Computations of 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.
