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]

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

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]