Some results in obstruction theory for projective modules

Abstract

Let A be a commutative noetherian ring and P be a finitely generated projective A-module. It is known that P can be written as Q⊕ A when rank(P) > dim( A). But this level of generality is far from sufficient when rank(P) = dim(A). The notion of the Euler Class group was developed to address this case, and it is known, at this level of generality, that when the Euler class of P vanishes, P can be written as Q⊕ A.

In this dissertation, we look at overrings of polynomial rings, B = A[X,1/f] where A is a commutative noetherian ring of dimension ≥2 and f is a non-zero divisor of A[X] so that dim(B) = dim(A[X]), and show, through the use of the Euler Class group, that every finitely generated projective B-module, P, with rank (P) = dim(B) can be written as Q⊕B.

We also prove, for a commutative noetherian ring A, which is the image of a regular ring, the equivalence of several conditions to ensure the vanishing of the entire Euler Class group over A, which will again indicate that finitely generated projective A-modules can be written as Q⊕A. We also give similar results for geometrically reduced affine algebras over an infinite field.

This dissertation also looks at the history behind the development of the Euler Class group for inspiration towards future development. Several examples are also given to show some of the difficulties that come in the process of generalization.