nLab Demazure, lectures on p-divisible groups, I.7. operations on formal schemes

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• Michel Demazure, lectures on p-divisible groups, web

A finite limit of $k$-formal schemes is $k$-formal. The category of $k$-formal schemes has arbitrary products.

A formal $k$-scheme is called local $k$-scheme if it is the formal spectrum of a local ring (=has a unique maximal ideal).

Any formal $k$-scheme is a coproduct of local $k$-schemes (Demazure, p.15).