On a theorem of Faltings on formal functions
Abstract
In 1980, Faltings proved, by deep local algebra methods, a local resultregarding formal functions which has the following global geometric fact
as a consequence. Theorem. − Let k be an algebraically closed field (of
any characteristic). Let Y be a closed subvariety of a projective irreducible
variety X defined over k. Assume that X ⊂ P^n , dim(X) = d > 2 and Y
is the intersection of X with r hyperplanes of P^n , with r ≤ d − 1. Then,
every formal rational function on X along Y can be (uniquely) extended to
a rational function on X . Due to its importance, the aim of this paper is to
provide two elementary global geometric proofs of this theorem.
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