Quillen-Suslin Theory

Let R be a commutative ring. A.A. Suslin proved that a unimodular row of length r+1, which is a factorial row, can be completed to an invertible matrix. In general, given two unimodular rows v, w of length r + 1 with v,w = 1, Suslin described an inductive procedure for constructing the Suslin matrix, whose size can be reduced by elementary row and column operations to get a matrix of determinant 1, which is a completion of a factorial unimodular row. This book begins with the study of Suslin matrices and define the Special Unimodular Vector Group (Gr) and Elementary Unimodular Vector Group (Hr). We establish the Quillen-Suslin Theory for the pair (Gr,Hr). We obtain the Fundamental Property of Suslin Matrices, which gives an action of Gr on the Suslin space of matrices, when r is even. This action induces an injective homomorphism from the quotient group of Gr to the quotient group of the Special Orthogonal group. Under this map Hr is mapped onto the Elementary Orthogonal group. Due to this, questions concerning unimodular rows can be reduced to the corresponding questions regarding special orthogonal matrices. Finally, we show that Gr/Hr is nilpotent of class dim(R).