Dualité sur un corps local de caractéristique positive à corps résiduel algébriquement clos

Let K be a complete discretely valued field with residue field k of characteristic p>0. There is a duality theory for cohomology with coefficients in commutative finite K-group schemes in the following cases : char(K)=0 and k finite (Tate), char(K)=p and k finite (Shatz), char(K)=0 and k algebraically closed (B\'egueri). In this paper, we settle the case where char(K)=p and k is algebraically closed.