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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 ...
[12] [13] He was one of the first to propose the idea of a quantum computer in 1980 with his book Computable and Uncomputable. [14] He wrote a book on cubic surfaces and cubic forms, showing how to apply both classical and contemporary methods of algebraic geometry, as well as nonassociative algebra. [15]
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.
The books in this series are published in hardcover and e-book formats. List of books ... 234 Homological Methods in Commutative Algebra, Andrea Ferretti ...
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).
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
An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259. Yekutieli, Amnon (2019). Derived Categories. Cambridge Studies in Advanced Mathematics. Vol. 183. Cambridge University Press. ISBN 978-1108419338
Homotopical algebra (published as a book and also sometimes called noncommutative homological algebra): The study of various model categories and the interplay between fibrations, cofibrations and weak equivalences in arbitrary closed model categories 1967: Daniel Quillen: Quillen axioms for homotopy theory in model categories: 1967: Daniel Quillen
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