In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel of the next.
Associated to a chain complex is its homology, which is (loosely speaking) a measure of the failure of a chain complex to be exact.
The homology of a cochain complex is called its cohomology.
In algebraic topology, the singular chain complex of a topological space X is constructed using continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex.
The homology of this chain complex is called the singular homology of X, and is a commonly used invariant of a topological space.
Chain complexes are studied in homological algebra, but are used in several areas of mathematics, including abstract algebra, Galois theory, differential geometry and algebraic geometry.
is a sequence of abelian groups or modules ..., A0, A1, A2, A3, A4, ... connected by homomorphisms (called boundary operators or differentials) dn : An → An−1, such that the composition of any two consecutive maps is the zero map.
Explicitly, the differentials satisfy dn ∘ dn+1 = 0, or with indices suppressed, d2 = 0.
It consists of a sequence of abelian groups or modules ..., A0, A1, A2, A3, A4, ... connected by homomorphisms dn : An → An+1 satisfying dn+1 ∘ dn = 0.
All the concepts and definitions for chain complexes apply to cochain complexes, except that they will follow this different convention for dimension, and often terms will be given the prefix co-.
In this article, definitions will be given for chain complexes when the distinction is not required.
A chain complex is bounded above if all modules above some fixed degree N are 0, and is bounded below if all modules below some fixed degree are 0.
The n-th (co)homology group Hn (Hn) is the group of (co)cycles modulo (co)boundaries in degree n, that is, An exact sequence (or exact complex) is a chain complex whose homology groups are all zero.
This means all closed elements in the complex are exact.
For example, the following chain complex is a short exact sequence.
for each n that commutes with the boundary operators on the two chain complexes, so
A chain map sends cycles to cycles and boundaries to boundaries, and thus induces a map on homology
Given two chain complexes A and B, and two chain maps f, g : A → B, a chain homotopy is a sequence of homomorphisms hn : An → Bn+1 such that hdA + dBh = f − g. The maps may be written out in a diagram as follows, but this diagram is not commutative.
One says f and g are chain homotopic (or simply homotopic), and this property defines an equivalence relation between chain maps.
Define Cn(X) for natural n to be the free abelian group formally generated by singular n-simplices in X, and define the boundary map
to be where the hat denotes the omission of a vertex.
That is, the boundary of a singular simplex is the alternating sum of restrictions to its faces.
Singular homology is a useful invariant of topological spaces up to homotopy equivalence.
The differential k-forms on any smooth manifold M form a real vector space called Ωk(M) under addition.
The cohomology of this complex is called the de Rham cohomology of M. Locally constant functions are designated with its isomorphism
with c the count of mutually disconnected components of M. This way the complex was extended to leave the complex exact at zero-form level using the subset operator.
Chain complexes of K-modules with chain maps form a category ChK, where K is a commutative ring.
is a chain complex with degree n elements given by and differential given by where a and b are any two homogeneous vectors in V and W respectively, and
The identity object with respect to this monoidal product is the base ring K viewed as a chain complex in degree 0.
Moreover, the category of chain complexes of K-modules also has internal Hom: given chain complexes V and W, the internal Hom of V and W, denoted Hom(V,W), is the chain complex with degree n elements given by