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Concept
Homological algebra
Summary
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 Henri Poincaré and David Hilbert.
Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of . A central concept is that of chain complexes, which can be studied through both their homology and cohomology.
Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.
It has played an enormous role in algebraic topology. Its influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.
Homological algebra began to be studied in its most basic form in the 1800s as a branch of topology, but it wasn't until the 1940s that it became an independent subject with the study of objects such as the ext functor and the tor functor, among others.
Chain complex
The notion of chain complex is central in homological algebra. An abstract chain complex is a sequence of abelian groups and group homomorphisms,
with the property that the composition of any two consecutive maps is zero:
The elements of Cn are called n-chains and the homomorphisms dn are called the boundary maps or differentials.
Official source
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Ontological neighbourhood