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 <math display=block>d(x,y) = 2^{-\sup{\{n \mid x-y\in\mathfrak{a}^nM\}}}.</math> The family <math display=block>\{x+\mathfrak{a}^nM:x\in M,n\in\mathbb{Z}^+\}</math> 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<math display=block>\bigcap_{n > 0}{\mathfrak{a}^nM} = 0\text{,}</math>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 <math>\bigcap_{n > 0}{\mathfrak{a}^n} = 0</math> 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

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} When Template:Mvar is Hausdorff, Template:Mvar can be completed as a metric space; the resulting space is denoted by <math>\widehat M</math> and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to): <math display=block>\widehat{M} = \varprojlim M/\mathfrak{a}^n M</math> where the right-hand side is an inverse limit of quotient modules under natural projection.Template:Sfn

For example, let <math>R = k[x_1, \ldots, x_n]</math> be a polynomial ring over a field Template:Mvar and Template:Math the (unique) homogeneous maximal ideal. Then <math>\hat{R} = kx_1, \ldots, x_n</math>, the formal power series ring over Template:Mvar in Template:Mvar variables.<ref>Template:Harvnb, problem 8.16.</ref>

Closed submodules

The Template:Math-adic closure of a submodule <math>N \subseteq M</math> is <math display=inline>\overline{N} = \bigcap_{n > 0}{(N + \mathfrak{a}^n M)}\text{.}</math><ref>Template:Harvnb, problem 8.4.</ref> This closure coincides with Template:Mvar whenever Template:Mvar is Template:Math-adically complete and Template:Mvar is finitely generated.<ref>Template:Harvnb, problem 8.8</ref>

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.<ref>Template:Harvnb, exercise 6.</ref>

References

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Sources

• Template:Cite book
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