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In
mathematics, particularly
homological algebra, the '''snake lemma''', a statement valid in every
Abelian category, is the crucial tool used to construct the
long exact sequences that are ubiquitous in
homological algebra and its applications, for instance in
algebraic topology.
== 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
Image:SnakeLemma01.png
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'':
Image:SnakeLemma02.png
Furthermore, if the morphism
f is a
monomorphism, then so is the morphism ker
a → ker
b, and if
g' is an
epimorphism, then so is coker
b → coker
c.
== Explanation of the name ==
To see where the snake lemma gets its name, expand the diagram above as follows:
Image:SnakeLemma03.png
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''). We defined ''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 choices of ''y'' and ''z''), that it is a homomorphism, and that the resulting long sequence is indeed exact.
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
:
Image:Snake lemma nat.png
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
:
Image:Snake lemma nat2.png
