A new proof is given of the connecting homomorphism

Fay, T. H.

Hardie, K. A.

Hilton, Peter

T. H. Fay

K. A. Hardie

P. J. Hilton

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Publicacions Matemàtiques

The two-square lemma

Fay, Temple H.

Keith, Hardie A.

Hilton, Peter J.

Diposit Digital de Documents de la UAB

Publicacions Matemátiques, Vol 33 (1989), 133-137.AbstractTHE TWO-SQUARE LEMMATEMPLE H. FAY, KEITH A. HARDIE, AND PETER J . HILTONA new proof is given of the connecting homomorphism .One of the most useful, and hence important, lemmas of homological algebrain an abelian category (say, the category of R-modules) is the Snake Lemma(see the references at the end of this note). This result is used to produceconnecting homomorphisms, building long exact sequences such as the familiarones involving the Hom - Ext and ® - Tor functors . According to Mac Lane[M1], early proofs of this lmma were obscure. One technique, used by Popescu[P], is to prove the result for abelian groups and use the Mitchell EmbeddingTheorem to obtain the result in other abelian categories . The essential point isthe construction of the homomorphism to do the connecting, not the exactnesswhich comes from simple diagram chasing. In this note we give an interestingnew result, which we call the Two-Square Lemma, that provides a simple andcompletely categorical construction of the snake connecting morphism . Thishas the pleasant property of avoiding all references to zig-zags or Switchbackformulas and to such powerful tools as Mitchell's Embedding Theorem.We shall do all our work in an abelian category, although some relaxing ofthe axioms might be possible, say to an exact category. We begin by statingtwo lemmas which are well-known and fundamental. Indeed apart from thesetwo lemmas we require only the notions of exactness, pullback and pushout ina balanced category.We record that Beyl [B] has also given a purely categorical description of theconnecting homomorphism. He bases his work on the Product Lemma which isour Lemma 4, and does not enunciate a Two-Square Lemma.Lemma 1 . Consider the commutative diagram05Bal pbm'A' --~ B'	.C7C'134	T.H . FAY, K.A . HARDIE, P.J . HILTONwith ¡he botiom roca exact. Then there exists a unique morphism : A' --.> Bsuch that(ZJ N Y' = Y',~and (ii) A' P) B -~-+ C is exactThe dual of this result is, of course, also valid.Lemma 2. (The Sharp 3 x 3 Lemma) . Consider ¡he following commutativediagram having exact columns and rocas .is exact.0 > A2There are uniquely determined morphisms Kl + K2 and K2 --> K3 such thatthe completed diagram is commutative and the sequence>B2 i. C2K3Again, the dual of this result is valid. Lemma 2 and its dual provide the twohalves of the long exact sequence which the Snake Lemma connects up . Thenext lemma is the one cae wish to emphasize .Lemma 3. (The Two-Square Lemma) Let the following commutative dia-gram have exact rocas. Form the pulí-back of y and 0',and the push-out of 0 and a, thusOTal p, Q fil oi17,,~~.% ~p, m, pbA' 3 B' ) C'TWO-SQUARE LEMMA	135Then(i) there exists a unique 0 : Q -+ B' such that 9r = /3, Br'(ii) there exists a unique p : B -~ P such that op = 0, o'p = /3 ;(iii) there exists a unique rl : Q -~ P such that r/r = p, o'i = 0, or7T' = 0 .Moreover, rl is monic if zo' is monic, and 17 is epic if 0 is epic .Proof Assertions (i) and (ii) are obvious. To prove (iii), we first use Lemma1 to find a unique tt : A' --> P such that o'p = ' and A' '> P -°> C is exact .We then claim that aa = po.ForotccY=0=00=opo, ando'pa=Tp'a=Po=o'pO .Thus there exists a unique 77 : Q --> P such that y/T = p, i7r' = ti .But then o77T' = op = 0, and o'r/r = o'p = (~ = Br, o'77r' = o'u _ o' = Br', sothat o'ir/ = 0 .Conversely, if rl : Q + P satisfies rlr = p, o'ij = 0, orlr' = 0, then ortr' = 0 =op, o'i7r' = BT' = o' = o'M, so that 177-' = Es, and the uniqueness of rl satisfying(iii) is established.Consider now the commutative diagramHere the exactness of the top row is obtained by dualizing the argument justgiven, noting that the definition of 77 is evidently self-dual .If 0' is monic, so is p = 71r' . Hence, by the above diagram, so is rl . Dually, if0 is epic, so is 77 .The final ingredient for the Snake Lemma is the following Triangle Lemma.Lemma 4 . (Triangle Lemma) Given the Tranglethere is an exact sequenceker (~ -> ker > coker l; -> cokerHere all morphisms, except possibly w, are obvious and w is ¡he compositeker ( -) L -> coker136	T.H . FAY, K.A . HARDIE, P.J . HILTONof ¡he embedding and the projection .Proof. Exactness at ker ( follows from Lemma 2 applied to the diagramis exaci .aBy duality we have also exactness at coker 1 .A' ) B' > C'We are now ready to show how the Snake Lemma follows easily from theTwo-Square Lemma.Lemma 5 . '(The Snake Lemma) Given the commutative diagram with exactrowsthere is a connecting morphism w : ker y -> coker a such that the sequenceker a --> ker f3 --.> ker y w > coker a -> coker /i -+ coker yProof: The exactness of ker a -> ker a -+ ker y and coker a --> coker /l ->coker -1 are just Lemma 2 and its dual. Adopting the notation of the Two-Square Lemma, we first apply Lemma 1 to obtain a (monic) te : ker y -> P suchthat urc is the embedding ker y -> C and ker y r> P -> B' is exact . Duallywe find an epic A : Q + coker a such that A-r' is the projection A' -> coker aand B T> Q IX) coker a is exact .Exploiting the isomorphism il : Q -> P and the commutative triangleB~277-p= 6rl-10 0 0ker ker coker11K L cokerM M - 0.and the proof is complete .TWO-SQUARE LEMMA	137we have coker p = coker T = coker a, ker u' = ker y, so that Lemma 4yields the exactness ofker ,(3 -+ ker ,y coker a -3 coker P,Acknowledgements . The second author acknowledges a grant to the To-pology Research Group from the South African Council for Scientific and In-dustrial Research .References[B] F . R . BEYL, The connecting morphism in the Kernel-Cokernel sequence,Archiv de Math . 32 (1979), 305-308 .[HS] P . J . HILTON AND U. STAMMBACH, "A Course in Homological Álge-bra," Springer, New York, Heidelberg, Berlin, 1970 .[M1] S . MAC LANE, "Categories for ¡he Working Mathematicáan," Springer,New York, Heidelberg, Berlin, 1971 .[M2] S . MAC LANE, "Homology," Springer, New York, Heidelberg, Berlin,1963.[M] B . MITCHELL, Theory of Categories, Academic Press (1965), London,New York .[P] N . POPESCU, Abelian Categories with Applications to Rings and Mo-dules, Academic Press (1973), London, New York.[H] H . SCHUBERT, "Categories," Springer, New York, Heidelberg, Berlin,1972 .Temple H. Fay: Department of MathematicsUniv . of Southern MississippiHattiesburg, MS 39406USAPeter J . Hilton : Department of Math . SciencesState Univ . of New YorkBinghamton, NY 13901USAKeith A. Hardie : Department of MathematicsUniv. of Cape TownRondebosch 7700SOUTH ÁFRICARebut el 16 de juny de 1988

