I-adic Topology and Completion

Completion of a ring at an ideal provides a way to study local algebraic properties by passing to a "limit" that captures all information about successive approximations modulo powers of the ideal.

The I-adic Topology

Definition

Let $R$ be a commutative ring and $I \subseteq R$ an ideal. The $I$ -adic topology on $R$ is the topology in which the sets $\{a + I^n : a \in R, n \geq 0\}$ form a basis. The ring $R$ becomes a topological ring: addition and multiplication are continuous. A sequence $(a_n)$ converges to $a$ if and only if for every $k \geq 0$ , there exists $N$ such that $a_n - a \in I^k$ for all $n \geq N$ .

Definition

The $I$ -adic completion of $R$ is the inverse limit $\hat{R} = \varprojlim_{n} R/I^n = \left\{(a_n)_{n \geq 0} \in \prod_{n \geq 0} R/I^n : a_{n+1} \equiv a_n \pmod{I^n}\right\}$ There is a natural ring homomorphism $\iota : R \to \hat{R}$ with $\ker \iota = \bigcap_{n \geq 0} I^n$ . The ring $R$ is $I$ -adically complete if $\iota$ is an isomorphism.

Basic Properties

Example$p$-adic integers

The $(p)$ -adic completion of $\mathbb{Z}$ is the ring of $p$ -adic integers $\mathbb{Z}_p = \varprojlim \mathbb{Z}/p^n\mathbb{Z}$ . This is an integral domain, a principal ideal domain, and a local ring with maximal ideal $(p)$ . Its field of fractions is the field of $p$ -adic numbers $\mathbb{Q}_p$ .

ExampleFormal power series

The $(x)$ -adic completion of the polynomial ring $k[x]$ is the formal power series ring $k[[x]]$ . More generally, the $(x_1, \ldots, x_n)$ -adic completion of $k[x_1, \ldots, x_n]$ is $k[[x_1, \ldots, x_n]]$ .

Completion of Modules

RemarkFlatness of completion

For a Noetherian ring $R$ and an ideal $I$ , the completion $\hat{R}$ is a flat $R$ -algebra. Moreover, the completion functor on finitely generated $R$ -modules is exact: if $0 \to M' \to M \to M'' \to 0$ is exact with finitely generated modules, then $0 \to \hat{M}' \to \hat{M} \to \hat{M}'' \to 0$ is exact. This exactness is crucial for many applications in local algebra.

Completion preserves many ring-theoretic properties and provides a bridge between algebra and analysis, as exemplified by the $p$ -adic numbers in number theory and formal power series in algebraic geometry.