Some preliminaries
In this section we define some of the terms used in the paper and state some results for later use. All rings considered in this paper are commutative and Noetherian and all modules considered are assumed to be finitely generated. For a module M over a ring, µ(M) will denote the minimal number of generators of M.
Definition 2.2. Let A be a Noetherian ring. Let P be a projective A-module. An element p ∈ P is said to be unimodular if there exists a linear map φ :
We now state a theorem of Serre [Se].
Theorem 2.3. Let A be a Noetherian ring with dim A = d. Then any projective A-module having rank > d has a unimodular element.
As an immediate consequence we have the following corollary.
Corollary 2.4. Let A be a Noetherian ring with dim A = 1. Then any projective A-module having trivial determinant is free.
The following lemma has been proved in [Bh].
Lemma 2.5. Let A be a ring and J ⊂ A be an ideal of height r. Let P , Q be projective A/J -modules of rank r and let α : P J/J 2 and β : Q J/J 2 be surjections. Let ψ : P → Q be a homomorphism such that βψ = α. Then ψ is an isomorphism.
The following lemma is easy to prove and hence we omit the proof.
Lemma 2.6. Let A be a Noetherian ring and P a finitely generated projective A-module.
Let P [T ] denote projective A[T ]-module P ⊗ A[T ]. Let α(T ) : P [T ] A[T ] and β(T ) : P [T ] A[T ] be two surjections such that α(0) = β(0). Suppose further that the projective A[T ]-modules ker α(T ) and ker β(T ) are extended from A. Then there exists an automorphism σ (T ) of P [T ] with σ (0) = id such that β(T )σ (T ) = α(T ).
The next lemma follows from the well-known Quillen Splitting Lemma [Qu, Lemma 1] and its proof is essentially contained in [Qu, Theorem 1].
Lemma 2.7. Let A be a Noetherian ring and P be a finitely generated projective A-module. Let s, t ∈ A be such that As + At = A. Let σ (T ) be an A st [T ]-automorphism of P st [T ] such that σ (0) = id. Then, σ (T ) = α(T ) s β(T ) t , where α(T ) is an A t [T ]-automorphism of P t [T ] such that α(T ) = id modulo the ideal (sT ) and β(T ) is an A s [T ]-automorphism of P s [T ] such that β(T ) = id modulo the ideal (tT ).