(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
The -lemma or nine lemma is one of the basic diagram chasing lemmas in homological algebra.
Let
be a short exact sequence of chain complexes. Then if two of the three complexes are exact, so is the remaining third.
Let
be a commuting diagram in some abelian category such that each of the three columns is an exact sequence. Then
If the two bottom rows are exact, then so is the top.
If the top two rows are exact, then so is the bottom.
If the top and bottom rows are exact and is the zero morphism, then also the middle row is exact.
A proof by way of the salamander lemma is spelled out in detail at Salamander lemma - Implications - 3x3 lemma.
An early appearance of the -lemma is as lemma (5.5) in
In
it appears as exercise 1.3.2.
The sharp -lemma appears as lemma 2 in
Also lemma 3.2-3.4 of
Discussion of generalization to non-abelian categories is in
Marino Gran, Diana Rodelo, Goursat categories and the -lemma, Applied Categorical Structures, Vol. 20, No 3, 2012, 229-238. (journal, pdf slides)
Marino Gran, Zurab Janelidze and Diana Rodelo, lemma for star-exact sequences, Homology, Homotopy and Applications, Vol. 14 (2012), No. 2, pp.1-22. (journal)
Dominique Bourn, -lemma and protomodularity, Journal of Algebra, Volume 236, Number 2, 15 February 2001 , pp. 778-795(18)
Dominique Bourn, The denormalized lemma, Journal of Pure and Applied Algebra, Volume 177, Issue 2, 24 January 2003, Pages 113-129, doi:10.1016/S0022-4049(02)00143-3
Last revised on August 4, 2019 at 11:36:18. See the history of this page for a list of all contributions to it.