Amitsur cohomology of quadratic extensions: formulas and number-theoretic examples

Abstract

Amitsur cohomology (in the units functor U) for extensions of rings of algebraic integers have been achieved in two ways: via Mayer-Vietoris sequences (by Morris and Mandelberg) and via cohomology in the functor UK/U (by the second-named author). One of the goals of these computations has been to shed light on the Chase-Rosenberg homomorphism from Amitsur cohomology to the split Brauer group. In this paper we obtain, for quadratic ring extensions, formulas for cohomology in U and in UK/U, which have wider application than the corresponding work of Morris and Mandelberg. Our formulas lead to examples showing that the Chase-Rosenberg homomorphism, arising from a quadratic extension of rings of algebraic integers, need not be injective or surjective.