diagram chasing in homological algebra
salamander lemma$\Rightarrow$
four lemma$\Rightarrow$ five lemma
snake lemma$\Rightarrow$ connecting homomorphism
(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
Diagram chasing is a common technique in homological algebra for proving properties of and constructing morphisms in abelian categories, where one traces elements in various ways around commutative diagrams.
Many basic lemmas in homological algebra, such as the five lemma, the 3x3 lemma and the snake lemma, are typically proven by diagram chases. See for instance the proof at five lemma or any book on homological algebra.
The salamander lemma can sometimes be used to give more conceptual proofs.
There are at least five approaches to performing diagram chases in general abelian categories (not assumed to be concrete like Ab or $R$Mod):
George Bergman, A note on abelian categories – translating element-chasing proofs, and exact embedding in abelian groups (1974) [pdf, pdf]
Daniel Murfet, Diagram Chasing in Abelian Categories (2006) (pdf)
George Bergman, On diagram-chasing in double complexes, Theory and Applications of Categories 26 (2012) 60-96 [arXiv:1108.0958, tac:26-03, pdf]
(introducing the salamander lemma)
Last revised on January 13, 2024 at 05:17:48. See the history of this page for a list of all contributions to it.