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