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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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