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Galois module structure of Galois cohomology and partial EulerPoincaré characteristics
, 2006
"... 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 BlochKato Conjecture we determine the structure of the cohomology group H n (U, Fp) as an Fp[GF /U]module for all n ∈ N. Previously this stru ..."
Abstract

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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 BlochKato Conjecture we determine the structure of the cohomology group H n (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 1 (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 EulerPoincaré characteristics of open subgroups N of the maximal prop quotient T of GF. We extend the notion of a partial EulerPoincaré characteristic to this case and we show that the nth partial EulerPoincaré characteristic Θn(N) is determined only by Θn(T) and the conorm in H n (T, Fp).