Homological algebra

Homological algebra studies homology and cohomology through algebraic structures such as chain complexes, derived functors, and spectral sequences.

Chain complex: A sequence of objects connected by maps whose consecutive composites are zero. Chain complexes are the basic structures whose homology homological algebra computes.
Derived functor: A sequence of functors that measures how a non-exact functor fails to preserve exact sequences. Derived functors recover homological information lost when ordinary algebraic operations are not exact.
Tor functor: A family of functors measuring the failure of tensor product to preserve exact sequences. Tor is computed from resolutions and detects hidden relations in tensor products.
Henri Poincaré: A French mathematician whose work helped establish topology and its algebraic study. Poincaré's topological work helped motivate the homology theories later treated algebraically.
Ordinary functor: A structure-preserving map between categories that assigns objects and morphisms consistently. Unlike a derived functor, an ordinary functor alone does not record its failure to preserve exactness.
Homology (mathematics): A sequence of invariants measuring cycles modulo boundaries in an algebraic or topological structure. Homology is the central invariant that motivates the subject's methods.
Projective resolution: An exact chain complex of projective objects mapping onto a given object. Applying a functor to this resolution makes its derived functors computable.
Ext functor: A family of functors classifying extensions and measuring the failure of Hom to be exact. Ext turns resolution methods into invariants of extensions between algebraic objects.
Emmy Noether: A German mathematician whose structural approach transformed abstract algebra. Noether helped establish the group-theoretic and module-theoretic foundations used in homological methods.
Tensor product: A construction combining modules or vector spaces into an object encoding bilinear interactions. The ordinary tensor product can fail to preserve exact sequences, prompting the derived invariant Tor.

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