Revistes Catalanes amb Accés Obert

Publicacions Matemátiques, Vol 33 (1989), 133-137.
Abstract
THE TWO-SQUARE LEMMA
TEMPLE H. FAY, KEITH A. HARDIE, AND PETER J . HILTON
A new proof is given of the connecting homomorphism .
One of the most useful, and hence important, lemmas of homological algebra
in an abelian category (say, the category of R-modules) is the Snake Lemma
(see the references at the end of this note). This result is used to produce
connecting homomorphisms, building long exact sequences such as the familiar
ones involving the Hom - Ext and ® - Tor functors . According to Mac Lane
[M1], early proofs of this lmma were obscure. One technique, used by Popescu
[P], is to prove the result for abelian groups and use the Mitchell Embedding
Theorem to obtain the result in other abelian categories . The essential point is
the construction of the homomorphism to do the connecting, not the exactness
which comes from simple diagram chasing. In this note we give an interesting
new result, which we call the Two-Square Lemma, that provides a simple and
completely categorical construction of the snake connecting morphism . This
has the pleasant property of avoiding all references to zig-zags or Switchback
formulas and to such powerful tools as Mitchell's Embedding Theorem.
We shall do all our work in an abelian category, although some relaxing of
the axioms might be possible, say to an exact category. We begin by stating
two lemmas which are well-known and fundamental. Indeed apart from these
two lemmas we require only the notions of exactness, pullback and pushout in
a balanced category.
We record that Beyl [B] has also given a purely categorical description of the
connecting homomorphism. He bases his work on the Product Lemma which is
our Lemma 4, and does not enunciate a Two-Square Lemma.
Lemma 1 . Consider the commutative diagram
05
B
al pb
m'A' --~ B'
	
.
C
7
C'
134
	
T.H . FAY, K.A . HARDIE, P.J . HILTON
with ¡he botiom roca exact. Then there exists a unique morphism : A' --.> B
such that
(ZJ N Y' = Y',~
and (ii) A' P) B -~-+ C is exact
The dual of this result is, of course, also valid.
Lemma 2. (The Sharp 3 x 3 Lemma) . Consider ¡he following commutative
diagram having exact columns and rocas .
is exact.
0 > A2
There are uniquely determined morphisms Kl + K2 and K2 --> K3 such that
the completed diagram is commutative and the sequence
>B2 i. C2
K3
Again, the dual of this result is valid. Lemma 2 and its dual provide the two
halves of the long exact sequence which the Snake Lemma connects up . The
next lemma is the one cae wish to emphasize .
Lemma 3. (The Two-Square Lemma) Let the following commutative dia-
gram have exact rocas. Form the pulí-back of y and 0',
and the push-out of 0 and a, thus
O
T
al p, Q fil oi
17,,~~.
% ~p, m, pb
A' 3 B' ) C'
TWO-SQUARE LEMMA
	
