I am going to get back shortly to discussing algebraic number theory and discrete valuation rings. But this tidbit from EGA 1 that I just learned today was too much fun to resist. Besides, it puts the material on completions in more context, so I think the digression is justified.
The theorem says we can lift “approximate idempotents” in complete rings to actual ones. In detail:
Theorem 1 Let be a ring complete with respect to the -adic filtration. Then if is idempotent (i.e. ) then there is an idempotent such that reduces to .
More elegantly, if for a ring denotes the set of idempotents, we have surjective.
The proof I knew of earlier is a fairly straightforward application of Hensel’s lemma, that more general result which I plan to cover in the future. But there is a proof using (a very little bit of) algebraic geometry.
The first step is to prove:
Proposition 2 Let be a ring and a nilpotent ideal. Then is surjective.
Indeed, the topological spaces of and are the same. The result then follows from the next section.
Idempotents and Connectedness
Idempotents measure the disconnectedness of for a ring :
Proposition 3 If , then there is a one-to-one correspondence between and the open and closed subsets of .
Given an open and closed , we have . Since , we can define a global section of the structure sheaf by on , on .
Similarly, if is an idempotent, then must be either or for , because there are no nontrivial idempotents in a local ring. (If were a nontrivial idempotent in a local ring, then , and either or is necessarily invertible.) So we can set . It can be checked that is open, and so is by symmetry. This establishes the bijection I claimed. (Another approach here is to note that idempotents decompose as a product in the category of rings, which corresponds by fully faithful contravariantness the contravariant equivalence of to a coproduct in the category of affine schemes. Except one has to check that an open and closed subscheme of an affine scheme is itself affine. Perhaps this is easy, but a quick proof wasn’t obvious to me at least.)
So the first proposition is now proved. Finally, we need to prove the theorem. Well, choose an idempotent . Lift it to , and inductively to . In the inverse limit, we get an idempotent reducing to .
There are some useful applications of this in representation theory, because one can look for idempotents in endomorphism rings; these tell you whether a module can be decomposed as a direct sum into smaller parts. Except, of course, that endomorphism rings aren’t necessarily commutative and this proof breaks down.