**Yoshifumi Tsuchimoto**

Let be a group. Let us consider a functor

The functor is left-exact. The derived functor of this functor

is called

Thus we have

To compute cohomologies of , it is useful to use -resolution of . For any tuples of , we introduce a symbol

and we consider the following sequence

( ) |

where are determined by the following rules.

To see that the sequence is acyclic, we consider a homotopy

- Each of the modules that appears in the sequence
admits an action
of
determined by

There are several choices for the -basis of . One such is clearly

It is traditional (and probably useful) to use another basis

where

Conversely we have

defined by

span

is called the homology of with coefficients in . We denote the homology group by .