Vol. 60, No. 1, 1975

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

Richard Thomas Bumby and David Earl Dobbs

Vol. 60 (1975), No. 1, 21–30

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.

Mathematical Subject Classification 2000
Primary: 13A20
Secondary: 12G05
Received: 9 May 1974
Revised: 5 August 1974
Published: 1 September 1975
Richard Thomas Bumby
David Earl Dobbs