I-adic topology

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.

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.

Completion

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

For example, let ${\displaystyle R=k[x_1,\dots ,x_n]}$ be a polynomial ring over a field k and 𝔞 = (x₁, ..., x_n) the (unique) homogeneous maximal ideal. Then ${\displaystyle \hat{R}=k[[x_1,\dots ,x_n]]}$, the formal power series ring over k in n variables.

Closed submodules

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

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.