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The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology .
The snake lemma shows how a commutative diagram with two exact rows gives rise to a longer exact sequence. The nine lemma is a special case. The five lemma gives conditions under which the middle map in a commutative diagram with exact rows of length 5 is an isomorphism; the short five lemma is a special case thereof applying to short exact ...
The name of the lemma arises from the "zig-zag" behavior of the maps in the sequence. A variant version of the zig-zag lemma is commonly known as the "snake lemma" (it extracts the essence of the proof of the zig-zag lemma given below).
If 0 → A → B → C → 0 is a short exact sequence in A, then applying F yields the exact sequence 0 → F(A) → F(B) → F(C) and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact ...
The above short exact sequences specifying the relative chain groups give rise to a chain complex of short exact sequences. An application of the snake lemma then yields a long exact sequence
The above-mentioned property of turning short exact sequences into long exact sequences is a consequence of the snake lemma. This tells us that the collection of derived functors is a δ-functor. If X is itself injective, then we can choose the injective resolution 0 → X → X → 0, and we obtain that R i F(X) = 0 for all i ≥ 1. In ...
All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequences, and derived functors. Important theorems that apply in all abelian categories include the five lemma (and the short five lemma as a special case), as well as the snake lemma (and the nine lemma as a special case).
Schanuel's lemma; Schur multiplier; Secondary calculus and cohomological physics; Section (category theory) Severi–Brauer variety; Shapiro's lemma; Sheaf cohomology; Short five lemma; Six operations; Snake lemma; Splitting lemma; Standard complex