nLab
snake lemma

Context

Diagram chasing lemmas

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

A basic lemma in homological algebra: it constructs connecting homomorphisms.

Statement

Lemma

Let

A B p C 0 f g h 0 A i B C \array{ && A' &\to & B' &\stackrel{p}{\to}& C' &\to & 0 \\ && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{h}} \\ 0 &\to& A &\stackrel{i}{\to} & B &\to& C }

be a commuting diagram in an abelian category 𝒜\mathcal{A} such that the two rows are exact sequences.

Then there is a long exact sequence of kernels and cokernels of the form

ker(f)ker(g)ker(h)coker(f)coker(g)coker(h). ker(f) \to ker(g) \to ker(h) \stackrel{\partial}{\to} coker(f) \to coker(g) \to coker(h) \,.

Moreover

  • if ABA \to B is a monomorphism then so is ker(f)ker(g)ker(f) \to ker(g)

  • if BCB \to C is an epimorphism, then so is coker(g)coker(h)coker(g) \to coker(h).

If 𝒜\mathcal{A} is realized as a (full subcategory of) a category of RR-modules, then the connecting homomorphism \partial here can be defined on elements cker(h)Cc' \in ker(h) \subset C' by

(c):=i 1gp 1(c), \partial (c') := i^{-1} \,g\, p^{-1}(c') \,,

where i 1()i^{-1}(-) and p 1()p^{-1}(-) denote any choice of pre-image (the total formula is independent of that choice).

Remark

The snake lemma derives its name from the fact that one may draw the connecting homomorphism \partial that it constructs diagrammatically as follows:

Snake lemma diagram

References

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)

In

it appears as lemma 1.3.2.

A purely category-theoretic proof is given in

and in

See also

Revised on April 9, 2014 06:32:44 by Tim Porter (2.26.27.237)