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 the case when 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 a filtered ring, a filtered module, or a filtered vector 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 to commutative and noncommutative rings equals zero, this produces a complete topological ring.

Krull topology

In commutative algebra, the filtration on a commutative ring R by the powers of a proper ideal I determines the Krull (after Wolfgang Krull) or I-adic topology on R. The case of a maximal ideal is especially important, for example the distinguished maximal ideal of a valuation ring. The basis of open neighbourhoods of 0 in R is given by the powers In, which are nested and form a descending filtration on R:

(Open neighborhoods of any r R are given by cosets r + In.) The (I-adic) completion is the inverse limit of the factor rings,

pronounced "R I hat". The kernel of the canonical map from the ring to its completion is the intersection of the powers of I. Thus 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 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 I-adic completion of an R-module M is the inverse limit of the quotients

This procedure converts any module over R into a complete topological module if I is finitely generated.

Examples

• The ring of p-adic integers of integers at the ideal (p).

  Let R = K[x1,...,xn] be the polynomial ring in n variables over a field K is the ring K''x''₁,...,''x''_''n'' of formal power series in n variables over K.

  is an image of a formal power series ring, specifically, the image of the surjection

Completions can also be used to analyze the local structure of singularities of a scheme and the nodal cubic plane curve have 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 More 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 of a Noetherian ring with respect to some ideal is a Noetherian ring.

• The completion of a Noetherian local ring with respect to the unique maximal ideal is a Noetherian local ring.

• uniquely extends to the map of the completions:

• The completion of a Noetherian ring is a flat module.

• can be obtained by extension of scalars:

• -modules is exact: it preserves short exact sequences, there is an isomorphism

• Cohen structure theorem be a complete local and residue field contains a field, then

• (Eisenbud, Theorem 7.7).

See also

• Formal scheme

• Profinite integer

• Locally compact field

• Zariski ring

• Linear topology

• Quasi-unmixed ring

Citations

References

• David Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp.;