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In homological algebra, the Cartan–Eilenberg resolution is in a sense, a resolution of a chain complex. It can be used to construct hyper-derived functors. It is named in honor of Henri Cartan and Samuel Eilenberg.
Cartan-Eilenberg: In their 1956 book "Homological Algebra", these authors used projective and injective module resolutions. 'Tohoku': The approach in a celebrated paper by Alexander Grothendieck which appeared in the Second Series of the Tohoku Mathematical Journal in 1957, using the abelian category concept (to include sheaves of abelian groups).
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.
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 .
Henri Paul Cartan (French:; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. [1] [2] [3]He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of composer Jean Cartan [fr; de], physicist Louis Cartan [] and mathematician Hélène Cartan [], and the son-in-law of physicist Pierre ...
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. Weibel, Charles (1999), "History of homological algebra", History of topology (PDF), Amsterdam: North-Holland, pp. 797– 836, MR 1721123
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
Let k be a field, A an associative k-algebra, and M an A-bimodule.The enveloping algebra of A is the tensor product = of A with its opposite algebra.Bimodules over A are essentially the same as modules over the enveloping algebra of A, so in particular A and M can be considered as A e-modules.