Galois Module Structure of Galois Cohomology and Partial Euler-poincaré Characteristics

Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group GF of F . Using the Bloch-Kato Conjecture we determine the structure of the cohomology group H(U, Fp) as an Fp[GF /U ]-module for all n ∈ N. Previously this structure was known only for n = 1, and until recently the structure even of H(U, Fp) was determined only for F a local field, a case settled by Borevič and Faddeev in the 1960s. We apply these results to study partial Euler-Poincaré characteristics of open subgroups N of the maximal pro-p quotient T of GF . We extend the notion of a partial Euler-Poincaré characteristic to this case and we show that the nth partial Euler-Poincaré characteristic Θn(N) is determined only by Θn(T ) and the conorm in H (T, Fp). Let F be a field containing a primitive pth root of unity ξp. Let GF be the absolute Galois group of F , U an open normal subgroup of GF of index p, and G = GF/U . In the 1960s Z. I. Borevič and D. K. Faddeev classified the possible G-module structures of the first cohomology groups H(U,Fp) in the case F a local field [Bo]. Recently this result was extended from local fields to all fields F as above [MS1], and in this more general context the result has been further developed and applied in [MS2], [MS3], Date: January 23, 2006. 2000 Mathematics Subject Classification. 12G05 (primary), 19D45 (secondary).