# 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.

How to Cite This Entry:
Complex (in homological algebra). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complex_(in_homological_algebra)&oldid=53375
This article was adapted from an original article by A.V. Mikhalev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article