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

$${\displaystyle d(x,y)=2^{-\sup {\{n\mid x-y\in {\mathfrak {a}}^{n}M\}}}.}$$

The family

$${\displaystyle \{x+{\mathfrak {a}}^{n}M:x\in M,n\in \mathbb {Z} ^{+}\}}$$

is a basis for this topology. ^[1]

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

$${\displaystyle \bigcap _{n>0}{{\mathfrak {a}}^{n}M}=0{\text{,}}}$$

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 ${\displaystyle \bigcap _{n>0}{{\mathfrak {a}}^{n}}=0}$ 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

Main article: Completion (algebra)

When M is Hausdorff, M can be completed as a metric space; the resulting space is denoted by ${\displaystyle {\widehat {M}}}$ and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to):

$${\displaystyle {\widehat {M}}=\varprojlim M/{\mathfrak {a}}^{n}M}$$

where the right-hand side is an inverse limit of quotient modules under natural projection. ^[3]

For example, let ${\displaystyle R=k[x_{1},\ldots ,x_{n}]}$ be a polynomial ring over a field k and 𝔞 = (x₁, ..., x_n) the (unique) homogeneous maximal ideal. Then ${\displaystyle {\hat {R}}=k[[x_{1},\ldots ,x_{n}]]}$, the formal power series ring over k in n variables. ^[4]

Closed submodules

As a consequence of the above, the 𝔞-adic closure of a submodule ${\displaystyle N\subseteq M}$ 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

1. Singh 2011, p. 147.
2. Singh 2011, p. 148.
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.

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.