|
|
| Line 1: |
Line 1: |
| In [[abstract algebra]], a '''completion''' is any of several related [[functor]]s on [[ring (mathematics)|ring]]s and [[module (mathematics)|modules]] that result in complete [[topological ring]]s and modules. Completion is similar to [[localization of a ring|localization]], and together they are among the most basic tools in analysing [[commutative ring]]s. 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 topology|Zariski closed]] [[algebraic variety|subvariety]] of its [[spectrum of a ring|spectrum]] Spec ''R''.
| | Andera is what you can call her but she by no means truly liked that title. To perform lacross is the thing I love most of all. Credit authorising is how she tends to make a residing. For many years he's been living in Mississippi and he doesn't strategy on changing it.<br><br>my web blog: psychic phone readings ([http://www.taehyuna.net/xe/?document_srl=78721 http://www.taehyuna.net/xe/?document_srl=78721]) |
| <!--- this appears to be incorrect: most completions are *defined* using projective limits, not Cauchy seq.
| |
| In [[algebra]], the completion of an ''R''-[[module]] ''M'', relative to a [[topology]] of a certain prescribed form defined on ''M'', is, in rough terms, an ''R''-module <math>\hat M</math> defined in a way analogous to the completion of a metric space using Cauchy sequences.
| |
| -->
| |
| | |
| == General construction ==
| |
| | |
| Suppose that ''E'' is an [[abelian group]] with a descending [[filtration (mathematics)|filtration]]
| |
| | |
| : <math> E = F^0{E} \supset F^1{E} \supset F^2{E} \supset \cdots \, </math>
| |
| | |
| of subgroups, one defines the completion (with respect to the filtration) as the [[inverse limit]]:
| |
| | |
| : <math> \hat{E}=\varprojlim (E/F^n{E}). \, </math>
| |
| | |
| 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 algebra|filtered ring]], a filtered [[module (mathematics)|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 ring|commutative]] and [[noncommutative ring]]s. As may be expected, this produces a [[complete metric space|complete]] [[topological ring]].
| |
| | |
| == Krull topology ==
| |
| | |
| In [[commutative algebra]], the filtration on a [[commutative ring]] ''R'' by the powers of a proper [[ideal (ring theory)|ideal]] ''I'' determines the '''[[Krull topology]]''' (after [[Wolfgang Krull]]) or ''' ''I''-adic topology''' on ''R''. The case of a [[maximal ideal|''maximal'' ideal]] <math>I=\mathfrak{m}</math> is especially important. The [[basis of open neighbourhoods]] of 0 in ''R'' is given by the powers ''I''<sup>''n''</sup>, which are ''nested'' and form a descending filtration on ''R'':
| |
| <!-- nested: ''I''<sup>''n''</sup>⊃''I''<sup>''n''+1</sup>-->
| |
| | |
| :<math> F^0{R}=R\supset I\supset I^2\supset\cdots, \quad F^n{R}=I^n.</math>
| |
| | |
| The completion is the [[inverse limit]] of the [[factor ring]]s,
| |
| | |
| : <math> \hat{R}_I=\varprojlim (R/I^n) </math>
| |
| | |
| (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 (mathematics)|module]] ''M'' is given by the sets of the form
| |
| | |
| :<math>x + I^n M \quad\text{for }x\in M. </math> | |
| | |
| The completion of an ''R''-module ''M'' the inverse limit of the quotients
| |
| | |
| : <math> \hat{M}_I=\varprojlim (M/I^n{M}). </math>
| |
| | |
| This procedure converts any module over ''R'' into a complete [[topological module]] over <math>\hat{R}_I</math>.
| |
| | |
| == Examples ==
| |
| | |
| 1. The ring of [[p-adic integers]] '''''Z'''''<sub>p</sub> is obtained by completing the ring '''''Z''''' of integers at the ideal (''p'').
| |
| | |
| 2. Let ''R'' = ''K''[''x''<sub>1</sub>,…,''x''<sub>''n''</sub>] be the [[polynomial ring]] in ''n'' variables over a field ''K'' and <math>\mathfrak{m}=(x_1,\ldots,x_n)</math> be the maximal ideal generated by the variables. Then the completion <math>R_{\mathfrak{m}}</math> is the ring ''K''[[''x''<sub>1</sub>,…,''x''<sub>''n''</sub>]] of [[formal power series]] in ''n'' variables over ''K''.
| |
| | |
| 3. Let ''R'' be the ring of [[holomorphic functions]] on a [[complex manifold]] and let ''I'' be the maximal ideal of functions vanishing at some point p. Then the completion of ''R'' at the ideal ''I'' is the ring of [[power series]] over '''C'''.
| |
| | |
| == 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,
| |
| | |
| : <math> \hat{f}: \hat{R}\to\hat{S}. </math>
| |
| | |
| 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:
| |
| | |
| : <math> \hat{f}: \hat{M}\to\hat{N},\quad </math> where <math>\hat{M},\hat{N}</math> are modules over <math>\hat{R}.</math>
| |
| | |
| 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'':
| |
| :<math> \hat{M}=M\otimes_R \hat{R}. </math>
| |
| Together with the previous property, this implies that the functor of completion on finitely generated ''R''-modules is [[exact functor|exact]]: it preserves [[short exact sequence]]s.
| |
| | |
| 4. '''[[Cohen structure theorem]]''' (equicharacteristic case). Let ''R'' be a complete [[local ring|local]] Noetherian commutative ring with maximal ideal <math>\mathfrak{m}</math> and [[residue field]] ''K''. If ''R'' contains a field, then
| |
| | |
| : <math> R\simeq K[[x_1,\ldots,x_n]]/I </math>
| |
| | |
| for some ''n'' and some ideal ''I'' (Eisenbud, Theorem 7.7).
| |
| | |
| == See also ==
| |
| | |
| * [[p-adic number]]s
| |
| | |
| == 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}}
| |
| | |
| <br />
| |
| | |
| [[Category:Commutative algebra]]
| |
| [[Category:Topological algebra]]
| |
Andera is what you can call her but she by no means truly liked that title. To perform lacross is the thing I love most of all. Credit authorising is how she tends to make a residing. For many years he's been living in Mississippi and he doesn't strategy on changing it.
my web blog: psychic phone readings (http://www.taehyuna.net/xe/?document_srl=78721)