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Let K be a complete discrete valuation field of characteristic 
$0$
, with not necessarily perfect residue field of characteristic 
$p>0$
. We define a Faltings extension of 
$\mathcal {O}_K$
over 
$\mathbb {Z}_p$
, and we construct a Hodge-Tate filtration for abelian varieties over K by generalizing Fontaine’s construction [Fon82] where he treated the perfect residue field case.
