Physical Sciences and Mathematics Commons^™

Stable Cohomology Of Local Rings And Castelnuovo-Mumford Regularity Of Graded Modules, Luigi Ferraro

Department of Mathematics: Dissertations, Theses, and Student Research

This thesis consists of two parts:

1) A bimodule structure on the bounded cohomology of a local ring (Chapter 1),

2) Modules of infinite regularity over graded commutative rings (Chapter 2).

Chapter 1 deals with the structure of stable cohomology and bounded cohomology. Stable cohomology is a $\mathbb{Z}$-graded algebra generalizing Tate cohomology and first defined by Pierre Vogel. It is connected to absolute cohomology and bounded cohomology. We investigate the structure of the bounded cohomology as a graded bimodule. We use the information on the bimodule structure of bounded cohomology to study the stable cohomology algebra as a trivial extension …

Algebraic Properties Of Ext-Modules Over Complete Intersections, Jason Hardin

Department of Mathematics: Dissertations, Theses, and Student Research

We investigate two algebraic properties of Ext-modules over a complete intersection R of codimension c. Given an R-module M, Ext(M,k) can be viewed as a graded module over a polynomial ring in c variables with an action given by the Eisenbud operators. We provide an upper bound on the degrees of the generators of this graded module in terms of the regularities of two associated coherent sheaves. In the codimension two case, our bound recovers a bound of Avramov and Buchweitz in terms of the Betti numbers of M. We also provide a description of the differential graded (DG) R-module …

Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer

Department of Mathematics: Dissertations, Theses, and Student Research

This work is primarily concerned with the study of artinian modules over commutative noetherian rings.

We start by showing that many of the properties of noetherian modules that make homological methods work seamlessly have analogous properties for artinian modules. We prove many of these properties using Matlis duality and a recent characterization of Matlis reflexive modules. Since Matlis reflexive modules are extensions of noetherian and artinian modules many of the properties that hold for artinian and noetherian modules naturally follow for Matlis reflexive modules and more generally for mini-max modules.

In the last chapter we prove that if the Betti …

Vanishing Of Ext And Tor Over Complete Intersections, Olgur Celikbas

Department of Mathematics: Dissertations, Theses, and Student Research

Let (R,m) be a local complete intersection, that is, a local ring whose m-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of Tor(M, N) and Ext(M, N). In this context, M satisfies Serre's condition (S_{n}) if and only if M is an nth syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r-1 for all sufficiently large n. We use this notion of …