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The formal functions have been introduced and studied systematically in algebraic geometry by Zariski in [151]. In particular, Zariski used them to prove his famous connectedness theorem. Later on this theory was considerably extended and deepened by Grothendieck in the framework of formal schemes, by providing powerful tools (such as Grothendieck's existence theorem, see [66] III, Theorem (5.1.4)) to study related questions (see [67], [68]).

Let K be an algebraically closed field of arbitrary characteristic. The term “variety” always means here an irreducible algebraic variety over K. The notations and the terminology are borrowed in general from EGA [4].

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