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 y, with not necessarily perfect residue field of characteristic p > 0. We define a Faltings extension of O-K over 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.