The Basic Calculus of Unimodular Rows

Abstract

The first part of this chapter presents two elementary proofs of Serre’s Conjecture, due to Suslin and Vaserstein respectively, that were found shortly after the first solutions of the Conjecture were given by Quillen and Suslin in 1976. These “elementary” proofs are both formulated in the language of unimodular rows, so their main focus is on stably free modules rather than general projective modules. Both proofs are highly ingenious and delightful, and should be of interest to anyone seeking a thorough understanding of Serre’s Conjecture and the variety of its many possible solutions. The presentation of these two elementary proofs occupies §1 and §2, which are followed by a section (§3) on the existence of monic polynomials (“Suslin’s Monic Polynomial Theorem”) in certain polynomial ideals.