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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).

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