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 number|Template:Mvar-adic topologies]] on the integers.

Definition

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

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

By Krull's intersection theorem, if Template:Mvar is a Noetherian ring which is an integral domain or a local ring, it holds that $\bigcap _{n>0}{𝔞}^n=0$ for any proper ideal Template:Mvar of Template:Mvar. Thus under these conditions, for any proper ideal Template:Mvar of Template:Mvar and any Template:Mvar-module Template:Mvar, the Template:Mvar-adic topology on Template:Mvar is separated.

For a submodule Template:Mvar of Template:Mvar, the canonical homomorphism to Template:Math induces a quotient topology which coincides with the Template:Math-adic topology. The analogous result is not necessarily true for the submodule Template:Mvar itself: the subspace topology need not be the Template:Math-adic topology. However, the two topologies coincide when Template:Mvar is Noetherian and Template:Mvar finitely generated. This follows from the Artin-Rees lemma.Template:Sfn

Completion

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

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

Closed submodules

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

Template:Mvar is called Zariski with respect to Template:Math if every ideal in Template:Mvar is Template:Math-adically closed. There is a characterization:

Template:Mvar is Zariski with respect to Template:Math if and only if Template:Math is contained in the Jacobson radical of Template:Mvar.

In particular a Noetherian local ring is Zariski with respect to the maximal ideal.

References

Sources

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