Binomial Rings and their Cohomology

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.