On Hopf Ore Extensions and Zariski Cancellation Problems

Abstract

In this thesis, we investigate Ore extensions of Hopf algebras and the Zariski Cancellation problem for noncommutative rings. In particular, we improve upon the existing conditions for when $T=R[x; \sigma, \delta]$ is a Hopf Ore extension of a Hopf algebra $R$, and we give noncommutative analogues of a cancellation theorem of Abhyankar, Eakin, and Heinzer. In Chapter 3, we study the relationship between prime ideals of $T=R[x; \sigma, \delta]$ and their contractions under $R$. In Chapter 4, we look at when $T$ is a Hopf algebra and by studying the coproduct of $x$, $\Delta(x)$, we provide a sequence of results that answers a question due to Panov; that is, given a Hopf algebra $R$, for which automorphisms $\sigma$ and $\sigma$-derivations $\delta$ does the Ore extension $T=R[x; \sigma, \delta]$ have a Hopf algebra structure extending the given Hopf algebra structure on $R$? In Chapter 5, we consider the question of cancellation for finitely generated not-necessarily-commutative domains of Gelfand-Kirillov dimension one and show that such algebras are necessarily cancellative when the characteristic of the base field is zero. In particular, this recovers the cancellation result of Abhyankar, Eakin, and Heinzer in characteristic zero when one restricts to the commutative case. We also provide examples that show affine domains of Gelfand-Kirillov dimension one need not be cancellative when the base field has a positive characteristic, giving a counterexample to a conjecture of Tang et al. In Chapter 6, we prove a skew analogue of the result of Abhyankar-Eakin-Heinzer, in which one works with skew polynomial extensions as opposed to ordinary polynomial rings.