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AuthorFrรถhlich, A. author
TitleClassgroups and Hermitian Modules [electronic resource] / by A. Frรถhlich
ImprintBoston, MA : Birkhรคuser Boston, 1984
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Descript XVIII, 226 p. online resource


These notes are an expanded and updated version of a course of lectures which I gave at King's College London during the summer term 1979. The main topic is the Hermitian classgroup of orders, and in particular of group rings. Most of this work is published here for the first time. The primary motivation came from the connection with the Galois module structure of rings of algebraic integers. The principal aim was to lay the theoretical basis for attacking what may be called the "converse problem" of Galois module structure theory: to express the symplectic local and global root numbers and conductors as algebraic invariants. A previous edition of these notes was circulated privately among a few collaborators. Based on this, and following a partial solution of the problem by the author, Ph. Cassou-Nogues and M. Taylor succeeded in obtaining a complete solution. In a different direction J. Ritter published a paper, answering certain character theoretic questions raised in the earlier version. I myself disapprove of "secret circulation", but the pressure of other work led to a delay in publication; I hope this volume will make amends. One advantage of the delay is that the relevant recent work can be included. In a sense this is a companion volume to my recent Springer-Ergebnisse-Bericht, where the Hermitian theory was not dealt with. Our approach is via "Hom-groups", analogous to that followed in recent work on locally free classgroups


I Preliminaries -- ยง1 Locally free modules and locally freely presented torsion modules -- ยง2 Determinants and the Hom language for classgroups -- ยง3 Supplement at infinity -- II Involution algebras and the Hermitian classgroup -- ยง1 Involution algebras and duality -- ยง2 Hermitian modules -- ยง3 Pfaffians of matrices -- ยง4 Pfaffians of algebras -- ยง5 Discriminants and the Hermitian classgroup -- ยง6 Some homomorphisms -- ยง7 Pulling back discriminants -- ยง8 Unimodular modules -- ยง8 Products -- III Indecomposable involution algebras -- ยง1 Dictionary -- ยง2 The map P -- ยง3 Discriminants once more -- ยง4 Norms of automorphisms -- ยง5 Unimodular classes once more -- IV Change of order -- ยง1 Going up -- ยง2 Going down -- V Groups -- ยง1 Characters -- ยง2 Character action. Ordinary theory -- ยง3 Character action. Hermitian theory -- ยง4 Special formulae -- ยง5 Special properties of the group ring involution -- ยง6 Some Frobenius modules -- ยง7 Some subgroups of the adelic Hermitian classgroup -- VI Applications in arithmetic -- ยง1 Local theory -- ยง2 The global discriminant -- Literature -- List of Theorems -- Some further notation

Mathematics Algebraic geometry Group theory K-theory Matrix theory Algebra Number theory Algebraic topology Mathematics K-Theory Algebraic Topology Number Theory Linear and Multilinear Algebras Matrix Theory Algebraic Geometry Group Theory and Generalizations


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