  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 in 1981, where he treated the perfect residue field case.
