Stable Cohomology of Local Rings and Castelnuovo–Mumford Regularity of Graded Modules

Authors

• Luigi Ferraro, University of Nebraska-LincolnFollow

First Advisor

Luchezar L. Avramov

Second Advisor

Srikanth B. Iyengar

Date of this Version

8-2017

Document Type

Dissertation

Comments

A dissertation Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy Major: Mathematics Under the Supervision of Professors L. L. Avramov and S. B. Iyengar Lincoln, Nebraska August, 2017

Copyright (c) 2017 Luigi Ferraro

Abstract

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 algebra and to study its commutativity.

In Chapter 2 it is proved that if a graded, commutative algebra $R$ over a field $k$ is not Koszul, then the nonzero modules $\mathfrak{m} M$, where $M$ is a finitely generated $R$-module and $\mathfrak{m}$ is the maximal homogeneous ideal of $R$, have infinite Castelnuovo-Mumford regularity.

Advisers: L. L. Avramov and S. B. Iyengar