View source for I-adic topology

{{DISPLAYTITLE:''I''-adic topology}}
{{Short description|Concept in commutative algebra}}

In [[commutative algebra]], the mathematical study of [[commutative ring]]s, '''adic topologies''' are a family of [[topological space|topologies]] on the underlying set of a [[module (mathematics)|module]], generalizing the [[p-adic number|{{mvar|p}}-adic topologies]] on the [[integer]]s.

==Definition==
Let {{mvar|R}} be a commutative ring and {{mvar|M}} an {{mvar|R}}-module. Then each [[ideal (ring theory)|ideal]] {{math|𝔞}} of {{mvar|R}} determines a topology on {{mvar|M}} called the {{math|𝔞}}-adic topology, characterized by the [[pseudometric space|pseudometric]] <math display=block>d(x,y) = 2^{-\sup{\{n \mid x-y\in\mathfrak{a}^nM\}}}.</math>  The family <math display=block>\{x+\mathfrak{a}^nM:x\in M,n\in\mathbb{Z}^+\}</math> is a [[basis (topology)|basis]] for this topology.{{sfn|Singh|2011|p=147}}

An {{math|𝔞}}-adic topology is a [[linear topology]] (a topology generated by some submodules).<!-- but the converse is false, I think -->

==Properties==
With respect to the topology, the module operations of addition and scalar multiplication are [[Continuity (topology)|continuous]], so that {{mvar|M}} becomes a [[topological module]].  However, {{mvar|M}} need not be [[Hausdorff space|Hausdorff]]; it is Hausdorff [[if and only if]]<math display=block>\bigcap_{n > 0}{\mathfrak{a}^nM} = 0\text{,}</math>so that {{mvar|d}} becomes a genuine [[metric (mathematics)|metric]]. Related to the usual terminology in topology, where a Hausdorff space is also called separated, in that case, the {{mvar|𝔞}}-adic topology is called ''separated''.{{sfn|Singh|2011|p=147}}

By [[Krull's intersection theorem]], if {{mvar|R}} is a [[Noetherian ring]] which is an [[integral domain]] or a [[local ring]], it holds that <math>\bigcap_{n > 0}{\mathfrak{a}^n} = 0</math> for any proper ideal {{mvar|𝔞}} of {{mvar|R}}. Thus under these conditions, for any proper ideal {{mvar|𝔞}} of {{mvar|R}} and any {{mvar|R}}-module {{mvar|M}}, the {{mvar|𝔞}}-adic topology on {{mvar|M}} is separated.   

For a submodule {{mvar|N}} of {{mvar|M}}, the [[1st isomorphism theorem|canonical]] [[homomorphism]] to {{math|''M''/''N''}} induces a [[quotient topology]] which coincides with the {{math|𝔞}}-adic topology. The analogous result is not necessarily true for the submodule {{mvar|N}} itself: the [[subspace topology]] need not be the {{math|𝔞}}-adic topology. However, the two topologies coincide when {{mvar|R}} is Noetherian and {{mvar|M}} [[finitely generated module|finitely generated]]. This follows from the [[Artin–Rees lemma]].{{sfn|Singh|2011|p=148}}

==Completion==
{{Main|Completion (algebra)}}
When {{mvar|M}} is Hausdorff, {{mvar|M}} can be [[completion of a metric space|completed]] as a metric space; the resulting space is denoted by <math>\widehat M</math> and has the module structure obtained by extending the module operations by continuity.  It is also the same as (or [[natural isomorphism|canonically isomorphic]] to): <math display=block>\widehat{M} = \varprojlim M/\mathfrak{a}^n M</math> where the right-hand side is an [[inverse limit]] of [[quotient module]]s under natural projection.{{sfn|Singh|2011|pp=148-151}}

For example, let <math>R = k[x_1, \ldots, x_n]</math> be a [[polynomial ring]] over a [[field (mathematics)|field]] {{mvar|k}} and {{math|𝔞 {{=}} (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}} the (unique) homogeneous [[maximal ideal]]. Then <math>\hat{R} = k[[x_1, \ldots, x_n]]</math>, the [[Formal power series ring#Power series in several variables|formal power series ring]] over {{mvar|k}} in {{mvar|n}} variables.<ref>{{harvnb|Singh|2011}}, problem&nbsp;8.16.</ref>

==Closed submodules==
The {{math|𝔞}}-adic closure of a submodule <math>N \subseteq M</math> is <math display=inline>\overline{N} = \bigcap_{n > 0}{(N + \mathfrak{a}^n M)}\text{.}</math><ref>{{harvnb|Singh|2011}}, problem&nbsp;8.4.</ref>  This closure coincides with {{mvar|N}} whenever {{mvar|R}} is {{math|𝔞}}-adically complete and {{mvar|M}} is finitely generated.<ref>{{harvnb|Singh|2011}}, problem&nbsp;8.8</ref>

{{mvar|R}} is called [[Zariski ring|Zariski]] with respect to {{math|𝔞}} if every ideal in {{mvar|R}} is {{math|𝔞}}-adically closed. There is a characterization:
:{{mvar|R}} is Zariski with respect to {{math|𝔞}} if and only if {{math|𝔞}} is contained in the [[Jacobson radical]] of {{mvar|R}}.
In particular a Noetherian local ring is Zariski with respect to the maximal ideal.<ref>{{harvnb|Atiyah|MacDonald|1969|p=114}}, exercise 6.</ref>

==References==
<references />

===Sources===
* {{cite book|first=Balwant|last=Singh|title=Basic Commutative Algebra|year=2011|publisher=World Scientific|location=Singapore/Hackensack, NJ|isbn=978-981-4313-61-2}}
* {{cite book|first=M.&nbsp;F.|last=Atiyah|author-link1=Michael Atiyah|first2=I.&nbsp;G.|last2=MacDonald|publisher=Addison-Wesley|location=Reading, MA|year=1969|title=Introduction to Commutative Algebra}}

[[category:Commutative algebra]]
[[category:Topology]]