Chain complex

A chain complex is a sequence of abelian groups (or modules) C_n together with boundary maps ∂_n: C_n → C_n−1 such that ∂_n−1 ∘ ∂_n = 0 for all n.¹ This condition implies that the image of each boundary map is contained in the kernel of the next, allowing the definition of homology groups H_n = ker ∂_n / im ∂_n+1. A complex is exact if all homology groups are trivial, and a chain map between complexes is a collection of homomorphisms commuting with the boundary maps. Chain complexes can be bounded, unbounded, or finite, and they form a category with morphisms given by chain maps.

Chain complexes are indispensable in algebraic topology, where they arise from geometric objects such as simplicial complexes or CW complexes. The singular chain complex of a topological space is built from continuous maps of simplices into the space, and its homology groups are topological invariants.² For example, the zeroth homology group counts path components, while the first homology group detects loops and holes. The Euler characteristic of a finite complex is the alternating sum of the ranks of its homology groups, a result known as the Euler–Poincaré formula. Chain complexes also enable the definition of cohomology by dualizing, leading to richer algebraic invariants like cup products.³

Beyond topology, chain complexes are fundamental in homological algebra, where they are used to define derived functors such as Ext and Tor. A free resolution of a module is a chain complex of free modules that is exact except at the zeroth term, and its homology yields these functors. Chain complexes also appear in group cohomology, where they encode the cohomology of groups, and in algebraic geometry, where they are used in sheaf cohomology and derived categories. In mathematical physics, chain complexes underlie the BRST formalism in gauge theory and the definition of topological quantum field theories.⁴

Chain complexes have several subtle and less-known features. For instance, the condition ∂² = 0 is sometimes relaxed to ∂² = 0 up to homotopy, leading to A_∞-algebras and homotopy coherent structures. The category of chain complexes over a field is equivalent to the category of graded vector spaces with a differential, and its derived category is a triangulated category. Chain complexes also appear in computational topology, where persistent homology tracks changes across a filtration of complexes. A notable historical curiosity is that the original definition by Poincaré used incidence matrices, and the modern algebraic formulation was developed later by Emmy Noether and others.⁵

Boundary map

A homomorphism between consecutive groups in a chain complex, satisfying ∂² = 0.

Homology group

The quotient of the kernel of one boundary map by the image of the next, measuring the failure of exactness.

Chain map

A collection of homomorphisms between two chain complexes that commute with the boundary maps.

Exact sequence

A chain complex where the image of each map equals the kernel of the next, so all homology groups are zero.

Free resolution

An exact chain complex of free modules used to compute derived functors.