TITLE: Usefulness of completion in commutative algebra
QUESTION [21 upvotes]: After studying about the completion of a module $M$ over a ring $A$ (e.g. $I$-adic completion), I am left with the following questions:
(i) What is the usefulness of the concept of completion in commutative algebra or algebraic geometry, apart from the philosophical point of every Cauchy sequence being convergent?
(ii) What does it allow us to achieve on the technical level and what type of geometric insights does it lead to?

REPLY [15 votes]: Matt's answer gives some justification from a geometric perspective.  However, I thought you might benefit from an algebraic point of view as well.  From this perspective, the main point is something Matt mentioned in passing -- the Cohen structure theorem.
Suppose your domain of inquiry is Noetherian rings, and that you are interested in using homological tools.  An arbitrary Noetherian ring can have very strange structural properties.  However, Cohen's structure theorem says that if $R$ is a complete Noetherian local ring, then

$R$ is isomorphic to a quotient (factor ring) of a power series ring in finitely many variables over a field (or over a complete DVR, in the mixed characteristic case), and
If $R$ contains a field, then $R$ is module-finite over a regular local subring.  Indeed, if $x_1, \ldots, x_d$ is any system of parameters for $R$, and $k$ is the coefficient field for $R$ (i.e. a subfield of $R$ that is isomorphic to its residue field, which always exists, again by Cohen's results), then the subring $A=k[[x_1, \ldots, x_d]]$ of $R$ is isomorphic to a power series ring in $d$ variables, and $R$ is finite as an $A$-module.

These two facts are incredibly useful if you happen to be working with a complete ring.  But your question was about the utility of the completion process.  The point here is that when $R$ is a local Noetherian ring, the morphism $R \rightarrow \hat R$ is faithfully flat, and for any finite $R$-module $M$ we have $\hat M \cong M \otimes_R \hat R$.  This means that to find homological (and other) information about $R$, its modules, and its ideals, one can often pass to the completion, which (as outlined above) is typically a much easier ring to work with than $R$ itself.