GREEN FUNCTOR CONSTRUCTIONS IN THE THEORY OF ASSOCIATIVE ALGEBRAS.

Persistent Link:
http://hdl.handle.net/10150/186082
Title:
GREEN FUNCTOR CONSTRUCTIONS IN THE THEORY OF ASSOCIATIVE ALGEBRAS.
Author:
JACOBSON, ELIOT THOMAS.
Issue Date:
1983
Publisher:
The University of Arizona.
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.
Abstract:
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.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Associative algebras.; Finite groups.; Burnside problem.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Pierce, Richard

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleGREEN FUNCTOR CONSTRUCTIONS IN THE THEORY OF ASSOCIATIVE ALGEBRAS.en_US
dc.creatorJACOBSON, ELIOT THOMAS.en_US
dc.contributor.authorJACOBSON, ELIOT THOMAS.en_US
dc.date.issued1983en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © 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.en_US
dc.description.abstractLet 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.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectAssociative algebras.en_US
dc.subjectFinite groups.en_US
dc.subjectBurnside problem.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorPierce, Richarden_US
dc.identifier.proquest8315288en_US
dc.identifier.oclc688638554en_US
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