**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

We note that the functor is a ``bifunctor''. We may thus consider the right derived functor of and that of . Fortunately, both coincide: 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

is called an

- .
- .
- .

- .
- For any , is a right exact functor.
- For any right ideal
of
and for any
-module
,
we have

By definition, may be computed by using projective resolutions of .