Hilbert functions of Cohen -Macaulay modules

Tony Joseph Puthenpurakal, Purdue University

Abstract

Let (A, [special characters omitted]) be a d-dimensional Noetherian local ring, M a finite Cohen-Macaulay A-module of dimension r and let I be an ideal of definition for M. In Chapter 2 we define the notion of minimal multiplicity of Cohen-Macaulay modules with respect to I and show that if M has minimal multiplicity with respect to I then the associated graded module GI(M) is Cohen-Macaulay. In Chapter 3 we assume that A is Cohen-Macaulay, M is maximal Cohen-Macaulay and I is [special characters omitted]-primary. We find a relation between the first Hilbert coefficient of M, A and [special characters omitted](M). Sharper results are found when I = [special characters omitted]. Set [special characters omitted](M) = e1(M) − e0(M) + μ(M). We prove that [special characters omitted] When A is Gorenstein then M is the first syzygy of SA(M) = ([special characters omitted](M*))*. A relation between the second Hilbert coefficient of M, A and SA( M) is found when G(M) is Cohen-Macaulay and depth G(A) ≥ d − 1. In Chapter 4 we study Hilbert functions of maximal Cohen-Macaulay modules over a hypersurface ring A. We show that if d > 0 then the Hilbert function of M with respect to [special characters omitted] is non-decreasing. If A = Q/( f) for some regular local ring Q, we determine a lower bound for e0(M) and e1(M). Furthermore we analyze the case when equality holds and prove that in this case G( M) is Cohen-Macaulay. Furthermore in this case we also determine the Hilbert function of M.

Degree

Ph.D.

Advisors

Avramov, Purdue University.

Subject Area

Mathematics

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