One of the basic concepts of homological algebra. Let
be an Abelian category. A
graded object
is a sequence
of objects
in
.
A sequence
of morphisms
is called a
morphism
of graded objects.
One defines the object
by setting
.
A morphism of graded objects
is called a
morphism of degree
from
into
.
A graded object is said to be
positive
if
for all
,
bounded from below
if
is positive for some
and
finite
or
bounded
if
for all but a finite number of integers
.
A
chain complex
in a category
consists of a graded object
and a morphism
of degree
such that
.
More precisely:
,
where
and
for any
.
A
morphism of chain complexes
is a morphism
of graded objects for which
.
A
cochain complex
is defined in a dual manner (as a graded object with a morphism
of degree
).
Most frequently, complexes are considered in categories of Abelian groups,
modules or sheaves of Abelian groups on a topological space. Thus,
a complex of Abelian groups is a graded
differential group the differential of which has degree
or
.
Associated with each complex
are the three graded objects:
the
boundaries
,
where
;
the
cycles
,
where
;
and
the
-dimensional
homology objects
(classes)
,
where
(see
Homology of a complex).
For a cochain complex, the analogous objects are called
coboundaries,
cocycles
and
cohomology objects
(notations
,
and
,
respectively).
If
,
then the complex
is said to be
acyclic.
A morphism
of complexes induces morphisms
and hence a homology or cohomology morphism
Two morphisms
are said to be
homotopic
(denoted by
)
if there is a morphism
(or
for cochain complexes) of graded objects (called a
homotopy),
such that
(which implies that
).
A complex
is said to be
contractible
if
,
in which case the complex
is acyclic.
If
is an exact sequence of complexes, then there exists a
connecting morphism
of degree
(
)
that is natural with respect to morphisms of exact sequences and is such that the
long homology sequence
(that is, the sequence
for a chain complex, and the sequence
for a cochain complex) is exact.
The
cone
of a morphism
of chain complexes is the complex
defined as follows:
with
The direct sum decomposition of the complex
leads to an exact sequence of complexes
for which the associated long homology sequence is isomorphic to the sequence
Hence the chain complex
is acyclic if and only if
is an isomorphism. Analogous notions and facts hold for cochain complexes.