$I$-adic completion

Core idea

Let $A$ be a commutative ring and $I\subseteq A$ an ideal. Its $I$ -adic completion is the inverse limit

\widehat A^{\,I}:=\varprojlim_{n\geq1}A/I^n

with respect to the quotient maps $A/I^{n+1}\to A/I^n$ . The canonical map

A\longrightarrow\widehat A^{\,I},\qquad a\longmapsto(a\bmod I^n)_n

has kernel $\bigcap_n I^n$ .

What an element records

An element of $\widehat A^{\,I}$ is a compatible system $(a_n)_{n\geq1}$ with $a_n\in A/I^n$ . It records an element to every finite $I$ -adic order, whether or not one compatible representative already lies in $A$ . The completion carries the inverse-limit topology with kernels

\ker(\widehat A^{\,I}\to A/I^n)

as a neighborhood basis of zero; it is complete and separated for this topology.

Important general-ring caution

In well-behaved settings, such as a Noetherian ring with any ideal $I$ , the inverse-limit topology agrees with the $I\widehat A$ -adic topology and the completion operation is idempotent. For arbitrary non-Noetherian rings and non-finitely generated ideals, these assertions can fail. Thus “ $\widehat A$ is complete” should specify the inverse-limit topology unless hypotheses identifying it with the $I$ -adic topology are available.

Examples

Completing $R[x]$ along $(x)$ gives $R[[x]]$ .
Completing $R[X_1,\ldots,X_n]$ along $(X_1,\ldots,X_n)$ gives $R[[X_1,\ldots,X_n]]$ .
Completing $\mathbb Z$ along $(p)$ gives $\mathbb Z_p$ .

Geometric role

Completion along an ideal keeps all infinitesimal neighborhoods of the closed subscheme cut out by $I$ . Its formal-geometric avatar is the formal spectrum $\operatorname{Spf}(\widehat A^{\,I})$ .

References

The Stacks Project Authors, “Topological rings and modules.” Section 15.37, Tag 07E7. Relevant: completion and the distinction between limit and adic topologies.
Hideyuki Matsumura, Commutative Ring Theory, Cambridge University Press, 1986. Relevant: Section 8, completions and the Noetherian case.