Editing Five lemma

{{Short description|Lemma in category theory about commutative diagrams}}
In [[mathematics]], especially [[homological algebra]] and other applications of [[abelian category]] theory, the '''five lemma''' is an important and widely used [[lemma (mathematics)|lemma]] about [[commutative diagram]]s.
The five lemma is not only valid 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 [[duality (category theory)|dual]] to each other.

==Statements==
Consider the following [[commutative diagram]] in any [[abelian category]] (such as the category of [[abelian group]]s or the category of [[vector space]]s over a given [[field (algebra)|field]]) or in the category of [[group (mathematics)|group]]s.

: [[file:5 lemma.svg]]

The five lemma states that, if the rows are [[exact sequence|exact]], ''m'' and ''p'' are [[isomorphism]]s, ''l'' is an [[epimorphism]], and ''q'' is a [[monomorphism]], then ''n'' is also an isomorphism.

The two four-lemmas state:{{ordered list
| If the rows in the commutative diagram
: [[file:4 lemma right.svg]]

are exact and ''m'' and ''p'' are epimorphisms and ''q'' is a monomorphism, then ''n'' is an epimorphism.
| If the rows in the commutative diagram
: [[file:4 lemma left.svg]]

are exact and ''m'' and ''p'' are monomorphisms and ''l'' is an epimorphism, then ''n'' is a monomorphism.
}}

==Proof==
The method of proof we shall use is commonly referred to as [[diagram chasing]].<ref>{{cite book|author=Massey |title=A basic course in algebraic topology|year=1991|url={{Google books|plainurl=y|id=svbU9nxi2xQC|page=184|text=diagram chasing}}|page=184}}</ref> We shall prove the five lemma by individually proving each of the two four lemmas.

To perform diagram chasing, we assume that we are in a category of [[module (mathematics)|modules]] over some [[ring (mathematics)|ring]], so that we may speak of ''elements'' of the objects in the diagram and think of the morphisms of the diagram as ''[[function (mathematics)|function]]s'' (in fact, [[homomorphism]]s) acting on those elements.
Then a morphism is a monomorphism [[if and only if]] it is [[injective]], and it is an epimorphism if and only if it is [[surjective]].
Similarly, to deal with exactness, we can think of [[kernel (algebra)|kernel]]s and [[image (function)|image]]s 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 of abelian group is never used.

So, to prove (1), assume that ''m'' and ''p'' are surjective and ''q'' is injective.
[[File:Four lemma epic and zero.gif|thumb|300px|A proof of (1) in the case where <math>t(c') = 0</math>]]
[[File:4lemma-epic-nonzero.gif|alt=An animation showing a diagram chase to prove (1) of the 4 lemma. This is the case where we assume c' gets sent to a nonzero element and want to show the map from B to B' is epic.|thumb|302x302px|A proof of (1) in the case where <math>t(c')\neq 0</math> is nonzero]]
: [[file:4 lemma right.svg]]
* Let ''c&prime;'' be an element of ''C&prime;''.
* Since ''p'' is surjective, there exists an element ''d'' in ''D'' with ''p''(''d'') =  ''t''(''c&prime;'').
* By commutativity of the diagram, ''u''(''p''(''d'')) = ''q''(''j''(''d'')).
* Since im ''t'' = ker ''u'' by exactness, 0 = ''u''(''t''(''c&prime;'')) = ''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&prime;'').  Since ''t'' is a homomorphism, it follows that ''t''(''c&prime;'' − ''n''(''c'')) = 0.
* By exactness, ''c&prime;'' − ''n''(''c'') is in the image of ''s'', so there exists ''b&prime;'' in ''B&prime;'' with ''s''(''b&prime;'') = ''c&prime;'' − ''n''(''c'').
* Since ''m'' is surjective, we can find ''b'' in ''B'' such that ''b&prime;'' = ''m''(''b'').
* By commutativity, ''n''(''g''(''b'')) = ''s''(''m''(''b'')) = ''c&prime;'' − ''n''(''c'').
* Since ''n'' is a homomorphism, ''n''(''g''(''b'') + ''c'') = ''n''(''g''(''b'')) + ''n''(''c'') = ''c&prime;'' − ''n''(''c'') + ''n''(''c'') = ''c&prime;''.
* Therefore, ''n'' is surjective.

Then, to prove (2), assume that ''m'' and ''p'' are injective and ''l'' is surjective.
[[File:Four lemma monic case.gif|thumb|300px|A proof of (2)]]
: [[file:4 lemma left.svg]]
* 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&prime;'' of ''A&prime;'' such that ''r''(''a&prime;'') = ''m''(''b'').
* Since ''l'' is surjective, there is ''a'' in ''A'' such that {{nowrap|1=''l''(''a'') = ''a&prime;''}}.
* By commutativity, {{nowrap|1=''m''(''f''(''a'')) = ''r''(''l''(''a'')) = ''m''(''b'')}}.
* Since ''m'' is injective, ''f''(''a'') = ''b''.
* So ''c'' = ''g''(''f''(''a'')).
* Since the composition of ''g'' and ''f'' is trivial, ''c'' = 0.
* Therefore, ''n'' is injective.

Combining the two four lemmas now proves the entire five lemma.

==Applications==
The five lemma is often applied to [[long exact sequence]]s: when computing [[homology (mathematics)|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.

==See also==
*[[Short five lemma]], a special case of the five lemma for [[short exact sequence]]s
*[[Snake lemma]], another lemma proved by diagram chasing
*[[Nine lemma]]

==Notes==
{{Reflist}}

==References==
* {{cite book |first=W.R. |last=Scott |title=Group Theory |orig-year=1964 |publisher=Dover |year=1987 |isbn=978-0-486-65377-8 |url={{GBurl|fAPLAgAAQBAJ|pg=PP11}}}}
* {{Citation| last=Massey| first=William S.| author-link=William S. Massey| date=1991| title=A basic course in algebraic topology| edition=3rd| volume = 127 | series=Graduate texts in mathematics | publisher=Springer | isbn = 978-0-387-97430-9}}

[[Category:Homological algebra]]
[[Category:Lemmas in category theory]]
[[Category:Articles containing proofs]]