# 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:

`This is again an abelian group. Usually`

*E*is an*additive*abelian group. If*E*has additional algebraic structure compatible with the filtration, for instance*E*is afiltered ring, a filteredmodule, or a filteredvector space, then its completion is again an object with the same structure that is complete in the topology determined by the filtration. This construction may be applied both tocommutativeandnoncommutative rings. As may be expected, when the intersection of theequals zero, this produces acompletetopological ring.Krull topology

`Incommutative algebra, the filtration on acommutative ring`

*R*by the powers of a properideal*I*determines the **Krull topology** (afterWolfgang Krull) or*on***I***R*. The case of a*maximal*idealis especially important, for example the distinguished maximal ideal of avaluation ring. Thebasis of open neighbourhoodsof 0 in*R*is given by the powers*I**n*, which are*nested*and form a descending filtration on*R*:(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

`This procedure converts any module over`

*R*into a completetopological moduleover.Examples

The ring of

*p*-adic integers is obtained by completing the ring of integers at the ideal (*p*).

Let

*R*=*K*[*x*1,...,*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*[[*x*1,...,*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]}

`Completions can also be used to analyze the local structure ofsingularitiesof ascheme. For example, the affine schemes associated toand the nodal cubicplane curvehave similar looking singularities at the origin when viewing their graphs (both look like a plus sign). Notice that in the second case, any Zariski neighborhood of the origin is still an irreducible curve. If we use completions, then we are looking at a "small enough" neighborhood where the node has two components. Taking the localizations of these rings along the idealand completing givesandrespectively, whereis the formal square root ofinMore explicitly, the power series:`

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:

`whereare modules over`

- 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*:

`Together with the previous property, this implies that the functor of completion on finitely generated`

*R*-modules isexact: it preservesshort exact sequences. In particular, taking quotients of rings commutes with completion, meaning that for any quotient*R*-algebra, there is an isomorphism

- **

*R*be a completelocalNoetherian commutative ring with maximal idealandresidue field*K*. If*R*contains a field, thenfor some *n* and some ideal *I* (Eisenbud, Theorem 7.7).

See also

Formal scheme

Profinite integer

Zariski ring

Linear topology

Quasi-unmixed ring

## References

*stacks.math.columbia.edu*. Retrieved 2017-01-14.