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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