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 .
A basic example is the singular chain complex of a topological space, or the de Rham complex of a smooth manifold. Another type of example occurs with the Dold-Kan correspondence as the Moore complex of a simplicial abelian group or similar. Both the first and the third of these types of example correspond, on the surface, to chain complexes in which the grading is by , not . Dually the de Rham complex example can be included by indexing by the non-positive integers, but by defining them to take trivial, that is zero, values in other dimensions they become chain complexes in the sense used here. The more general definition is important as it is (i) more inclusive and (ii) leads to objects that behave well with respect to shift / translation operators, (see below).
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