Formal Properties of Completion

Completion preserves and reflects many algebraic properties of Noetherian local rings, making it a central tool for reducing problems to the well-understood complete case.

Preservation of Ring Properties

Theorem7.10Preservation Under Completion

Let $(R, \mathfrak{m})$ be a Noetherian local ring and $\hat{R}$ its $\mathfrak{m}$ -adic completion. Then:

$\dim \hat{R} = \dim R$
$\operatorname{depth} \hat{R} = \operatorname{depth} R$
$R$ is regular $\iff$ $\hat{R}$ is regular
$R$ is Cohen-Macaulay $\iff$ $\hat{R}$ is Cohen-Macaulay
$R$ is Gorenstein $\iff$ $\hat{R}$ is Gorenstein
$R$ is a domain $\implies$ $\hat{R}$ is a domain (but not conversely in general)

The proofs rely on faithful flatness of $R \to \hat{R}$ and the isomorphism $\hat{R}/\mathfrak{m}\hat{R} \cong R/\mathfrak{m}$ .

ExampleCompletion can create zero divisors

The local ring $R = k[x, y]_{(x,y)} / (xy)$ is not a domain. Its completion $\hat{R} = k[[x, y]]/(xy)$ is also not a domain. However, even for domains, completion can introduce complications: $R = \mathbb{R}[x, y]_{(x,y)} / (x^2 + y^2)$ is a domain (since $x^2 + y^2$ is irreducible over $\mathbb{R}$ ), but $\hat{R} = \mathbb{R}[[x,y]]/(x^2+y^2) \cong \mathbb{R}[[x,y]]/((x+iy)(x-iy))$ has zero divisors after extending to $\mathbb{C}$ .

Completion and Tensor Products

Theorem7.11Completed Tensor Product

For finitely generated modules $M$ over a Noetherian local ring $(R, \mathfrak{m})$ : $\hat{M} \cong M \otimes_R \hat{R}$ Moreover, $\hat{R} \otimes_R (M/N) \cong \hat{M}/\hat{N}$ for any submodule $N \subseteq M$ .

RemarkFormal geometry

In algebraic geometry, completion at a point $x \in X$ of a variety gives the formal neighborhood $\hat{\mathcal{O}}_{X,x}$ , which captures the infinitesimal structure of $X$ near $x$ . Two varieties have the same formal neighborhood at corresponding points if and only if they are locally analytically isomorphic. This is formalized in Grothendieck's theory of formal schemes, where completion plays the role that localization plays in scheme theory.

The interplay between algebraic properties of $R$ and $\hat{R}$ is at the heart of deformation theory, formal geometry, and the study of singularities in algebraic geometry.