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 -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 $\$ 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$\bigcap_ = 0\text$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 $\bigcap_ = 0$ 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

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

For example, let R = k _1, \ldots, x_n/math> be a polynomial ring over a field and the (unique) homogeneous maximal ideal. Then \hat = k x_1, \ldots, x_n, the formal power series ring over in variables.

Closed submodules

The -adic closure of a submodule $N \subseteq M$ is $\overline = \bigcap_\text$ This closure coincides with whenever is -adically complete and is finitely generated.

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., exercise 6.

References

Sources

*
* {{cite book, first=M. F., last=Atiyah, author-link1=Michael Atiyah, first2=I. G., last2=MacDonald, publisher=Addison-Wesley, location=Reading, MA, year=1969, title=Introduction to Commutative Algebra

category:Commutative algebra
category:Topology