White Rose University Consortium logo
University of Leeds logo University of Sheffield logo York University logo

Binomial Rings and their Cohomology

Kareem, Shadman (2018) Binomial Rings and their Cohomology. PhD thesis, University of Sheffield.

[img]
Preview
Text
Submission draft.pdf
Available under License Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 UK: England & Wales.

Download (1066Kb) | Preview

Abstract

A binomial ring is a Z-torsion free commutative ring R, in which all the binomial operations in R tenser Q; actually lie in R; for all r in R and n greater than and equal 0. It is a special type of Lambda-ring in which the Adams operations on it all are the identity and the Lambda-operations are given by the binomial operations. This thesis studies the algebraic properties of binomial rings, considers examples from topology and begins a study of their cohomology. The first two chapters give an introduction and some background material. In Chapter 3 and Chapter 4 we study the algebraic structure and properties of binomial rings, focusing on the notion of a binomial ideal in a binomial ring. We study some classes of binomial rings. We show that the ring of integers Z is a binomially simple ring. We give a characterisation of binomial ideals in the ring of integer valued-polynomials on one variable. We apply this to prove that ring of integer valued-polynomials on one variable is a binomially principal ring and rings of polynomials that are integer valued on a subset of the integers are also binomially principal rings. Also, we prove that the ring of integer-valued polynomials on two variables is a binomially Noetherian ring. The ring of integer valued-polynomials on one variable and its dual appear as certain rings of operations and cooperations in topological K-theory. We give some non-trivial examples of binomial rings that come from topology such as stably integer-valued Laurent polynomials on one variable and stably integer-valued polynomials on one variable. We study generalisations of these rings to a set X of variables. We show that in the one variable case both rings are binomially principal rings and in the case of finitely many variables both are binomially Noetherian rings. As a main result we give new descriptions of these examples. In Chapter 5 and Chapter 6 we define cohomology of binomial rings as an example of a cotriple cohomology theory on the category of binomial rings. To do so, we study binomial modules and binomial derivations. Our cohomology has coeffcients given by the contravariant functor Der_Bin(-,M) of binomial derivations to a binomial module M: We give some examples of binomial module structures and calculate derivations for these examples. We define homomorphisms connecting the cohomology of binomial rings to the cohomology of Lambda-rings and to the Andre-Quillen cohomology of the underlying commutative rings.

Item Type: Thesis (PhD)
Academic Units: The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Identification Number/EthosID: uk.bl.ethos.739881
Depositing User: Mr Shadman Kareem
Date Deposited: 05 Apr 2018 15:16
Last Modified: 12 Oct 2018 09:54
URI: http://etheses.whiterose.ac.uk/id/eprint/19921

You do not need to contact us to get a copy of this thesis. Please use the 'Download' link(s) above to get a copy.
You can contact us about this thesis. If you need to make a general enquiry, please see the Contact us page.

Actions (repository staff only: login required)