135
Then
(i) there exists a unique 0 : Q -+ B' such that 9r = /3, Br'
(ii) there exists a unique p : B -~ P such that op = 0, o'p = /3 ;
(iii) there exists a unique rl : Q -~ P such that r/r = p, o'i = 0, or7T' = 0 .
Moreover, rl is monic if zo' is monic, and 17 is epic if 0 is epic .
Proof Assertions (i) and (ii) are obvious. To prove (iii), we first use Lemma
1 to find a unique tt : A' --> P such that o'p = ' and A' '> P -°> C is exact .
We then claim that aa = po.
ForotccY=0=00=opo, ando'pa=Tp'a=Po=o'pO .
Thus there exists a unique 77 : Q --> P such that y/T = p, i7r' = ti .
But then o77T' = op = 0, and o'r/r = o'p = (~ = Br, o'77r' = o'u _ o' = Br', so
that o'ir/ = 0 .
Conversely, if rl : Q + P satisfies rlr = p, o'ij = 0, orlr' = 0, then ortr' = 0 =
op, o'i7r' = BT' = o' = o'M, so that 177-' = Es, and the uniqueness of rl satisfying
(iii) is established.
Consider now the commutative diagram
Here the exactness of the top row is obtained by dualizing the argument just
given, noting that the definition of 77 is evidently self-dual .
If 0' is monic, so is p = 71r' . Hence, by the above diagram, so is rl . Dually, if
0 is epic, so is 77 .
The final ingredient for the Snake Lemma is the following Triangle Lemma.
Lemma 4 . (Triangle Lemma) Given the Trangle
there is an exact sequence
ker (~ -> ker > coker l; -> coker
Here all morphisms, except possibly w, are obvious and w is ¡he composite
ker ( -) L -> coker
136
	
T.H . FAY, K.A . HARDIE, P.J . HILTON
of ¡he embedding and the projection .
Proof. Exactness at ker ( follows from Lemma 2 applied to the diagram
is exaci .
a
By duality we have also exactness at coker 1 .
A' ) B' > C'
We are now ready to show how the Snake Lemma follows easily from the
Two-Square Lemma.
Lemma 5 . '(The Snake Lemma) Given the commutative diagram with exact
rows
there is a connecting morphism w : ker y -> coker a such that the sequence
ker a --> ker f3 --.> ker y w > coker a -> coker /i -+ coker y
Proof: The exactness of ker a -> ker a -+ ker y and coker a --> coker /l ->
coker -1 are just Lemma 2 and its dual. Adopting the notation of the Two-
Square Lemma, we first apply Lemma 1 to obtain a (monic) te : ker y -> P such
that urc is the embedding ker y -> C and ker y r> P -> B' is exact . Dually
we find an epic A : Q + coker a such that A-r' is the projection A' -> coker a
and B T> Q IX) coker a is exact .
Exploiting the isomorphism il : Q -> P and the commutative triangle
B
~
277-
p
= 6rl-1
0 0 0
ker ker coker
11
K L coker
M M - 0.
and the proof is complete .
TWO-SQUARE LEMMA
	
137
we have coker p = coker T = coker a, ker u' = ker y, so that Lemma 4
yields the exactness of
ker ,(3 -+ ker ,y coker a -3 coker P,
Acknowledgements . The second author acknowledges a grant to the To-
pology Research Group from the South African Council for Scientific and In-
dustrial Research .
References
[B] F . R . BEYL, The connecting morphism in the Kernel-Cokernel sequence,
Archiv de Math . 32 (1979), 305-308 .
[HS] P . J . HILTON AND U. STAMMBACH, "A Course in Homological Álge-
bra," Springer, New York, Heidelberg, Berlin, 1970 .
[M1] S . MAC LANE, "Categories for ¡he Working Mathematicáan," Springer,
New York, Heidelberg, Berlin, 1971 .
[M2] S . MAC LANE, "Homology," Springer, New York, Heidelberg, Berlin,
1963.
[M] B . MITCHELL, Theory of Categories, Academic Press (1965), London,
New York .
[P] N . POPESCU, Abelian Categories with Applications to Rings and Mo-
dules, Academic Press (1973), London, New York.
[H] H . SCHUBERT, "Categories," Springer, New York, Heidelberg, Berlin,
1972 .
Temple H. Fay: Department of Mathematics
Univ . of Southern Mississippi
Hattiesburg, MS 39406
USA
Peter J . Hilton : Department of Math . Sciences
State Univ . of New York
Binghamton, NY 13901
USA
Keith A. Hardie : Department of Mathematics
Univ. of Cape Town
Rondebosch 7700
SOUTH ÁFRICA
Rebut el 16 de juny de 1988


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The two-square lemma

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