CHAPTER 1

Class-groups of group-rings

1. Horn Descriptions

DEFINITION 1.1. Adeles and Ideles

A more extensive reference for the material of this section is ([17] II,p.334 et

seq).

Let K be a number field. The adele ring of K is defined to be the ring given

by the restricted product

J(K)

= n

K

F prime

where Y\ signifies that we take those elements of the topological ring, Y\p Kp, for

whom almost all entries lie in the ring of integers, OKP • When P is an Archimedean

prime we adopt the convention that OKP = Kp The group of ideles, J*(K), is

the group of units in J(K),

J*(K) — {(xp) G J(K) | xp 7^ 0 and almost everywhere xp £ 0*Kp}

where, as usual, (D*K denotes the multiplicative group of units in

OKP

The unit

ideles is the subgroup in which every entry is a unit,

u{oK)= n °KP-

P prime

Now let G be a finite group. We may extend the adeles and ideles to the

group-rings, C?x[G] and jftT[G]. Define

J(K[G\)= U'pPrimeKp[G},

J*(K[G}) = {(aP) e J{K[G\) | aP € 0Kp[G]* for almost

all P and ap € Kp[G}* otherwise},

U(OK[G})= X\PwimeOKp\G\*.

1.2. Let K be a field of characteristic zero. The K-representation ring of

G, denoted by

RK(G),

is defined to be the free abelian group on the irreducible

^-representations of G, made into a commutative ring via the tensor product of

representations. A ^-representation of G may be envisioned either as an isomor-

phism class of a finite-dimensional K-vector space with a linear G-action or as a

conjuacy class of homorphisms into matrix groups of the form

G — GLn(K).

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