2. NON ADDITIVE EXACT FUNCTORS 13
Now let M be any object of A, and consider first an exact sequence in A
with P and Q in V. By the above arguments there are maps a : P
b : Q » Q M , a' : PM P and &' : Q M —• Q and a commutative diagram
Q
p
i
b'
Q
M
VM
PI
M
1.
M
M
M
0
0
M
As i\) o a! o a ipM °fl = ^ there is a map c : P M Q such that l - a ' o a ^ ^ o c .
Similarly, there is a map c' : P Q M such that 1 - a o a' = /?M ° c' Now there is
a commutative diagram
P ( Q 0 P ) ^ P(P) ^ F'(M)
(2.6)
6 0
0 a
* »
1
P ( Q M
©
-PM)

^ ( ^ M )
F (M)
0
0
in which the bottom line is exact by construction of
Ff(M).
Now F(
,coa)
is a map
from F{PM) to F(Q © P), and
AF{ip)oF( ,C ) = (F(P,1)-F(0,1))OF( ,
o a
... = F((poC +
afoa)-F(afoa)=F(l)-F(a')oF(a)
=
l-F(af)oF{a)
Similarly, the map P(afa,) : F(P) - P ( Q M © P M ) is such that
AF(^M)OF(
C'
\ =l-F{a)oF{a')
\a o a'J
This shows that F(a) and ^(a') induce mutual inverse isomorphisms between the
cokernel of AF(p) and the cokernel of
AF((/?M),
equal to F'{M) by definition. In
other words, the top line in 2.6 is exact.
In particular, if ij; : P » M is an epimorphism from an object of P to M, then
F'(ip) is also an epimorphism.
Now let
be an arbitrary short exact sequence in A. It is well-known (see [Wei94] Horseshoe
lemma 2.2.8) that it is possible to find a resolution
Q
I "PN -nf ^N jy _^
Q
N N
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