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…)