**Yoshifumi Tsuchimoto**

Let be an abelian category. For any object of , the extension group is defined to be the derived functor of the ``hom'' functor

Let be a group. Let us consider a functor

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

is called the -th cohomology of with coefficients in . Let us consider as a -module with trivial -action. Then we may easily verify that

Thus we have

The extension group may be calculated by using either an injective resolution of the second variable or a projective resoltuion of the first variable .

- We may compute them by using an injective resolution
- We may compute them by using a free resolution

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