$I$-adic topology - Knowlpedia

Definition

Let $A$ be a commutative ring and $I\subseteq A$ an ideal. The $I$ -adic topology on $A$ is the ring topology for which

A\supseteq I\supseteq I^2\supseteq\cdots

is a neighborhood basis of $0$ . Thus a sequence $a_j$ converges to $0$ exactly when, for every $N$ , it eventually lies in $I^N$ .

For an $A$ -module $M$ , the $I$ -adic topology similarly has $\{I^NM\}_{N\geq0}$ as a neighborhood basis of $0$ .

Separation and completion

The topology on $A$ is separated (Hausdorff) precisely when

\bigcap_{N\geq0}I^N=0.

Separatedness does not by itself mean that every Cauchy system has a limit. The $I$ -adic completion $\widehat A=\varprojlim A/I^N$ measures and remedies that second issue, subject to a general-ring subtlety about which topology is placed on the completion.

Continuous maps

If $B$ has its $J$ -adic topology, a ring homomorphism $\varphi:A\to B$ is continuous exactly when, for every $m$ , there is an $n$ such that

\varphi(I^n)\subseteq J^m.

The common sufficient condition $\varphi(I)\subseteq J$ makes this immediate. Continuity is the condition needed for homomorphisms between formal power series rings and formal spectra.

Examples

The $0$ -adic topology is discrete.
The $x$ -adic topology on $R[[x]]$ records agreement to increasing order in $x$ .
For $A=\mathbb Z$ and $I=(p)$ , completion produces the ring of $p$ -adic integers.

References

The Stacks Project Authors, “Topological rings and modules.” Section 15.37, Tag 07E7. Relevant: Definition 15.37.1 and the discussion of $I$ -adic topologies.
Hideyuki Matsumura, Commutative Ring Theory, Cambridge University Press, 1986. Relevant: Section 8, completions.