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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