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Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology ) and abstract algebra (theory of modules and syzygies ) at the end of the 19th century, chiefly by ...
Research there allowed him to put homological algebra on an axiomatic basis, by introducing the abelian category concept. [5] [6] A textbook treatment of homological algebra, "Cartan–Eilenberg" after the authors Henri Cartan and Samuel Eilenberg, appeared in 1956. Grothendieck's work was largely independent of it.
In abstract algebra, one uses homology to define derived functors, for example the Tor functors. Here one starts with some covariant additive functor F and some module X. The chain complex for X is defined as follows: first find a free module and a surjective homomorphism :.
Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulated categories was to extend properties of sheaves on a space X to complexes of sheaves, viewed as objects of the derived category of sheaves on X ...
In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra, and possibly the abelian aspects as special cases. . The homotopical nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory, as in nonabelian algebraic topology, and in particular the theory of closed model categor
Grothendieck elegantly defined and characterized sheaf cohomology in the language of homological algebra. The essential point is to fix the space X and think of sheaf cohomology as a functor from the abelian category of sheaves on X to abelian groups. Start with the functor taking a sheaf E on X to its abelian group of global sections over X, E(X).
Homological algebra is a collection of algebraic techniques that originated in the study of algebraic topology but has also found applications to group theory and algebraic geometry The main article for this category is Homological algebra .
In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology.In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that we can talk about cohomology, then the cone of a map f being acyclic means that the map is a quasi ...