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

6 - Residue maps

Summary

Residue maps constitute a fundamental technical tool for the study of the cohomological symbol. Their definition is not particularly enlightening at a first glance, but the reader will see that they emerge naturally during the computation of Brauer groups of function fields or power series fields. When one determines these, a natural idea is to pass to a field extension having trivial Brauer group, so one needs some sufficient condition that ensures this property. The C1 condition introduced by Emil Artin and baptized by Serge Lang furnishes such a sufficient condition via the vanishing of low-degree polynomials. There are three famous classes of C1-fields: finite fields, function fields of curves and Laurent series fields, the latter two over an algebraically closed base field. Once we know that the Brauer groups of these fields vanish, we are able to compute the Brauer groups of function fields and Laurent series fields over an arbitrary perfect field. The central result here is Faddeev's exact sequence for the Brauer group of a rational function field. We give two important applications of this theory: one to the class field theory of curves over finite fields, the other to constructing counterexamples to the rationality of the field of invariants of a finite group acting on some linear space. Following this ample motivation, we finally attack residue maps with finite coefficients, thereby preparing the ground for the next two chapters.

Residue maps for the Brauer group first appeared in the work of the German school on class field theory; the names of Artin, Hasse and F. K. Schmidt are the most important to be mentioned here.

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