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Cohomologies.
Yoshifumi Tsuchimoto
By employing the following axiom
instead of the axiom (functor2) above, we obtain a definition of
a contravariant functor:
(functor
)
for any composable morphisms
DEFINITION 05.2
Let
be a functor between additive categories.
We call
additive if for any objects
in
,
is additive.
DEFINITION 05.3
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.
LEMMA 05.4
Let
be a (unital associative but not necessarily commutative) ring.
Then for any
module
, the following conditions are equivalent.

is a direct summand of free modules.

is projective
COROLLARY 05.5
For any ring
, the category
of
modules
have enough projectives. That means, for any object
,
there exists a projective object
and a surjective morphism
.
DEFINITION 05.6
Let
be a commutative ring.
We assume
is a domain (that means,
has no zerodivisors except for 0
.)
An
module
is said to be divisible if
for any
, the multplication map
is surjective.
DEFINITION 05.7
Let
be a commutative ring.
We assume
is a domain (that means,
has no zerodivisors except for 0
.)
An
module
is said to be divisible if
for any
, the multplication map
is epic.
DEFINITION 05.8
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.
DEFINITION 05.10
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 05.11
The derived functor is indeed a functor.
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