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  • Print publication year: 2006
  • Online publication date: November 2009

9 - Symbols in positive characteristic

Summary

In the preceding chapters, when working with Galois cohomology groups or K-groups modulo some prime, a standing assumption was that the groups under study were torsion groups prime to the characteristic of the base field. We now remove this restriction. In the first part of the chapter the central result is Teichmüller's theorem, according to which the p-primary torsion subgroup in the Brauer group of a field of characteristic p > 0 is generated by classes of cyclic algebras – a characteristic p ancestor of the Merkurjev–Suslin theorem. We shall give two proofs of this statement: a more classical one due to Hochschild which uses central simple algebras, and a totally different one based on a presentation of the p-torsion in Br (k) via logarithmic differential forms. The key tool here is a famous theorem of Jacobson–Cartier characterizing logarithmic forms. The latter approach leads us to the second main topic of the chapter, namely the study of the differential symbol. This is a p-analogue of the Galois symbol which relates the Milnor K-groups modulo p to a certain group defined using differential forms. As a conclusion to the book, we shall prove the Bloch–Gabber–Kato theorem establishing its bijectivity.

Teichmüller's result first appeared in the ill-famed journal Deutsche Mathematik (Teichmüller [1]); see also Jacobson [3] for an account of the original proof. The role of derivations and differentials in the theory of central simple algebras was noticed well before the Second World War; today the most important work seems to be that of Jacobson [1].

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