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Pareigis, Bodo und Rosenberg, A. (1964): Addendum to "Amitsur's complex for purely inseparable fields". In: Osaka Journal of Mathematics, Vol. 1: S. 33-44




We begin this note by pointing out that a few modifications in some of the notations and arguments of C131 will make these fit in more closely with results in the literature. We also complete the results of C131 in several points. In particular we point out that the spectral sequence used in C131 is not quite a genuine generalization of the Hochschild-Serre spectral sequence in Galois cohomology. However with a slightly different spectral sequence the results of C131 can also be obtained and we shall show in section 2 that this is indeed a genuine generalization of the Hochschild-Serre sequence for Galois cohomology. In section 3 we shall use some of the results of [13] to derive an exact sequence complementary to that of Proposition 7.8 of [13] from which we deduce the following result first pointed out to us by S. Shatz: Let C be a field, C, its separable algebraic closure and its algebraic closure. Then if X is the lift map [2, Def. 2. 3.1, we have that X : Hr(C,/C)- . ~ ' ( 6 1is~ ) an isomorphism for r = 1,2, ...