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

In commutative algebra, the mathematical study of commutative rings, '''adic topologies''' are a family of topologies on the underlying set of a module, generalizing the {{mvar|p}}-adic topologies on the integers.

==Definition== Let {{mvar|R}} be a commutative ring and {{mvar|M}} an {{mvar|R}}-module. Then each ideal {{math|𝔞}} of {{mvar|R}} determines a topology on {{mvar|M}} called the {{math|𝔞}}-adic topology, characterized by the 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 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 continuous, so that {{mvar|M}} becomes a topological module. However, {{mvar|M}} need not be 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. 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 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. 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 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 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 modules 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 {{mvar|k}} and {{math|𝔞 {{=}} (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}} the (unique) homogeneous maximal ideal. Then <math>\hat{R} = kx_1, \ldots, x_n</math>, the 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 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|author-link2=Ian G. Macdonald|publisher=Addison-Wesley|location=Reading, MA|year=1969|title=Introduction to Commutative Algebra}}

category:Commutative algebra Category:Topological spaces