I-adic topology#Completion
{{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 p-adic number on the integers.
Definition
Let {{mvar|R}} be a commutative ring and {{mvar|M}} an {{mvar|R}}-module. Then each ideal {{math|π}} of {{mvar|R}} determines a topology on {{mvar|M}} called the {{math|π}}-adic topology, characterized by the pseudometric The family is a basis for this topology.{{sfn|Singh|2011|p=147}}
An {{math|π}}-adic topology is a linear topology (a topology generated by some submodules).
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 ifso 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|π}}-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 for any proper ideal {{mvar|π}} of {{mvar|R}}. Thus under these conditions, for any proper ideal {{mvar|π}} of {{mvar|R}} and any {{mvar|R}}-module {{mvar|M}}, the {{mvar|π}}-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|π}}-adic topology. The analogous result is not necessarily true for the submodule {{mvar|N}} itself: the subspace topology need not be the {{math|π}}-adic topology. However, the two topologies coincide when {{mvar|R}} is Noetherian and {{mvar|M}} finitely generated. This follows from the ArtinβRees 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 and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to): where the right-hand side is an inverse limit of quotient modules under natural projection.{{sfn|Singh|2011|pp=148-151}}
For example, let be a polynomial ring over a field {{mvar|k}} and {{math|π {{=}} (x1, ..., xn)}} the (unique) homogeneous maximal ideal. Then , the formal power series ring over {{mvar|k}} in {{mvar|n}} variables.{{harvnb|Singh|2011}}, problem 8.16.
Closed submodules
The {{math|π}}-adic closure of a submodule is {{harvnb|Singh|2011}}, problem 8.4. This closure coincides with {{mvar|N}} whenever {{mvar|R}} is {{math|π}}-adically complete and {{mvar|M}} is finitely generated.{{harvnb|Singh|2011}}, problem 8.8
{{mvar|R}} is called Zariski with respect to {{math|π}} if every ideal in {{mvar|R}} is {{math|π}}-adically closed. There is a characterization:
:{{mvar|R}} is Zariski with respect to {{math|π}} if and only if {{math|π}} is contained in the Jacobson radical of {{mvar|R}}.
In particular a Noetherian local ring is Zariski with respect to the maximal ideal.{{harvnb|Atiyah|MacDonald|1969|p=114}}, exercise 6.
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. F.|last=Atiyah|author-link1=Michael Atiyah|first2=I. G.|last2=MacDonald|publisher=Addison-Wesley|location=Reading, MA|year=1969|title=Introduction to Commutative Algebra}}