Complex (In Homological Algebra)

One of the basic concepts of homological algebra. Let $ A $ be an Abelian category. A graded object is a sequence $ K = ( K _ {n} ) _ {n \in \mathbf Z } $ of objects $ K _ {n} $ in $ A $. A sequence $ \alpha = ( a _ {n} ) $ of morphisms $ a _ {n} : K _ {n} ^ { \prime } \rightarrow K _ {n} $ is called a morphism $ a : K ^ { \prime } \rightarrow K _ {n} $ of graded objects. One defines the object $ K ( h) $ by setting $ K ( h) _ {n} = K _ {n+h}$. A morphism of graded objects $ K ^ { \prime } \rightarrow K ( h) $ is called a morphism of degree $ h $ from $ K ^ { \prime } $ into $ K $. A graded object is said to be positive if $ K _ {n} = 0 $ for all $ n < 0 $, bounded from below if $ K ( h) $ is positive for some $ h $ and finite or bounded if $ K _ {n} = 0 $ for all but a finite number of integers $ n $. A chain complex in a category $ A $ consists of a graded object $ K $ and a morphism $ d : K \rightarrow K $ of degree $ - 1 $ such that $ d ^ {2} = 0 $. More precisely: $ d = ( d _ {n} ) $, where $ d _ {n} : K _ {n} \rightarrow K _ {n-1} $ and $ d _ {n-1} d _ {n} = 0 $ for any $ n $. A morphism of chain complexes

$$ ( K ^ { \prime } , d ^ { \prime } ) \rightarrow ( K , d ) $$

is a morphism $ a : K ^ { \prime } \rightarrow K $ of graded objects for which $ a d ^ { \prime } = d a $. A cochain complex is defined in a dual manner (as a graded object with a morphism $ d $ of degree $ + 1 $).

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 $ - 1 $ or $ + 1 $.

Associated with each complex $ K $ are the three graded objects:

the boundaries $ B = B ( K) $, where $ B _ {n} = \mathop{\rm Im} ( K _ {n+1} \rightarrow ^ {d _ {n+1} } K _ {n} ) $;

the cycles $ Z = Z ( K) $, where $ Z _ {n} = \mathop{\rm Ker} ( K _ {n} \rightarrow ^ {d _ {n} } K _ {n-1} ) $; and

the $ n $- dimensional homology objects (classes) $ H = H ( K) $, where $ H _ {n} = Z _ {n} / B _ {n} $( see Homology of a complex).

For a cochain complex, the analogous objects are called coboundaries, cocycles and cohomology objects (notations $ B ^ {n} $, $ Z ^ {n} $ and $ H ^ {n} $, respectively).

If $ H ( K) = 0 $, then the complex $ K $ is said to be acyclic.

A morphism $ a : K ^ { \prime } \rightarrow K $ of complexes induces morphisms

$$ Z ( K ^ { \prime } ) \rightarrow Z ( K) ,\ \ B ( K ^ { \prime } ) \rightarrow B ( K) , $$

and hence a homology or cohomology morphism

$$ H ( a) : H ( K ^ { \prime } ) \rightarrow H ( K) . $$

Two morphisms $ a , b : K ^ { \prime } \rightarrow K $ are said to be homotopic (denoted by $ a \simeq b $) if there is a morphism $ s : K ^ { \prime } \rightarrow K ( 1) $( or $ s : K ^ { \prime } \rightarrow K ( - 1 ) $ for cochain complexes) of graded objects (called a homotopy), such that

$$ a - b = ds + sd ^ \prime $$

(which implies that $ H ( a) = H ( b) $). A complex $ K $ is said to be contractible if $ 1 _ {K} \simeq 0 $, in which case the complex $ K $ is acyclic.

If $ 0 \rightarrow K ^ { \prime } \rightarrow K \rightarrow K ^ { \prime\prime } \rightarrow 0 $ is an exact sequence of complexes, then there exists a connecting morphism $ \partial : H ( K ^ { \prime } ) \rightarrow H ( K) $ of degree $ - 1 $( $ + 1 $) that is natural with respect to morphisms of exact sequences and is such that the long homology sequence (that is, the sequence

$$ \dots \rightarrow H _ {n} ( K ^ { \prime } ) \rightarrow H _ {n} ( K) \rightarrow \ H _ {n} ( K ^ { \prime\prime } ) \mathop \rightarrow \limits ^ \partial $$

$$ \mathop \rightarrow \limits ^ \partial H _ {n-1} ( K ^ { \prime } ) \rightarrow H _ {n-1} ( K) \rightarrow H _ {n-1} ( K ^ { \prime\prime } ) \rightarrow \dots $$

for a chain complex, and the sequence

$$ \dots \rightarrow H ^ {n} ( K ^ { \prime } ) \rightarrow H ^ {n} ( K) \rightarrow \ H ^ {n} ( K ^ { \prime\prime } ) \mathop \rightarrow \limits ^ \partial $$

$$ \mathop \rightarrow \limits ^ \partial H ^ {n+1} ( K ^ { \prime } ) \rightarrow H ^ {n+1} ( K) \rightarrow H ^ {n+1} ( K ^ { \prime\prime } ) \rightarrow \dots $$

for a cochain complex) is exact.

The cone of a morphism $ a : K ^ { \prime } \rightarrow K $ of chain complexes is the complex $ MC ( a) $ defined as follows:

$$ MC ( A) _ {n} = K _ {n} \oplus K _ {n-1} ^ \prime $$

with

$$ d ( a) _ {n+1} = \ \left ( \begin{array}{cr} d _ {n+1} &a _ {n} \\ 0 &- d _ {n} ^ { \prime } \\ \end{array} \right ) : MC ( a) _ {n+1} \rightarrow MC ( a) _ {n} . $$

The direct sum decomposition of the complex $ MC ( a) $ leads to an exact sequence of complexes

$$ 0 \rightarrow K \rightarrow MC ( a) \rightarrow K ^ { \prime } ( - 1 ) \rightarrow 0 , $$

for which the associated long homology sequence is isomorphic to the sequence

$$ \dots \rightarrow H _ {n} ( K) \rightarrow H _ {n} ( MC ( a) ) \rightarrow \ H _ {n-1} ( K ^ { \prime } ) \rightarrow ^ { {H _ n-1} ( a) } $$

$$ \rightarrow ^ { {H _ n-1} ( a) } H _ {n-1} ( K) \rightarrow H _ {n-1} ( MC ( a) ) \rightarrow \dots . $$

Hence the chain complex $ MC ( a) $ is acyclic if and only if $ H ( a) $ is an isomorphism. Analogous notions and facts hold for cochain complexes.

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