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 simpler structure than the general ones and Hensel's lemma applies to them. Geometrically, a completion of a commutative ring R concentrates on a formal neighborhood of a point or a Zariski closed subvariety of its spectrum Spec R.

General construction

Suppose that E is an abelian group with a descending filtration

E=F^{0}{E}\supset F^{1}{E}\supset F^{2}{E}\supset \cdots \,

of subgroups. One then defines the completion (with respect to the filtration) as the inverse limit:

{\hat {E}}=\varprojlim (E/F^{n}{E}).\,

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. As may be expected, when the intersection of the $F^{i}{E}$ 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 topology (after Wolfgang Krull) or I-adic topology on R. The case of a maximal ideal $I={\mathfrak {m}}$ is especially important. 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:

F^{0}{R}=R\supset I\supset I^{2}\supset \cdots ,\quad F^{n}{R}=I^{n}.

The completion is the inverse limit of the factor rings,

{\hat {R}}_{I}=\varprojlim (R/I^{n})

(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

x+I^{n}M\quad {\text{for }}x\in M.

The completion of an R-module M the inverse limit of the quotients

{\hat {M}}_{I}=\varprojlim (M/I^{n}{M}).

This procedure converts any module over R into a complete topological module over ${\hat {R}}_{I}$ .

Examples

1. The ring of p-adic integers Z_p is obtained by completing the ring Z of integers at the ideal (p).

2. Let R = K[x₁,...,x_n] be the polynomial ring in n variables over a field K and ${\mathfrak {m}}=(x_{1},\ldots ,x_{n})$ be the maximal ideal generated by the variables. Then the completion ${\hat {R}}_{{{\mathfrak {m}}}}$ is the ring K[[''x''<sub>1</sub>,...,''x''<sub>''n''</sub>]] of formal power series in n variables over K.

Properties

1. The completion is a functorial operation: a continuous map f: R → S of topological rings gives rise to a map of their completions,

{\hat {f}}:{\hat {R}}\to {\hat {S}}.

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:

{\hat {f}}:{\hat {M}}\to {\hat {N}},\quad

where

{\hat {M}},{\hat {N}}

are modules over

{\hat {R}}.

2. The completion of a Noetherian ring R is a flat module over R.

3. The completion of a finitely generated module M over a Noetherian ring R can be obtained by extension of scalars:

{\hat {M}}=M\otimes _{R}{\hat {R}}.

Together with the previous property, this implies that the functor of completion on finitely generated R-modules is exact: it preserves short exact sequences.

4. Cohen structure theorem (equicharacteristic case). Let R be a complete local Noetherian commutative ring with maximal ideal ${\mathfrak {m}}$ and residue field K. If R contains a field, then

R\simeq K[[x_{1},\ldots ,x_{n}]]/I

for some n and some ideal I (Eisenbud, Theorem 7.7).

References

David Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp. ISBN 0-387-94268-8; ISBN 0-387-94269-6 MR 1322960

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