Completion (algebra)
Completion (algebra)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry, a completion of a ring of functions R on a space X concentrates on a formal neighborhood of a point of X: heuristically, this is a neighborhood so small that all Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in case R has a metric given by a non-Archimedean absolute value.
General construction
Suppose that E is an abelian group with a descending filtration
of subgroups. One then defines the completion (with respect to the filtration) as the inverse limit:
Krull topology
(Open neighborhoods of any r ∈ R are given by cosets r + I**n.) The completion is the inverse limit of the factor rings,
pronounced "R I hat". The kernel of the canonical map π from the ring to its completion is the intersection of the powers of I. Thus π is injective if and only if this intersection reduces to the zero element of the ring; by the Krull intersection theorem, this is the case for any commutative Noetherian ring which is either an integral domain or a local ring.
There is a related topology on R-modules, also called Krull or I-adic topology. A basis of open neighborhoods of a module M is given by the sets of the form
The completion of an R-module M is the inverse limit of the quotients
Examples
The ring of p-adic integers is obtained by completing the ring of integers at the ideal (p).
Let R = K[x1,...,x**n] be the polynomial ring in n variables over a field K and be the maximal ideal generated by the variables. Then the completion is the ring K[[x1,...,x**n]] of formal power series in n variables over K.
Given a noetherian ring and an ideal the -adic completion of is an image of a formal power series ring, specifically, the image of the surjection[1]
![{\displaystyle {\begin{cases}R[[x_{1},\ldots ,x_{n}]]\to {\widehat {R}}_{I}\\x_{i}\mapsto f_{i}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85b5e292125fef414582568fbc97ed00df88bcb)
![{\displaystyle \mathbb {C} [x,y]/(xy)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fc69a7bc41282794b565e67a32951d9641e5ba0)
![{\displaystyle \mathbb {C} [x,y]/(y^{2}-x^{2}(1+x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3021597d95733e4b4dee274ca8d64c3b5068f7)
![{\displaystyle \mathbb {C} [[x,y]]/(xy)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08cf65954ee33c31b3a40d4c4b5fb869efb6564c)
![{\displaystyle \mathbb {C} [[x,y]]/((y+u)(y-u))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56d941d09994208da2ebcff093dc1faaba837c43)
![{\displaystyle \mathbb {C} [[x,y]].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b14351a5c1646dd706d382fcdc3a3b4b0e669d)
Since both rings are given by the intersection of two ideals generated by a homogeneous degree 1 polynomial, we can see algebraically that the singularities "look" the same. This is because such a scheme is the union of two non-equal linear subspaces of the affine plane.
Properties
- The completion is a functorial operation: a continuous map f: R → S of topological rings gives rise to a map of their completions,
Moreover, if M and N are two modules over the same topological ring R and f: M → N is a continuous module map then f uniquely extends to the map of the completions:
- The completion of a Noetherian ring R is a flat module over R.
- The completion of a finitely generated module M over a Noetherian ring R can be obtained by extension of scalars:
- **
![R\simeq K[[x_{1},\ldots ,x_{n}]]/I](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9ccb0c3a17cda05ca6dda98992a58030e12ed1)
for some n and some ideal I (Eisenbud, Theorem 7.7).
See also
Formal scheme
Profinite integer
Zariski ring
Linear topology
Quasi-unmixed ring