Duality over a local field of positive characteristic with algebraically closed residue field

Let K be a complete discretely valued field with residue field k of characteristic p > 0. There exists a duality theory for the cohomology of finite commutative K-group schemes in the following cases: K has characteristic 0 and k is finite (J. Tate, Duality theorems in Galois cohomology over number fields, in: Proceedings ICM 1962), K has characteristic p and k is finite (S.S. Shatz, Cohomology of Artinian group schemes over local fields, Ann. of Math. (2) 79 (3) (1964) 411-449), K has characteristic 0 and k is algebraically closed (L. Begueri, Dualite sur un corps local a corps residuel algebriquement clos, Mem. Soc. Math. Fr. 108 (4) (1980)). Here we present the case where K has characteristic p and k is algebraically closed; this is a summary of the detailed text (C. Pepin, Dualite sur un corps local de caracteristique positive a corps residuel algebriquement clos, prepublication, arXiv:1411.0742v1). An independent approach has been given recently by Suzuki (Duality for local fields and sheaves on the category of fields, prepublication, arXiv:1310.4941v2, 2.7.6 (1) (a)). (C) 2015 Academie des sciences. Publie par Elsevier Masson SAS. Tous droits reserves.