Page 75 - Curvature and Homology

2.1. COMPLEXES 57

(iii) There are only a finite number of simplexes Sr-I such that
[R : S?-'] # 0;
(iv) For every pair of simplexes Sf+', Sr-I whose dimensions differ
by two

We associate with K an integer dim K (which may be infinite) called
its dimension which is defined as the maximum dimension of the
simplexes of K.
An algebraic structure is imposed on K as follows: The p-simplexes
are taken as free generators of an abelian group. A (formal) finite sum

where G is an abelian group is called a p-did~al chain or, simply
a p-chain. Two p-chains may be added, their sum being defined in the
obvious manner:

In this way, the p-chains form an abelian group which is denoted by
Cp(K, G). This group can be shown to be isomorphic with Cp(K, 2) @ G
where Z denotes the ring of integers, that is, the tensor product (see
below) of the free abelian group generated by K with the abelian group G.
Let A be a ring with unity 1. A A-module is an (additive) abelian
group A together with a map (A, a) -+ ha of A x A -+ A satisfying

(iii) (hlh2)a = hl(&a),

(iv) lcc = a.
Since the ring A operates on the group A on the left such a module is
called a left A-module. A rkht A-module is defined similarly; indeed,
one need only replace ha by ah and (iii) becomes
(iii)' a(hlX2) = (ah#,.

For commutative rings no distinction is made between left and right
A-modules.