I have finished reading Commutative Algebra / Atiyah-Macdonald. Reviewing, I found that I don't know what completions are good for.
As a comparison, localization at a prime ideal $\mathfrak{p}$ of a ring $R$ allows me to focus attention on prime ideals contained in $\mathfrak{p}$. This was clearly useful when proving Going-up and Going-down, and in other places.
But what are completions of rings good for (I'm talking about $\mathfrak{a}$-adic completions of rings or modules for an ideal $\mathfrak{a}$)? Do they also allow me to focus attention on some information? If so, what information?
An answer does not have to be completely general (i.e. you can assume the ring if noetherian or local or whatever helps, or maybe that $\mathfrak{a}$ is a maximal ideal).
I've studied classical algebraic geometry and some scheme theory. I prefer an answer of an algebraic nature, but a geometric answer is acceptable as well.
What (algebraic) information about a ring or module becomes more accessible after $\mathfrak{a}$-adic completion?
Maybe the answer is that I'm looking for the wrong type of utility from completions. Maybe it's not about focusing attention on some information. That's an answer too if true (Maybe completions are more like algebraic closures of fields? That is, map a ring to a nicer ring, where more things work, and hopefully conclude something about the original ring?)
Best Answer
This answer is in the context of Noetherian local rings, where we complete at the maximal ideal. Some of the words below are beyond the level of Atiyah-Macdonald. This is, of course, not an exhaustive list of the utility of completions, but just some things that I know and love about completions.
One extremely useful fact about completions is the Cohen structure theorem. A complete local ring is module-finite over a power series ring over a field (or DVR in the mixed characteristic case) which makes many facts easy to discern. Then, many properties descend from the completion to the ring itself, since the completion map is faithfully flat. For example, local duality is often most neat in the setting of complete rings, and in many cases one can freely complete to study the problem.
Another fact is that completions can sometimes detect singularities in more simplistic ways. The nodal cubic $R=\mathbb{R}[x,y]/(x^2-y^2-y^3)$ is a domain, but has an isolated singularity at the maximal ideal $(x,y)$, corresponding to the point $(0,0)$ of self-intersection of the curve $x^2=y^2+y^3$ -- one can see this with the Jacobian criterion, for example. The completion $S=\widehat{R}$ is isomorphic to $\mathbb{R}[[x,y]]/J$, where $J = \left(x+y\sqrt{1+y}\right)\left(x-y\sqrt{1+y}\right)$, so $S$ is clearly not a domain and cannot be regular. So, in some sense, the completion is capturing geometric information about branches of the curve that does not seem to be readily apparent in the original ring without appealing to high-power theorems.
In this example, $R$ was a domain but $\widehat{R}$ was not since it has two minimal primes. In general, if $\varphi:R\rightarrow S$ is a faithfully flat map, then $\tilde{\varphi}:\operatorname{Spec}(S)\rightarrow \operatorname{Spec}(R)$ given by $\tilde{\varphi}(\mathfrak{p}) = \varphi^{-1}(\mathfrak{p})$ is a surjective map, so if $\widehat{R}$ is a domain, then $R$ must be a domain as well.
If $(R,\mathfrak{m})$ is our local ring, and you are studying a finite length module over $R$, then it is often extremely useful to pass to the completion of $R$ because the class of finite length modules over $R$ and $\widehat{R}$ are the same. So, for example, if one is studying $\mathfrak{m}$-primary ideals, one can complete to obtain a ``nicer" ring as you note, prove something about the $\mathfrak{m}$-primary ideals of $\widehat{R}$ which may be simpler since $\widehat{R}$ is complete, and then use that there is a natural one-to-one correspondence between the set of $\mathfrak{m}$-primary ideals of $R$ and $\mathfrak{m}\widehat{R}$-primary ideals of $\widehat{R}$.
One thing I use often about the completion is that the local cohomology modules of $R$ and $\widehat{R}$ supported at $\mathfrak{m}$ are the same. In other words, there is a canonical isomorphism $H^j_\mathfrak{m}(R) \simeq H^j_{\mathfrak{m} \widehat{R}}(\widehat{R})$, and again the local cohomology modules of $\widehat{R}$ might be a priori easier to study because $\widehat{R}$ is complete. This says that natural properties like depth and Cohen-Macaulayness are the same for $R$ and $\widehat{R}$.
In summary, many more things are true of complete rings than non-complete rings, and facts about the completion can often ``descend" to the base ring.
A downside to completions is that it can often be very difficult to know exactly what a completion might look like. For example, even in the nice case of $R=K[x_1,\cdots,x_n]/(f_1,\cdots,f_r)$, it can be quite difficult to explicitly write out $\widehat{R}$ as a quotient of a power series. So, completion is often more useful ``in the abstract," rather than in the specific case.