Properties of Complete Rings

Complete local rings enjoy many strong structural properties not shared by arbitrary Noetherian rings. These properties make completion an indispensable tool in both commutative algebra and algebraic geometry.

Hensel's Lemma

Definition

A local ring $(R, \mathfrak{m})$ is Henselian if it satisfies Hensel's lemma: whenever $f(x) \in R[x]$ is a monic polynomial and $\bar{f}(x) = \bar{g}(x)\bar{h}(x)$ in $(R/\mathfrak{m})[x]$ with $\gcd(\bar{g}, \bar{h}) = 1$ , then there exist monic polynomials $g, h \in R[x]$ with $f = gh$ and $g \equiv \bar{g}$ , $h \equiv \bar{h}$ modulo $\mathfrak{m}$ .

Theorem7.3Hensel's Lemma

Every complete local ring $(R, \mathfrak{m})$ is Henselian. In particular, if $f(x) \in R[x]$ and $a \in R$ satisfies $f(a) \in \mathfrak{m}$ and $f'(a) \notin \mathfrak{m}$ (i.e., $\bar{a}$ is a simple root of $\bar{f}$ ), then there exists a unique $b \in R$ with $f(b) = 0$ and $b \equiv a \pmod{\mathfrak{m}}$ .

ExampleSquare roots in $\mathbb{Z}_p$

In $\mathbb{Z}_7$ , consider $f(x) = x^2 - 2$ . Since $3^2 = 9 \equiv 2 \pmod{7}$ and $f'(3) = 6 \not\equiv 0 \pmod{7}$ , Hensel's lemma guarantees a unique $b \in \mathbb{Z}_7$ with $b^2 = 2$ and $b \equiv 3 \pmod{7}$ . Thus $\sqrt{2} \in \mathbb{Q}_7$ , even though $\sqrt{2} \notin \mathbb{Q}$ .

Completeness and Noetherian Property

Theorem7.4Completions are Noetherian

If $R$ is a Noetherian ring and $I$ an ideal, then the $I$ -adic completion $\hat{R}$ is Noetherian. Moreover, $\hat{R}/I\hat{R} \cong R/I$ and the natural map $I^n \hat{R} \to \hat{I}^n$ is an isomorphism.

RemarkFaithful flatness

The completion map $R \to \hat{R}$ for a Noetherian local ring $(R, \mathfrak{m})$ is faithfully flat: a finitely generated $R$ -module $M$ is zero if and only if $\hat{M} = M \otimes_R \hat{R}$ is zero. This means properties like depth, regularity, and Cohen-Macaulay-ness can be checked after completion. In particular, $R$ is regular $\iff$ $\hat{R}$ is regular, and $\dim R = \dim \hat{R}$ .

Faithful flatness of completion means that many questions in local algebra can be reduced to the complete case, where the powerful structure theory (Cohen's theorem) is available.