Duality for local fields and sheaves on the category of fields. (English) Zbl 1522.11069
Summary: Duality for complete discrete valuation fields with perfect residue field with coefficients in (possibly \(p\)-torsion) finite flat group schemes was obtained by Bégueri, Bester, and Kato. In this paper, we give another formulation and proof of this result. We use the category of fields and a Grothendieck topology on it. This simplifies the formulation and proof and reduces the duality to classical results on Galois cohomology. A key point is that the resulting site correctly captures extension groups between algebraic groups.
MSC:
| 11G45 | Geometric class field theory |
| 14F20 | Étale and other Grothendieck topologies and (co)homologies |
| 14L15 | Group schemes |
| 13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |
| 12F20 | Transcendental field extensions |
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