomorphic local rings, so are the varieties and already birationally equivalent. The local ring carries almost all information about the variety in itself, while the completion of the local ring of intuition about local behavior comes closer.
There is the following sentence:
Be a Noetherian local ring with maximal ideal and its completion. Then:
- Is regular if it is.
Cohen's structure theorem makes a statement about the complete local rings of varieties:
Is a regular local ring which is complete with respect to its maximal ideal and includes a body, then:
Where the residue field of is.
Regular points on algebraic varieties of the same dimension so have isomorphic completions, similar points on manifolds of equal dimensions homeomorphic environments have.
Functorial properties
Are and rings and as well as ideals and
A ring homomorphism with:
( Such a ring homomorphism is constantly called ) then there exists a homomorphism
" " Is thus a functor with continuous maps as morphisms
Design alternatives and generalizations
Generalizations to modules by filtrations
A filtration of a module is a consequence
So that
The now play the role in the definition of zero sequence and Cauchy sequence. The definitions can be transferred literally. It is
And is completely (relative to the filtration ), when the image
Is an isomorphism.
Rings as a ( pseudo-) metric spaces
The completion of a ring respect of an ideal can be understood as a special case of the completion of a metric space if a suitable metric is defined on the ring.
Is a ring and an ideal, so this ring can be defined by the ideal of a distance by:
This is a pseudo- metric, because it is:
If the following applies:
Thus, the distance function is a metric, i.e., it is in addition:
With respect to this ( pseudo-) metric agree the above terms Cauchy sequence, zero sequence and completion with those of metric spaces.
Completion as an inverse limit
An inverse system of rings (or modules ) is ( here ) is a sequence of rings (or modules ) and homomorphisms
So that
So:
The inverse limit of this inverse system is:
Is now an ideal and
Then the following isomorphism holds: