Editing Snake lemma

{{Short description|Theorem in homological algebra}}
The '''snake lemma''' is a tool used in [[mathematics]], particularly [[homological algebra]], to  construct [[long exact sequence]]s. 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 group]]s or the category of [[vector space]]s over a given [[field (algebra)|field]]), consider a [[commutative diagram]]:
:[[File:Snake lemma origin.svg]]

where the rows are [[exact sequence]]s and 0 is the [[zero object]].

Then there is an exact sequence relating the [[kernel (category theory)|kernels]] and [[cokernel]]s of ''a'', ''b'', and ''c'':

:<math>\ker a ~{\color{Gray}\longrightarrow}~ \ker b ~{\color{Gray}\longrightarrow}~ \ker c ~\overset{d}{\longrightarrow}~ \operatorname{coker}a ~{\color{Gray}\longrightarrow}~ \operatorname{coker}b ~{\color{Gray}\longrightarrow}~ \operatorname{coker}c</math>

where ''d'' is a homomorphism, known as the ''connecting homomorphism''.

Furthermore, if the morphism ''f'' is a [[monomorphism]], then so is the morphism <math>\ker a ~{\color{Gray}\longrightarrow}~ \ker b</math>, and if ''g''' is an [[epimorphism]], then so is <math>\operatorname{coker} b ~{\color{Gray}\longrightarrow}~ \operatorname{coker} c</math>.

The cokernels here are: <math>\operatorname{coker}a = A'/\operatorname{im}a</math>, <math>\operatorname{coker}b = B'/\operatorname{im}b</math>, <math>\operatorname{coker}c = C'/\operatorname{im}c</math>.

== Explanation of the name ==
To see where the snake lemma gets its name, expand the diagram above as follows:
:[[File:Snake lemma complete.svg]]

and then 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 ==
[[File:Snake lemma map construction.gif|alt=An animation of the diagram chase to construct the map d by finding d(x) given some x in ker c|thumb|361x361px|An animation of the construction of the map d]]
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'' <math>d</math> exists which completes the exact sequence.

In the case of abelian groups or [[module (mathematics)|modules]] over some [[ring (mathematics)|ring]], the map <math>d</math> can be constructed as follows:

Pick an element <math>x</math> in <math>\operatorname{ker} c</math> and view it as an element of <math>C</math>. Since <math>g</math> is [[surjective]], there exists <math>y</math> in <math>B</math> with <math>g(y)=x</math>. By commutativity of the diagram, we have <math>g'(b(y)) = c(g(y)) = c(x) = 0</math> (since <math>x</math> is in the kernel of <math>c</math>), and therefore <math>b(y)</math> is in the kernel of <math>g'</math>. Since the bottom row is exact, we find an element <math>z</math> in <math>A'</math> with <math>f'(z)=b(y)</math>. By injectivity of <math>f'</math>, <math>z</math> is unique. We then define <math>d(x)=z+\operatorname{im}(a)</math>. Now one has to check that <math>d</math> is well-defined (i.e., <math>d(x)</math> only depends on <math>x</math> and not on the choice of <math>y</math>), 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 <ref>{{harvnb|Lang|2002|p=159}}</ref>).

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 transformation]]s). This follows from the naturality of the sequence produced by the snake lemma.

If
:[[File:Snake lemma nature.svg|commutative diagram with exact rows]]

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
:[[File:snake lemma nat2.svg|commutative diagram with exact rows]]

== Example ==

Let <math>k</math> be field, <math>V</math> be a <math>k</math>-vector space. <math>V</math> is <math>k[t]</math>-module by <math>t:V \to V</math> being a <math>k</math>-linear transformation, so we can tensor <math>V</math> and <math>k</math> over <math>k[t]</math>.

: <math>V \otimes_{k[t]} k = V \otimes_{k[t]} (k[t]/(t)) = V/tV = \operatorname{coker}(t) .</math>

Given a short exact sequence of <math>k</math>-vector spaces <math>0 \to M \to N \to P \to 0</math>, we can induce an exact sequence <math>M \otimes_{k[t]} k \to N \otimes_{k[t]} k \to P \otimes_{k[t]} k \to 0</math> by right exactness of tensor product. But the sequence <math>0 \to M \otimes_{k[t]} k \to N \otimes_{k[t]} k \to P \otimes_{k[t]} k \to 0</math> is not exact in general. Hence, a natural question arises. Why is this sequence not exact?

[[File:Snklem.png|430x430px]]

According to the diagram above, we can induce an exact sequence <math>\ker(t_M) \to \ker(t_N) \to \ker(t_P) \to M \otimes_{k[t]} k \to N \otimes_{k[t]} k \to P \otimes_{k[t]} k \to 0</math> by applying the snake lemma. Thus, the snake lemma reflects the tensor product's failure to be exact.

== In the category of groups ==
Whether the snake lemma holds in the category of groups depends on the definition of cokernel. If <math>f: A \to B</math> is a homomorphism of groups, the universal property of the cokernel is satisfied by the natural map <math>B \to B / N(\operatorname{im} f)</math>, where <math>N(\operatorname{im} f)</math> is the normalization of the image of <math>f</math>. The snake lemma fails with this definition of cokernel: 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.

If one simply replaces the cokernels in the statement of the snake lemma with the (right) cosets <math>A' / \operatorname{im} a, B' / \operatorname{im} b, C' / \operatorname{im} c'</math>, the lemma is still valid. The quotients however are not groups, but pointed sets (a short sequence <math> (X, x) \to (Y, y) \to (Z, z)</math> of pointed sets with maps <math>f: X \to Y</math> and <math>g: Y \to Z</math> is called exact if <math> f(X) = g^{-1}(z)</math>).

=== Counterexample to snake lemma with categorical cokernel ===
Consider the [[alternating group]] <math>A_5</math>: this contains a subgroup isomorphic to the [[symmetric group]] <math>S_3</math>, which in turn can be written as a semidirect product of [[cyclic group]]s: <math>S_3\simeq C_3\rtimes C_2</math>.<ref>{{cite web|url=https://people.maths.bris.ac.uk/~matyd/GroupNames/1/e2/C2byC3.html#s1|title=Extensions of C2 by C3|accessdate=2021-11-06|work=GroupNames}}</ref> This gives rise to the following diagram with exact rows:
:<math>\begin{matrix} & 1 & \to & C_3 & \to & C_3 & \to 1\\
& \downarrow && \downarrow && \downarrow \\
1 \to & 1 & \to & S_3 & \to & A_5
\end{matrix}
</math>
Note that the middle column is not exact: <math>C_2</math> is not a normal subgroup in the semidirect product.

Since <math>A_5</math> is [[simple group|simple]], the right vertical arrow has trivial cokernel. Meanwhile the quotient group <math>S_3/C_3</math> is isomorphic to <math>C_2</math>. The sequence in the statement of the snake lemma is therefore
:<math>1 \longrightarrow 1 \longrightarrow 1  \longrightarrow 1  \longrightarrow C_2  \longrightarrow 1</math>,
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]]''.<ref>{{cite journal |first=C. L. |last=Schochet |title=The Topological Snake Lemma and Corona Algebras |journal=New York Journal of Mathematics |volume=5 |year=1999 |pages=131–7 |url=http://www.emis.de/journals/NYJM/j/1999/5-11.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.emis.de/journals/NYJM/j/1999/5-11.pdf |archive-date=2022-10-09 |url-status=live |citeseerx=10.1.1.73.1568 }}</ref>

==See also==
* [[Zig-zag lemma]]

==References==
{{Reflist}}
{{refbegin}}
*{{cite book |author-link=Serge Lang |first=Serge |last=Lang |title=Algebra |publisher=Springer |edition=3rd |date=2002 |isbn=978-0-387-95385-4 |pages=157–9 |chapter=III §9 The Snake Lemma |chapter-url={{GBurl|Fge-BwqhqIYC|pg=PA157}} }}
*{{cite book |author1-link=Michael Francis Atiyah |first1=M.F. |last1=Atiyah |author2-link=Ian G. Macdonald |first2=I. G. |last2=Macdonald |title=Introduction to Commutative Algebra |publisher=Addison–Wesley |date=1969 |isbn=0-201-00361-9 }}
*{{cite book |first1=P. |last1=Hilton |first2=U. |last2=Stammbach |title=A course in homological algebra |publisher=Springer |series=[[Graduate Texts in Mathematics]] |date=1997 |isbn=0-387-94823-6 |page=99 }}
{{refend}}

==External links==
*{{MathWorld|title=Snake Lemma|urlname=SnakeLemma}}
*[http://planetmath.org/encyclopedia/SnakeLemma.html Snake Lemma] at [[PlanetMath]]
*[https://www.youtube.com/watch?v=etbcKWEKnvg Proof of the Snake Lemma] in the film [https://www.imdb.com/title/tt0080936/ It's My Turn]

[[Category:Homological algebra]]
[[Category:Lemmas in category theory]]