Faltings extension and Hodge-Tate filtration for abelian varieties over p-adic local fields with imperfect residue fields

Let K be a complete discrete valuation field of characteristic, with not necessarily perfect residue field of characteristic 0$ ]]>. We define a Faltings extension of over, 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.