DEFINITION 06.1
Let
be a functor between additive categories.
We call
additive if for any objects
in
,
is additive.
DEFINITION 06.2
Let
be an additive functor from an abelian category
to
.
is said to be left exact (respectively, right exact )
if for any exact sequence
the corresponding map
(respectively,
is exact
is said to be exact if it is both left exact and right exact.
DEFINITION 06.3
Let
,
be complexes of objects
of an additive category
.
A morphism of complex
is a family
of morphisms in
such that
commutes with
. That means,
holds.
A homotopy between two morphisms
of complexes is a family of morphisms
such that
holds.
LEMMA 06.4Let
be an abelian category that has enough injectives. Then:
For any object
in
,
there exists an injective resolution of
. That means, there exists
an complex
and a morphism
such that
For any morphism
of
, and for any injective resolutions
,
of
and
(respectively), There exists a morphism
of complexes which commutes with
.
Forthermore, if there are two such morphisms
and
, then
the two are homotopic.
DEFINITION 06.5
Let
be an abelian category which has enough injectives.
Let
be a left exact functor to an abelian category.
Then for any object
of
we take an injective resolution
of
and define
and call it the derived functor of
.
LEMMA 06.6The derived functor is indeed a functor.