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# Categories, abelian categories and cohomologies.

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 .

EXAMPLE 08.1   Let us compute the extension groups .
1. We may compute them by using an injective resolution

of .
2. We may compute them by using a free resolution

of .

EXERCISE 08.1   Compute an extension group for modules of your choice. (Please choose a non-trivial example).

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

EXERCISE 08.2   Show that

LEMMA 08.2
1. Each of the modules that appears in the sequence admits an action of determined by

2. is -free

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

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2010-06-01