Five lemma
In
mathematics, especially
homological algebra and other applications of
Abelian category theory, the
five lemma is an important and widely used
lemma about
commutative diagrams.The five lemma is valid
not only for abelian categories but also works in the
category of groups, for example.
The five lemma can be thought of as a combination of two other theorems, the
four lemmas, which are
dual to each other.
Consider the following
commutative diagram in any
Abelian category (such as the category of
Abelian groups or the category of
vector spaces over a given
field) or in the category of
groups.
 |
FiveLemma.png |
The five lemma states that, if the rows are
exact,
m and
p are
isomorphisms,
l is an
epimorphism, and
q is a
monomorphism, then
n is also an isomorphism.
The two four-lemmas state:
(1) If the rows in the commutatve diagram
 |
FourLemma01.png |
are exact and
m and
p are epimorphisms and
q is a monomorphism, then
n is an epimorphism.
(2) If the rows in the commutatve diagram
 |
FourLemma02.png |
are exact and
m and
p are monomorphisms and
l is an epimorphism, then
n is a monomorphism.
The method of proof we shall use is commonly referred to as
diagram chasing. Although it may boggle the mind at first, once one has some practice at it, it is actually fairly routine. We shall prove the five lemma by individually proving each of the 2 four lemmas.
To perform diagram chasing, we assume that we are in a category of
modules over some
ring, so that we may speak of
elements of the objects in the diagram and think of the morphisms of the diagram as
functions (in fact,
homomorphisms) acting on those elements.Then a morphism is a monomorphism
if and only if it is
injective, and it is an epimorphism iff it is
surjective.Similarly, to deal with exactness, we can think of
kernels and
images in a function-theoretic sense.The proof will still apply to any (small) Abelian category because of
Mitchell's embedding theorem, which states that any small Abelian category can be represented as a category of modules over some ring.For the category of groups, just turn all additive notation below into multiplicative notation, and note that commutativity is never used.
So, to prove (1), assume that
m and
p are surjective and
q is injective.
|
FourLemma01.png |
* Let
c be an element of C.
* Since
p is surjective, there exists an element
d in
D with
p(
d) =
t(
c).
* By commutativity of the diagram, u
(p
(d
)) = q
(j
(d
)).
* Since im t
= ker u
by exactness, 0 = u
(t
(c)) =
u(
p(
d)) =
q(
j(
d)).
* Since
q is injective,
j(
d) = 0, so
d is in ker
j = im
h.
* Therefore there exists
c in
C with
h(
c) =
d.
* Then
t(
n(
c)) =
p(
h(
c)) =
t(
c), and since t
is a homomorphism it follows t
(c −
n(
c)) = 0.
* By exactness,
c − n
(c
) must be in the image of s
and therefore there exists b in
B with s
(b) =
c− n
(c
).
* Since m
is surjective, we can find b
in B
such that b =
m(
b).
* By commutativity,
n(
g(
b)) =
s(
m(
b)) =
c − n
(c
).
* Since n
is a homomorphism, n
(g
(b
) + c
) = n
(g
(b
)) + n
(c
) = c −
n(
c) +
n(
c) =
c'.
* Therefore,
n is surjective.
Then, to prove (2), assume that
m and
p are injective and
l is surjective.
|
FourLemma02.png |
* Let
c in
C be such that
n(
c) = 0.
*
t(
n(
c)) is then 0.
* By commutativity,
p(
h(
c)) = 0.
* Since
p is injective,
h(
c) = 0.
* By exactness, there is an element
b of
B such that
g(
b) =
c.
* By commutativity,
s(
m(
b)) =
n(
g(
b)) =
n(
c) = 0.
* By exactness, there is then an element
a of A such that
r(
a) = m
(b
).
* Since l
is surjective, there is a
in A
such that l
(a
) = a.
* By commutativity,
m(
f(
a)) =
r(
l(
a)) =
m(
b).
* Since
m is injective,
f(
a) =
b.
* So
c =
g(
f(
a)).
* By exactness, ker
g = im
f, so
c = 0.
* Therefore,
n is injective.
Combining the 2 four lemmas now proves the entire five lemma.
The five lemma is often applied to
long exact sequences: when computing
homology or cohomology of a given object, one typically employs a simpler subobject whose homology/cohomology is known, and arrives at a long exact sequence which involves the unknown homology groups of the original object. This alone is often not sufficient to determine the unknown homology groups, but if one can compare the original object and sub object to well-understood ones via morphisms, then a morphism between the respective long exact sequences is induced, and the five lemma can then be used to determine the unknown homology groups.
*
Short five lemma, a special case of the five lemma for
short exact sequences
*
Snake lemma, another lemma proved by diagram chasing
*
Nine lemma
