Dualit\'e sur un corps local de caract\'eristique positive \`a corps r\'esiduel alg\'ebriquement clos

TL;DR

This paper establishes a duality theory for cohomology of finite group schemes over a local field with algebraically closed residue field of characteristic p, filling a gap in existing duality results.

Contribution

It proves the duality theory for the case where the residue field is algebraically closed and the characteristic of the field is p, which was previously unresolved.

Findings

Established duality for cohomology with finite group schemes in the specified case.

Extended the existing duality theories to new residue field conditions.

Provided a comprehensive framework for duality in positive characteristic local fields.

Abstract

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.