I-adic topology explained

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 -adic topologies on the integers.

Definition

Let be a commutative ring and an -module. Then each ideal of determines a topology on called the -adic topology, characterized by the pseudometric $$d(x,y)\; =\; 2^.$$ The family $$\backslash$$ is a basis for this topology.

An -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 becomes a topological module. However, need not be Hausdorff; it is Hausdorff if and only if$$\backslash bigcap\_\; =\; 0\backslash text$$so that becomes a genuine metric. Related to the usual terminology in topology, where a Hausdorff space is also called separated, in that case, the -adic topology is called separated.

By Krull's intersection theorem, if is a Noetherian ring which is an integral domain or a local ring, it holds that

c ap _{n} { ak{a} ^{n}} = 0 for any proper ideal of . Thus under these conditions, for any proper ideal of and any -module, the -adic topology on is separated.

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

Completion

See main article: Completion (algebra). When is Hausdorff, can be completed as a metric space; the resulting space is denoted by

\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): $$\backslash widehat\; =\; \backslash varprojlim\; M/\backslash mathfrak^n\; M$$ where the right-hand side is an inverse limit of quotient modules under natural projection.

For example, let

R = k[x _{1,} \ldots, x _{n]} be a polynomial ring over a field and the (unique) homogeneous maximal ideal. Then \hat{R} = k[[x _{1,} \ldots, x _{n]]} , the formal power series ring over in variables.^[1]

Closed submodules

The -adic closure of a submodule

N \subseteq M is $\backslash overline\; =\; \backslash bigcap\_\backslash text$^[2] This closure coincides with whenever is -adically complete and is finitely generated.^[3]

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

is Zariski with respect to if and only if is contained in the Jacobson radical of .In particular a Noetherian local ring is Zariski with respect to the maximal ideal.^[4]

References

1. , problem 8.16.
2. , problem 8.4.
3. , problem 8.8
4. , exercise 6.

Sources

• Book: Singh, Balwant. Basic Commutative Algebra. 2011. World Scientific. Singapore/Hackensack, NJ. 978-981-4313-61-2.
• Book: Atiyah, M. F.. Michael Atiyah. I. G.. MacDonald. Addison-Wesley. Reading, MA. 1969. Introduction to Commutative Algebra.