Suslin's algorithms for reduction of unimodular rows

摘要

A well-known lemma of Suslin says that for a commutative ring A if (v_1(X),…, v_n(X)) ∈ (A[X])~n is unimodular where v_1 is monic and n ≥ 3, then there exist γ_1,…, γ_l ∈ E_(n-1) (A[X]) such that the ideal generated by Res(v_1, e_1.γ_1 ~t(v_2,….,v_n)),…, Res(v_1,e_1.γ_l ~t(v_2,…, v_n)) equals A. This lemma played a central role in the resolution of Serre's Conjecture. In the case where A contains a set E of cardinality greater than deg v_1 + 1 such that y — y′ is invertible for each y ≠ y′ in E, we prove that the γ_i can simply correspond to the elementary operations L_1 → L_1 + y_1 ∑_(J=2)~(n-1)u _(j+1) L_j, 1 ≤ i ≤ l = deg v_1 + 1, where u_1 v_1 + …+ u_n v_n = 1. These efficient elementary operations enable us to give new and simple algorithms for reducing unimodular rows with entries in K[X_1…, X_k] to ~t(1,0,…, 0) using elementary operations in the case where K is an infinite field. Another feature of this paper is that it shows that the concrete local-global principles can produce competitive complexity bounds.