Cores, Rees valuations and indecomposable modules over a two-dimensional regular local ring

Radha Mohan, Purdue University

Abstract

In chapter 1, we explicitly determine the core of a finitely generated, torsion-free, integrally closed module over a two-dimensional regular local ring. The main result asserts that the core of a finitely generated, torsion-free, integrally closed module over a two-dimensional regular local ring is the product of the module and the r + 1-st Fitting ideal of the module, where r is the rank of the module. This result is analogous to a result of Huneke and Swanson which determines the core of an integrally closed ideal. We make use of results of Kodiyalam in applying the technical tools of quadratic transforms and Buchsbaum-Rim multiplicity. In analogy with work of Lipman involving the adjoint of an integrally closed ideal, we define the adjoint of an integrally closed or complete module. We prove that given any complete module and a minimal reduction of this module the quotient module is independent of the reduction chosen. In chapter 2 we show that the Rees valuation rings of a finitely generated, torsion-free module over a two-dimensional regular local ring are precisely the Rees valuation rings of the ideal which is the r-th Fitting Invariant of the module, where r is the rank of the module. As before the main tools are quadratic transforms and Buchsbaum-Rim multiplicity. In chapter 3 we show that the possible ordered pairs of integers that can occur as the 0-th and 1-st Betti numbers of some finitely generated, torsion-free complete, indecomposable module, over a two-dimensional regular local ring, have the form ($t,t-r$) where $t\ge2r$ and $r\ge1$. To do this we construct finitely generated, complete indecomposable modules of rank r minimally generated by t elements where $t\ge2r$.

Degree

Ph.D.

Advisors

Heinzer, Purdue University.

Subject Area

Mathematics

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