CHAPTER 0
PRELIMINARIES
In this chapter we will give a brief survey of some results in homo-
logical algebra and transformation groups which has been mentioned in the
following chapters. We have only sketched some proofs and given the basic
ideas in other proofs; the details are referred to the main sources. Some
definitions are also included for the convenience of the reader. The
material of this chapter can also serve as a historical review of part of the
previous work done in this area which has been of importance in the develop-
ment of the material in Chapters I and II. Since in the later chapters we
have made references only to the original sources, the informed reader can
skip this -chapter and refer to it later if any need arises.
1_. Let R be a commutative ring with unit, and let G be a finite group.
The norm N of G is defined by N = Eg. For any RG-module M,
gcG
N determines an endomorphism which is also denoted by N, and is called the
"norm homomorphism". Thus for x e M, Nx = E g*x.
geG
Suppose R = Z, and ZZG -^ TL is the unit augmentation with the
augmentation ideal I. For any Z-module M, define NL = M/IM, and
p
M = {x M : g»x = x}, the set of invariant elements. The kernel of
N : M —• M is denoted by ..M., Since N-(g - 1) = 0 and g*N = N for all
P
g e G, it follows that I*M c ker N, and Im N c M . Then the "complete
derived sequence" of G, denoted by
Hn(G,
M), n °° in the sense
of Eilenberg-Cartan is defined as follows:
Received by the editor October 7, 1980 and, in revised form March 18, 1981.
Research supported by Princeton University and partially by an NSF grant
through The Institute for Advanced Study.
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