Chain complex

A chain complex is a sequence of abelian groups or modules connected by boundary maps such that each pair of successive maps composes to zero. This condition makes boundaries cycles and supports the definition of homology.

Abelian group: A group whose operation is commutative. Chain groups are abelian groups, making sums of chains and their boundaries well-defined.
Kernel (algebra): The set of elements mapped to zero by a homomorphism. Cycles in each degree are precisely the kernel of its boundary map.
Simplicial homology: A homology theory computed from the simplices and boundary maps of a simplicial complex. Its chain groups are generated by oriented simplices.
Cochain complex: A graded sequence with maps that raise degree and whose successive composites are zero. It reverses the direction of the degree-changing maps and supports cohomology.
Module: An algebraic structure that generalizes a vector space by using a ring as its scalars. Modules replace abelian groups when chain groups carry scalars from a ring.
Image (mathematics): The set of outputs of a function or homomorphism. Boundaries in a degree are the image of the boundary map from the next degree.
Singular homology: A homology theory built from continuous maps of simplices into a topological space. Its singular chains form a chain complex for every topological space.
Differential graded algebra: A graded algebra equipped with a degree-changing differential satisfying a graded product rule and square-zero identity. It adds multiplication and compatibility conditions to a chain-complex differential.
Boundary operator: A homomorphism in a chain complex that sends chains in one degree to chains in the preceding degree. These maps define the sequence and must compose to zero.
Exact sequence: A sequence of homomorphisms in which each map's image equals the next map's kernel. A chain complex requires image contained in kernel; exactness demands equality.

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