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.

Lifting Idempotents

The theorem says we can lift “approximate idempotents” in complete rings to actual ones. In detail:

Theorem 1 Let ${A}$ be a ring complete with respect to the ${I}$-adic filtration. Then if ${\bar{e} \in A/I}$ is idempotent (i.e. ${\bar{e}^2=\bar{e}}$) then there is an idempotent ${ e \in A}$ such that ${e}$ reduces to ${\bar{e}}$  (more…)