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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).
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 A. (1999), "History of homological algebra" (PDF), History of topology, Amsterdam: North-Holland, pp. 797– 836, ISBN 9780444823755, MR 1721123
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 F 1 {\displaystyle F_{1}} and a surjective homomorphism p 1 : F 1 → X . {\displaystyle p_{1}:F_{1 ...
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.
He worked on the axiomatic treatment of homology theory with Norman Steenrod (and the Eilenberg–Steenrod axioms are named for the pair), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory. Eilenberg was a member of Bourbaki and, with Henri Cartan, wrote the 1956 book Homological ...
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