Séminaire de Géométrie Algébrique et Singularités

Natural homomorphism of Witt rings of a certain cubic order

by Prof. Pawel Gladki (University of Silesia in Katowice, Pologne)

Salle Pellos (IMT, bat. 1R2)

Salle Pellos

IMT, bat. 1R2


Let $K$ be a number field and $\mathcal{O}_K$ its ring of integers. A famous result by Knebusch asserts that the natural homomorphism of Witt rings $W\mathcal{O}_K \rightarrow W K$ is injective. This, however, fails to be true if we replace $\mathcal{O}_K$ with an arbitrary ring $R$ whose field of fractions is equal to $K$. We shall consider one particular class of such rings here, namely orders, that is subrings $\mathcal{O}$ of $\mathcal{O}_K$ which are also $\mathbb{Z}$-modules of rank $n = [K :\mathbb{Q}]$. The case of orders in quadratic number fields is relatively well understood with both examples of natural homomorphisms of Witt rings being injective and not. In this talk we shall take a closer look at orders in cubic number fields. While orders in quadratic number fields are easy to describe and classify, cubic orders are considerably more difficult to handle. Nevertheless, we manage to exhibit an example of a cubic order $\mathcal{O}$ whose Witt ring $W\mathcal{O}$ naturally embedds into the Witt ring of its field of fractions.