I-adic topology

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 topologies on the integers.

Definition

Let $R$ be a commutative ring and $M$ an $R$ -module. Then each ideal $𝔞$ of $R$ determines a topology on $M$ called the $𝔞$ -adic topology, characterized by the pseudometric $d(x,y)=2^{-\sup {\{n\mid x-y\in {\mathfrak {a}}^{n}M\}}}.$ The family $\{x+{\mathfrak {a}}^{n}M:x\in M,n\in \mathbb {Z} ^{+}\}$ is a basis for this topology.^[1]

An $𝔞$ -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 $M$ becomes a topological module. However, $M$ need not be Hausdorff; it is Hausdorff if and only if $\bigcap _{n>0}{{\mathfrak {a}}^{n}M}=0{\text{,}}$ so that $d$ becomes a genuine metric. Related to the usual terminology in topology, where a Hausdorff space is also called separated, in that case, the $𝔞$ -adic topology is called separated.^[1]

By Krull's intersection theorem, if $R$ 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 $𝔞$ of $R$ . Thus under these conditions, for any proper ideal $𝔞$ of $R$ and any $R$ -module $M$ , the $𝔞$ -adic topology on $M$ is separated.

For a submodule $N$ of $M$ , the canonical homomorphism to $M / N$ induces a quotient topology which coincides with the $𝔞$ -adic topology. The analogous result is not necessarily true for the submodule $N$ itself: the subspace topology need not be the $𝔞$ -adic topology. However, the two topologies coincide when $R$ is Noetherian and $M$ finitely generated. This follows from the Artin-Rees lemma.^[2]

Completion

When $M$ is Hausdorff, $M$ 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.^[3]

For example, let $R=k[x_{1},\ldots ,x_{n}]$ be a polynomial ring over a field $k$ and $𝔞 = (x 1, ..., x n)$ the (unique) homogeneous maximal ideal. Then ${\hat {R}}=k[[x_{1},\ldots ,x_{n}]]$ , the formal power series ring over $k$ in $n$ variables.^[4]

Closed submodules

The $𝔞$ -adic closure of a submodule $N\subseteq M$ is ${\textstyle {\overline {N}}=\bigcap _{n>0}{(N+{\mathfrak {a}}^{n}M)}{\text{.}}}$ ^[5] This closure coincides with $N$ whenever $R$ is $𝔞$ -adically complete and $M$ is finitely generated.^[6]

$R$ is called Zariski with respect to $𝔞$ if every ideal in $R$ is $𝔞$ -adically closed. There is a characterization:

R

is Zariski with respect to

𝔞

if and only if

𝔞

is contained in the Jacobson radical of

R

.

In particular a Noetherian local ring is Zariski with respect to the maximal ideal.^[7]

References

^ ^a ^b Singh 2011, p. 147.
^ Singh 2011, p. 148.
^ Singh 2011, pp. 148–151.
^ Singh 2011, problem 8.16.
^ Singh 2011, problem 8.4.
^ Singh 2011, problem 8.8
^ Atiyah & MacDonald 1969, p. 114, exercise 6.

Sources

Singh, Balwant (2011). Basic Commutative Algebra. Singapore/Hackensack, NJ: World Scientific. ISBN 978-981-4313-61-2.
Atiyah, M. F.; MacDonald, I. G. (1969). Introduction to Commutative Algebra. Reading, MA: Addison-Wesley.

[FOOTNOTESingh2011147-1] Singh 2011, p. 147.

[FOOTNOTESingh2011148-2] Singh 2011, p. 148.

[FOOTNOTESingh2011148–151-3] Singh 2011, pp. 148–151.

[4] Singh 2011, problem 8.16.

[5] Singh 2011, problem 8.4.

[6] Singh 2011, problem 8.8

[7] Atiyah & MacDonald 1969, p. 114, exercise 6.

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