I-adic topology#Completion

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|𝔞}} of {{mvar|R}} determines a topology on {{mvar|M}} called the {{math|𝔞}}-adic topology, characterized by the pseudometric $d(x,y) = 2^{-\sup{\{n \mid x-y\in\mathfrak{a}^nM\}}}.$ The family $\{x+\mathfrak{a}^nM:x\in M,n\in\mathbb{Z}^+\}$ 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).

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 $\bigcap_{n > 0}{\mathfrak{a}^nM} = 0\text{,}$ 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 $\bigcap_{n > 0}{\mathfrak{a}^n} = 0$ 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

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

For example, let $R = k[x_1, \ldots, x_n]$ be a polynomial ring over a field {{mvar|k}} and {{math|𝔞 {{=}} (x₁, ..., x_n)}} the (unique) homogeneous maximal ideal. Then $\hat{R} = k x_1, \ldots, x_n$ , the formal power series ring over {{mvar|k}} in {{mvar|n}} variables.{{harvnb|Singh|2011}}, problem 8.16.

Closed submodules

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

{{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.{{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}}

category:Commutative algebra

category:Topology