Completion (ring theory)

From Wikipedia, the free encyclopedia

In commutative algebra, the term completion refers to several related functors on 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, in large part, due to Hensel's lemma. Geometrically, completion of a commutative ring R concentrates on a formal neighborhood of a point or a Zariski closed subscheme of its spectrum Spec R.

Contents

[edit] I-adic topology

The most important case is the so-called I-adic topology on a commutative ring R associated to an ideal I, which is often maximal. In it, a basis of open neighbourhoods of 0 is given by the powers

In,

which are nested (they are smaller as n gets larger, because I is an ideal). Speaking more formally they form a descending filtration by :

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

The completion \hat{R}_I (pronounced "R eye hat") of the ring R is the inverse limit of the rings

R/InR

as n goes to infinity. As may be expected, this produces a complete topological ring.

There is a related topology also on R-modules (also called I-adic): a basis of open neighborhoods of a module M is given by the sets of the form

x + InM

for x ∈ M. The completion of an R-module M is an \hat{R}-module \hat{M}_I obtained as the inverse limit of the quotients

M/InM.

This converts any module over R into a complete topological module over \hat{R}_I.

[edit] Examples

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

2. Let R be the ring of polynomials in n variables over a field K and I be the maximal ideal generated by the variables. Then the completion of R at the ideal I is the ring of the formal power series over K.

[edit] 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.

[edit] General case

Completion can be defined more generally for a noncommutative filtered ring R and (left or right) modules over it:

\hat{R}=\varprojlim R/F^n{R}, \quad \hat{M}=\varprojlim M/F^n{M}    for a left R-module M.

[edit] References


Languages