Multiplicities in Commutative Algebra

Abstract

This dissertation explores the notion of multiplicity and its generalizations within the theory of commutative algebra. Chapter 2 is dedicated to calculating the limits which give rise to Buchsbaum-Rim multiplicities. We provide a formula for the multiplicity of a module in terms of its summands. An example demonstrates that multiplicity, dimension and rank alone are insufficient to give the multiplicity of a specified power of the module. In the following chapter, we examine lengths given by a filtration, rather than powers of an ideal. We consider filtrations as generalizations of both valuations and ideal powers and identify a class of Rees valuations as Noetherian filtrations. We define a general $j$-multiplicity for Noetherian filtrations and prove some of its characteristics. Chapter 4 is dedicated to ideals of submaximal analytic spread which remain nonzero when we apply the local cohomology functor. The final chapter is joint work with Tony Se. We analyze subrings of two-variable polynomial rings generated over a field, $k$, by monomials. We explicitly describe the asympototic behavior of a system of parameters and relate this to the Cohen-Macaulay property of the ring.