Diagram chasing lemmas
A basic lemma in homological algebra: it constructs connecting homomorphisms.
be a commuting diagram in an abelian category such that the two rows are exact sequences.
Then there is a long exact sequence of kernels and cokernels of the form
if is a monomorphism then so is
if is an epimorphism, then so is .
If is realized as a (full subcategory of) a category of -modules, then the connecting homomorphism here can be defined on elements by
where and denote any choice of pre-image (the total formula is independent of that choice).
An early occurence of the snake lemma is as lemma (5.8) of
- D. A. Buchsbaum, Exact categories and duality, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 (JSTOR)
it appears as lemma 1.3.2.
A purely category-theoretic proof is given in
- Temple Fay, Keith Hardie, Peter Hilton, The two-square lemma, Publicacions Matemàtiques, Vol 33 (1989) (pdf)
Revised on September 18, 2012 23:43:24
by Urs Schreiber