Let G be a finite group. Given a contravariant, product preserving functor F:G-sets → AB, we construct a Green-functor A(F):G-sets → CRNG which specializes to the Burnside ring functor when F is trivial. A(F) permits a natural addition and multiplication between elements in the various groups F(S), S ∈ G-sets. If G is the Galois group of a field extension L/K, and SEP denotes the category of K-algebras which are isomorphic with a finite product of subfields of L, then any covariant, product preserving functor ρ:SEP → AB induces a functor Fᵨ:G → AB, and thus the Green-functor Aᵨ may be obtained. We use this observation for the case ρ = Br, the Brauer group functor, and show that Aᵦᵣ(G/G) is free on K-algebra isomorphism classes of division algebras with center in SEP. We then interpret the induction theory of Mackey-functors in this context. For a certain class of functors F, the structure of A(F) is especially tractable; for these functors we deduce that (DIAGRAM OMITTED), where the product is over isomorphism class representatives of transitive G-sets. This allows for the computation of the prime ideals of A(F)(G/G), and for an explicit structure theorem for Aᵦᵣ, when G is the Galois group of a p-adic field. We finish by considering the case when G = Gal(L/Q), for an arbitrary number field L.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/186082 |
| Date | January 1983 |
| Creators | JACOBSON, ELIOT THOMAS. |
| Contributors | Pierce, Richard |
| Publisher | The University of Arizona. |
| Source Sets | University of Arizona |
| Language | English |
| Detected Language | English |
| Type | text, Dissertation-Reproduction (electronic) |
| Rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. |
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