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On the Galois cohomology of ideal class groups. (English) Zbl 1139.11046
Let \(K/k\) be a finite Galois extension of number fields with Galois group \(G\) and put \(R = \mathbb Z [\frac 12][G]\). Using étale cohomology, the authors derive the isomorphism of Tate cohomology groups,
\[ \widehat H^{a+2} (J, e (\mathcal O_{K,S}^{\times})') \simeq \widehat H^a (J, e\operatorname{Pic} (\mathcal O_{K,S})'), \] where \(J\) is any subgroup of \(G\), \(e\) a central idempotent of \(R\), and the primes mean that the modules are considered as Galois modules over \(R\). This result is specialized for the case that \(K\) is a CM-field and furthermore for \(K = \mathbb Q (\zeta_{p^n})^+\), where \(p\) is an odd prime. This yields alternative proofs for results of P. Cornacchia and C. Greither [J. Number Theory 73, 459–471 (1998; Zbl 0926.11085)], R. Schoof [Math. Comput. 72, 913–937 (2003; Zbl 1052.11071)] and the second author [Acta Arith. 120, 337–348 (2005; Zbl 1139.11047)].
11R34 Galois cohomology
11S40 Zeta functions and \(L\)-functions
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