SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL ...

SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES 89
E contains all isomorphisms, applying Lemma 2.4(i) to the diagram
X ′ f ′ Y ′ q v
q u
Q u
f ′ Q
Q v
we obtain the desired factorization of f ′ Q ; since f ′ Q 1
is in E and f ′ Q 2
is a monomorphism,
we conclude that the kernel of f ′ Q exists in C. Let k f ′ Q : K f ′ Q → Q u be the kernel of f ′ Q
and let e f ′
Q
: K w → K f ′
Q
be the induced unique morphism with e f ′
Q
k f ′
Q
= d, it remains
to prove that e f ′
Q
is in E. Since q u is in E, the pullback (K × Qu X ′ , p 1 , p 2 ) of k f ′
Q
and q u
exists in C and p 1 is in E; therefore, we have the commutative diagram
p
Y ′′ v 2
K × Qu X ′
p 2 X ′ f ′
Y ′ q v
p 1
q u
K f ′
Q
kf ′
Q
Q u
f ′ Q
Q v
in which v 2 = ker(q v ) (recall, that since the second column of the diagram (3.4) is E-
exact at Y ′ , we have v = v 2 v 1 where v 1 ∈ E and v 2 = ker(q v )) and p : K × Qu X ′ → Y ′′
is the induced morphism. Using the fact that v 2 and p 2 are monomorphisms and that
k f ′
Q
= ker(f Q ′ ), an easy diagram chase proves that the square f ′ p 2 = v 2 p is the pullback
of f ′ and v 2 .
Next, consider the commutative diagram
Y
π 1
Y × Z K w
π 2
K w
ϕ
ψ
v
v 1
Y ′′
p
K × Qu X ′
p 1
e f ′
Q
K f ′
Q
d
v 2
p 2
Y ′ X ′
f ′
q u
k f ′
Q
Q u
were ψ = 〈ϕ, v 1 π 1 〉; since k f ′
Q
is a monomorphism and the equalities
k f ′
Q
p 1 ψ = q u p 2 ψ = q u ϕ = dπ 2 = k f ′
Q
e f ′
Q
π 2