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Commutative algebra
Yoshifumi Tsuchimoto
DEFINITION 08.1
Let
be a ring.
A
cochain complex of
modules
is a sequence of
modules
such that
.
The
th
cohomology of the cochain complex is defined to be the
module
Elements of
(respectively,
)
are often referred to as
cocycles (respectively,
coboundaries).
EXERCISE 08.1
Let
be a ring. Let
be an exact sequence of
modules.
Assume furthermore that
is projective.
Then show that
the sequence
is exact.
LEMMA 08.3
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 08.4
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 08.5
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 08.6
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 epic.
LEMMA 08.7
Let
be a (commutative) principal ideal domain (PID).
Then an
module
is injective if and only if it is divisible.
A bit of category theory:
DEFINITION 08.9
A
category
is a collection of the following data
 A collection
of objects of
.
 For each pair of objects
, a set
of morphisms.
 For each triple of objects
,
a map(``composition (rule)'')
satisfying the following axioms

unless
.
 (Existence of an identity) For any
,
there exists an element
such that
holds for any
(
).
 (Associativity)
For any objects
, and for any morphisms
,
we have
Morphisms are the basic actor/actoress in category theory.
An additive category is a category in which one may ``add'' some morphisms.
DEFINITION 08.10
An additive category
is said to be
abelian if it satisfies the
following axioms.
 A41.
 Every morphism
in
has a kernel
.
 A42.
 Every morphism
in
has a
cokernel
.
 A43.
 For any given morphism
, we have
a suitably defined isomorphism
in
.
More precisely,
is a morphism which is defined by the following relations:
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 08.12
Let
be a functor between additive categories.
We call
additive if for any objects
in
,
is additive.
DEFINITION 08.13
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 08.14
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 08.15
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 08.16
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 08.17
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 08.18
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 08.20
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 08.21
The derived functor is indeed a functor.
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