The Exact Sequence of Low Degree and Normal Algebras

The exact sequence of low degree associated to a first quadrant bicomplex (five terms long in [4, 1.4.5.1] seven terms long in [2, Lemma 7.5]) has been used in a number of situations, for example, in obtaining a cohomological description of the Brauer group of a commutative ring R [2]. In this note we observe that the sequence may be extended to an infinitely long exact sequence. The terms arising from the homology of the total complex are not F~"H(tot)i the (n— l) th filtration group of H, for n>2, but map onto it. As an application we embed the seven term Galois cohomology sequence of [l, 5.5] into an infinite sequence, and sketch a map from normal Azumaya algebras into the eighth term which extends the Teichmüller cocycle map of [3].