Homological algebra

Summary

Homological algebra is the part of algebra that studies chain complexes, their homology, and the exact sequences and functors built from them. It grew out of algebraic topology, where homology groups of spaces are computed from chain complexes, and out of the study of syzygies of modules. Its main tools include abelian categories, projective and injective objects, resolutions, derived functors such as Ext and Tor, spectral sequences and derived categories. It provides a common language for algebraic topology, algebraic geometry, commutative algebra and representation theory.