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On the Hodge-Tate decomposition in the imperfect residue field case. (English) Zbl 0571.14004
In this paper the Hodge-Tate decomposition in the imperfect residue field case is studied. Continuous cohomology groups \(H^ q(K,C_ p(r))\) are determined for all \(q\geq 0\), \(r\in {\mathbb{Z}}\) and for any complete discrete valuation field K of mixed characteristics (0,p). (When the residue field \(\bar K\) of K is perfect, this was done by Tate.) Contrary to the case \(\bar K\) is perfect, \(H^ 1(K,C_ p(1))\) does not vanish when \(\bar K\) is not perfect. By using this result, it is shown that there are many abelian varieties whose Tate module does not admit a Hodge-Tate decomposition when \(\bar K\) is not perfect.

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14K05 Algebraic theory of abelian varieties
14G15 Finite ground fields in algebraic geometry
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