When is Galois cohomology free or trivial?

摘要

Let p be a prime, F a field containing a primitive pth root ofunity, and E/F a cyclic extension of degree p. Using theBloch-Kato Conjecture we determine precise conditions for thecohomology group Hn(E):=Hn(GE,Fp) to be free or trivial as anFp[Gal(E/F)]-module, and we examine when these properties forHn(E) are inherited by Hk(E), kn. By analogy withcohomological dimension, we introduce notions of cohomologicalfreeness and cohomological triviality, and we give examples ofHn(E) free or trivial for each n∈ N with prescribedcohomological dimension.