Ratliff-Rush closures, coefficient modules, and Rees algebras of modules

Jung-Chen Liu, Purdue University

Abstract

Let (R, m) be a d-dimensional Noetherian local domain. Suppose M is a finitely generated torsion-free R-module and F is a free R-module containing M. In analogy with a result of Ratliff and Rush [RR] concerning ideals, in the first part of Chapter 1, we define and prove existence and uniqueness of the Ratliff-Rush closure of M in F. We also discuss properties of Ratliff-Rush closure. In the second part of Chapter 1, in addition to the assumptions above, suppose F/M has finite length as an R-module. Then we define the Buchsbaum-Rim polynomial of M in F. In analogy with work of K. Shah [Sh], we define coefficient modules of M in F. Under the assumption that R is quasi-unmixed, we prove existence and uniqueness of coefficient modules of M in F. In Chapter 2, we concentrate in the case where (R, m) is a two-dimensional regular local ring. Let M be a finitely generated torsion-free R-module. If M is a complete module, then Katz and Kodiyalam show M satisfies five conditions, one of these being that the Rees algebra S( M) of M is Cohen-Macaulay and another being that the “associated graded ring” S(M)/ IS(M) of M is Cohen-Macaulay. They ask whether these five conditions are equivalent without assuming M to be complete. We investigate implications among these five properties in the case where M is not complete. We prove in general that the depth of S(M) is greater than or equal to the depth of S(M)/IS( M), and that if a module has reduction number at most one, then a direct summand also has reduction number at most one. We present an example where M is a direct sum of two submodules each of which has reduction number at most one while M has reduction number at least two.

Degree

Ph.D.

Advisors

Heinzer, Purdue University.

Subject Area

Mathematics

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