Title:

Binomial rings and their cohomology

A binomial ring is a Ztorsion 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 Lambdaring in which the Adams operations on it all are the identity and the Lambdaoperations 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 valuedpolynomials on one variable. We apply this to prove that ring of integer valuedpolynomials 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 integervalued polynomials on two variables is a binomially Noetherian ring. The ring of integer valuedpolynomials on one variable and its dual appear as certain rings of operations and cooperations in topological Ktheory. We give some nontrivial examples of binomial rings that come from topology such as stably integervalued Laurent polynomials on one variable and stably integervalued 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 Lambdarings and to the AndreQuillen cohomology of the underlying commutative rings.
