Making the use of maximal ideals constructive

摘要

The purpose of this paper is to decipher constructively a lemma of Suslin which played a central role in his second solution of Serre's problem on projective modules over polynomial rings. This lemma says that for a commutative ring A if {v{sub}1(X),..., v{sub}n(X)} = A[X] where v{sub}1 is monic and n ≥ 3, then there exist γ{sub}1,..., γ{sub}l ∈ E{sub}(n-1)(A[X]) such that, denoting by ω{sub}i the first coordinate of γ{sub}it{sup left}(v{sub}2, ..., v{sub}n), we have = A. By the constructive proof we give, Suslin's proof of Serre's problem becomes fully constructive. Moreover, the new method with which we treat this academic example may be a model for miming constructively abstract proofs in which one works modulo a generic maximal ideal in order to prove that an ideal contains 1.