Snake lemma

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. Homomorphisms constructed with its help are generally called "connecting homomorphisms".
Statement.
In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), consider a commutative diagram:
where the rows are exact sequences and 0 is the zero object.
Then there is an exact sequence relating the kernels and cokernels of "a", "b", and "c":
where "d" is a homomorphism, known as the "connecting homomorphism".
Furthermore, if the morphism "f" is a monomorphism, then so is the morphism formula_2, and if "g"' is an epimorphism, then so is formula_3.
The cokernels here are: formula_4, formula_5, formula_6.
Explanation of the name.
To see where the snake lemma gets its name, expand the diagram above as follows:
and then note that the exact sequence that is the conclusion of the lemma can be drawn on this expanded diagram in the reversed "S" shape of a slithering snake.
Construction of the maps.
The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given (horizontal) maps because of the diagram's commutativity. The exactness of the two induced sequences follows in a straightforward way from the exactness of the rows of the original diagram. The important statement of the lemma is that a "connecting homomorphism" "d" exists which completes the exact sequence.
In the case of abelian groups or modules over some ring, the map "d" can be constructed as follows:
Pick an element "x" in ker "c" and view it as an element of "C"; since "g" is surjective, there exists "y" in "B" with "g"("y") = "x". Because of the commutativity of the diagram, we have "g"'("b"("y")) = "c"("g"("y")) = "c"("x") = 0 (since "x" is in the kernel of "c"), and therefore "b"("y") is in the kernel of "g' ". Since the bottom row is exact, we find an element "z" in "A' " with "f" '("z") = "b"("y"). "z" is unique by injectivity of "f" '. We then define "d"("x") = "z" + "im"("a"). Now one has to check that "d" is well-defined (i.e., "d"("x") only depends on "x" and not on the choice of "y"), that it is a homomorphism, and that the resulting long sequence is indeed exact. One may routinely verify the exactness by diagram chasing (see the proof of Lemma 9.1 in ).
Once that is done, the theorem is proven for abelian groups or modules over a ring. For the general case, the argument may be rephrased in terms of properties of arrows and cancellation instead of elements. Alternatively, one may invoke Mitchell's embedding theorem.
Naturality.
In the applications, one often needs to show that long exact sequences are "natural" (in the sense of natural transformations). This follows from the naturality of the sequence produced by the snake lemma.
If
is a commutative diagram with exact rows, then the snake lemma can be applied twice, to the "front" and to the "back", yielding two long exact sequences; these are related by a commutative diagram of the form
Example.
Let formula_7 be field, formula_8 be a formula_7-vector space. formula_8 is formula_11-module by formula_12 being a formula_7-linear transformation, so we can tensor formula_8 and formula_7 over formula_11.
formula_17
Given a short exact sequence of formula_7-vector spaces formula_19, we can induce an exact sequence formula_20 by right exactness of tensor product. But the sequence formula_21 is not exact in general. Hence, a natural question arise. Why is this sequence not exact?
According to the diagram above, we can induce an exact sequence formula_22 by applying the snake lemma. Thus we can see that snake lemma reflects the failure of tensor product being exact.
In popular culture.
The proof of the snake lemma is taught by Jill Clayburgh's character at the very beginning of the 1980 film "It's My Turn".