Hilbert functions of ideals in Cohen-Macaulay rings
Abstract
The Hilbert function of a zero-dimensional ideal in a d-dimensional Cohen-Macaulay local ring is studied when the depth of the associated graded ring of the ideal is at least d $-$ 1. In this case it is shown that the Hilbert function, Hilbert polynomial, and Hilbert coefficients are very well-behaved, while examples of bad behavior are given when the depth condition is not satisfied. Also, an interesting relationship between the reduction number and the Hilbert coefficients is demonstrated. Later, these results are generalized to a wider class of functions, especially one studied by Rees which involves the integral closure of power of an ideal. The last chapter deals with the unrelated topic of Bass numbers of a finitely generated module over an arbitrary local ring. It is proved that unmixed local rings of type two are Cohen-Macaulay, answering a question posed by Costa, Huneke and Miller.
Degree
Ph.D.
Advisors
Huneke, Purdue University.
Subject Area
Mathematics
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