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 1Let 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…)

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