snake lemma

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The snake lemma is a tool used in
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, particularly
homological algebra Homological algebra is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
, to construct
long exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules) such that the image An SAR radar imaging, radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teid ...
s. The snake lemma is valid in every
abelian category In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
and is a crucial tool in homological algebra and its applications, for instance in
algebraic topology 250px, A torus, one of the most frequently studied objects in algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathemat ...
. Homomorphisms constructed with its help are generally called ''connecting homomorphisms''.

# Statement

In an
abelian category In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
(such as the category of
abelian group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s or the category of
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s over a given
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
), consider a commutative diagram: : where the rows are exact sequences and 0 is the zero object. Then there is an exact sequence relating the kernel (category theory), kernels and cokernels of ''a'', ''b'', and ''c'': :$\ker a ~~ \ker b ~~ \ker c ~\overset~ \operatornamea ~~ \operatornameb ~~ \operatornamec$ where ''d'' is a homomorphism, known as the ''connecting homomorphism''. 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 $\operatorname b ~~ \operatorname c$. The cokernels here are: $\operatornamea = A\text{'}/\operatornamea$, $\operatornameb = B\text{'}/\operatornameb$, $\operatornamec = C\text{'}/\operatornamec$.

# 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 module (mathematics), modules over some ring (mathematics), 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 Commutative diagram#Diagram chasing, 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 $k$ be field, $V$ be a $k$-vector space. $V$ is $k\left[t\right]$-module by $t:V \to V$ being a $k$-linear transformation, so we can tensor $V$ and $k$ over $k\left[t\right]$. : $V \otimes_ k = V \otimes_ \left(k\left[t\right]/\left(t\right)\right) = V/tV = \operatorname\left(t\right) .$ Given a short exact sequence of $k$-vector spaces $0 \to M \to N \to P \to 0$, we can induce an exact sequence $M \otimes_ k \to N \otimes_ k \to P \otimes_ k \to 0$ by right exactness of tensor product. But the sequence $0 \to M \otimes_ k \to N \otimes_ k \to P \otimes_ k \to 0$ is not exact in general. Hence, a natural question arises. Why is this sequence not exact? According to the diagram above, we can induce an exact sequence $\ker\left(t_M\right) \to \ker\left(t_N\right) \to \ker\left(t_P\right) \to M \otimes_ k \to N \otimes_ k \to P \otimes_ k \to 0$ by applying the snake lemma. Thus, the snake lemma reflects the tensor product's failure to be exact.

# In the category of groups

While many results of homological algebra, such as the five lemma or the nine lemma, hold for abelian categories as well as in the category of groups, the snake lemma does not. Indeed, arbitrary cokernels do not exist. However, one can replace cokernels by (left) cosets $A\text{'}/\operatorname a$, $B\text{'}/\operatorname b$, and $C\text{'}/\operatorname c$. Then the connecting homomorphism can still be defined, and one can write down a sequence as in the statement of the snake lemma. This will always be a chain complex, but it may fail to be exact. Exactness can be asserted, however, when the vertical sequences in the diagram are exact, that is, when the images of ''a'', ''b'', and ''c'' are normal subgroups.

## Counterexample

Consider the alternating group $A_5$: this contains a subgroup isomorphic to the symmetric group $S_3$, which in turn can be written as a semidirect product of cyclic groups: $S_3\simeq C_3\rtimes C_2$. This gives rise to the following diagram with exact rows: :$\begin & 1 & \to & C_3 & \to & C_3 & \to 1\\ & \downarrow && \downarrow && \downarrow \\ 1 \to & 1 & \to & S_3 & \to & A_5 \end$ Note that the middle column is not exact: $C_2$ is not a normal subgroup in the semidirect product. Since $A_5$ is simple group, simple, the right vertical arrow has trivial cokernel. Meanwhile the quotient group $S_3/C_3$ is isomorphic to $C_2$. The sequence in the statement of the snake lemma is therefore :$1 \longrightarrow 1 \longrightarrow 1 \longrightarrow 1 \longrightarrow C_2 \longrightarrow 1$, which indeed fails to be 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 (film), It's My Turn''.

* Zig-zag lemma

# References

* Serge Lang: ''Algebra''. 3rd edition, Springer 2002, , pp. 157–159 () * Michael Francis Atiyah, M. F. Atiyah; Ian G. Macdonald, I. G. Macdonald: ''Introduction to Commutative Algebra''. Oxford 1969, Addison–Wesley Publishing Company, Inc. . *P. Hilton; U. Stammbach: ''A course in homological algebra.'' 2. Auflage, Springer Verlag, Graduate Texts in Mathematics, 1997, , p. 99 ()