Homology Theory - Homological Algebra and Homology of Other Objects

Homological Algebra and Homology of Other Objects

A chain complex consists of groups (for all ) and homomorphisms satisfying . This condition shows that the groups are contained in the groups, so one can form the quotient groups, which are called the homology groups of the original complex. There is a similar theory of cochain complexes, consisting of groups and homomorphisms . The simplicial, singular, Čech and Alexander–Spanier groups are all defined by first constructing a chain complex or cochain complex, and then taking its homology. Thus, a substantial part of the work in setting up these groups involves the general theory of chain and cochain complexes, which is known as homological algebra.

One can also associate (co)chain complexes to a wide variety of other mathematical objects, and then take their (co)homology. For example, there are cohomology modules for groups, Lie algebras and so on.