math lessons - Chain complex

In mathematics, in the field of homological algebra, a chain complex $(A_\bullet, d_\bullet)$ is a sequence of abelian groups or modules A₀, A₁, A₂... connected by homomorphisms d_n : A_n→A_n-1, such that the composition of any two consecutive maps is zero: d_n o d_n+1 = 0 for all n. They tend to be written out like so:

$\ldots \to A_{n+1} \begin{matrix} d_{n+1} \\ \to \\ \, \end{matrix} A_n \begin{matrix} d_n \\ \to \\ \, \end{matrix} A_{n-1} \begin{matrix} d_{n-1} \\ \to \\ \, \end{matrix} A_{n-2} \to \ldots \to A_2 \begin{matrix} d_2 \\ \to \\ \, \end{matrix} A_1 \begin{matrix} d_1 \\ \to \\ \, \end{matrix} A_0 \begin{matrix} d_0 \\ \to \\ \, \end{matrix} 0.$

A variant on the concept of chain complex is that of cochain complex. A cochain complex $(A^\bullet, d^\bullet)$ is a sequence of abelian groups or modules A₀, A₁, A₂... connected by homomorphisms d_n : A_n→A_n+1, such that the composition of any two consecutive maps is zero: d_n+1 o d_n = 0 for all n:

$0 \to A_0 \begin{matrix} d_0 \\ \to \\ \, \end{matrix} A_1 \begin{matrix} d_1 \\ \to \\ \, \end{matrix} A_2 \to \ldots \to A_{n-1} \begin{matrix} d_{n-1} \\ \to \\ \, \end{matrix} A_n \begin{matrix} d_n \\ \to \\ \, \end{matrix} A_{n+1} \to \ldots.$

The idea is basically the same.

Applications of chain complexes usually define and apply their homology groups (cohomology groups for cochain complexes); in more abstract settings various equivalence relations are applied to complexes (for example starting with the chain homotopy idea). Chain complexes are easily defined in abelian categories.

A bounded complex is one in which almost all the A_i are 0 — so a finite complex extended to the left and right by 0's. An example is the complex defining the homology theory of a (finite) simplicial complex.

Examples

Singular homology

Suppose we are given a topological space X.

Define C_n(X) for natural n to be the free abelian group formally generated by singular simplices in X, and define the boundary map

$\partial_n: C_n(X) \to C_{n-1}(X): \, (\sigma: [v_0,\ldots,v_n] \to X) \mapsto (\partial_n \sigma = \sum_{i=0}^n (-1)^i \sigma|[v_0,\ldots, \hat v_i, \ldots, v_n]),$

where the hat denotes the omission of a vertex. That is, the boundary of a singular simplex is alternating sum of restrictions to its faces. It can be shown ∂² = 0, so $(C_\bullet, \partial_\bullet)$ is a chain complex; the singular homology $H_\bullet(X)$ is the homology of this complex; that is,

$H_n(X) = \ker \partial_n / \mbox{im } \partial_{n+1}$ .

de Rham cohomology

The differential k-forms on any smooth manifold M form an abelian group (in fact an R-vector space) called Ω^k(M) under addition. The exterior derivative d = d_k maps Ω^k(M) → Ω^k+1(M), and d² = 0 follows essentially from symmetry of second derivatives, so the vector spaces of k-forms along with the exterior derivative are a cochain complex:

$\Omega^0(M) \to \Omega^1(M) \to \Omega^2(M) \to \Omega^3(M) \to \ldots.$

The homology of this complex is the de Rham cohomology

$H^k_{\mathrm{DR}}(M) = \ker d_{k+1} / \mbox{im } d_k$ .

Categories: Homological algebra

Mathematics Encyclopedia and Lessons

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Chain complex

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Mathematics Encyclopedia and Lessons

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Chain complex

Examples

Singular homology

de Rham cohomology

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