Homological algebra studies chain complexes, homology, cohomology, and the derived constructions that measure failure of exactness.
A cochain complex is a sequence of abelian groups or modules and maps satisfying . Its cohomology is .
The Hom functor sends two -modules to the abelian group of -linear maps between them. It is contravariant in its first argument and covariant in its second.
The Ext functors are the right derived functors of the Hom functor. They can be computed by applying to a projective resolution of and taking cohomology.
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Homological algebra is a branch of mathematics that studies algebraic structures and their relationships using concepts and methods from homology and cohomology. It originated from the study of algebraic topology but has since become a central area in various fields of mathematics, including algebra, geometry, and category theory.