Chain complexes are the basic objects of study in homological algebra.
A chain complex is a sequence of abelian groups or modules (for instance vector spaces) or similar equipped with linear maps such that , i.e. the composite of two consecutive such maps is the zero morphism .
Chain complexes crucially come with their chain homology groups. When regarding chain maps between them that induce isomorphisms on these groups (quasi-isomorphisms) as weak equivalences, chain complexes form a useful presentation for aspects of stable homotopy theory. More on this aspect below.
In particular this reasoning shows that connective chain complexes of abelian groups are a model for abelian ∞-groups (aka Kan complexes with abelian group structure). This correspondence figures as part of the cosmic cube-classification scheme of n-categories.
Let be an abelian category.
A (-graded) chain complex in is
(the zero morphism) for all .
One uses the following terminology for the components of a chain complex, all deriving from the example of a singular chain complex:
For a chain complex
the elements in the kernel
of are called the -cycles;
the elements in the image
of are called the -boundaries;
Notice that due to we have canonical inclusions
The dual notion:
One also says homologically graded complex, for the case that the differentials lower degree, and cohomologically graded complex for the case where they raise degree.
Frequently one also considers -graded (or nonnegatively graded) chain complexes (for instance in the Dold-Kan correspondence), which can be identified with -graded ones for which when . Similarly, an -graded cochain complex is a cochain complex for which when , or equivalently a chain complex for which when .
Note that in particular, a chain complex is a graded object with extra structure. This extra structure can be codified as a map of graded objects , where is the ‘shift’ endofunctor of the category of graded objects in , such that . More generally, in any pre-additive category with translation , we can define a chain complex to be a differential object such that is the zero morphism. When this recovers the original definition.
Common choices for the ambient abelian category include Ab, Vect (for a field) and generally Mod (for a ring), in which case we obtain the familiar definition of an (unbounded) chain complex of abelian groups, vector spaces and, generally, of modules.
In Vect a chain complex is also called a differential graded vector space, consistent with other terminology such as differential graded algebra over . This is also a special case in other ways: every chain complex over a field is formal: quasi-isomorphic to its homology (the latter considered as a chain complex with zero differentials). Nothing of the sort is true for chain complexes in more general categories.
A chain complex in a category of chain complexes is a double complex.
A basic discussion is for instance in section 1.1 of
A more comprehensive discussion is in section 11 of