6 SERGE BOUC
2.3. Basic properties.
2.3.1. Composition. The class of right exact functors is closed by composition:
PROPOSITION 2.9. Let F : A B and G : B C be right exact functors
between abelian categories. Then G o F is right exact.
PROOF.
Let M ^ N -^ L 0 be an exact sequence in A. Since F is right
exact, the sequence
F{M © N)
A F
^ \ F{N)
F
^ F(L) - 0
is exact. And since G is right exact, the sequence
G (F(M © N) 0 F(A0)
A G
(
A F
( ^
G 0
^(TV)
GoF^)
G 0 F
(
L
) _+ o
is exact. Moreover
AG(AF(^)) = G(F{p, 1) - F(0,1), l) - G(0,1)
On the other hand, the functor G o F is right exact if and only if the sequence
A ( G o F ) M GoF(xb)
(2.4) G o F( M © N) ^ 4 G o F(iV) ^ 4 G o F(L) - 0
is exact. Let
D - AG(AF(^)) = G(F{^ 1) - F(0,1), l ) - G(0,1)
£' = A(G o F)(y) = G o F(/, 1) - G o F(0,1)
I will show that Im D = Im D'.
Let a be the morphism from F(M © AT) to F(M © JV) © F(Af) defined by
V o i
F(0,1)
and let A = G(a). Let /? be the morphism from F(M © N) © F(AT) to F( M © AT)
defined by
and let B = G{0). Then D' o B = G(v) - G(v'), where
v = F{y,l)op v' = F(0,1) o /?
Note that 1/ is obtained from v by replacing ip by 0. But
^ = (F(^I)(I-F([5
^
^ ^ ( ^ ^ F Q )
= (.p^, 1)-^(o, 1), 1)
Sou' = (0,1), and
D'oB = G(F(p, 1) - F(0,1), l ) - G(0,1) = D
Moreover B o A = G(s), with
"0-'(xi)-'(D)tlv
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