2. NON ADDITIVE EXACT FUNCTORS 5

is also exact.

REMARK

2.7. Suppose that the sequence

is exact and split, in the following sense: there exist a morphism a : N — M and

a morphism /3 : L — N such that

-0 o/? = 1 (/)oa + /3oi/; = l

(note that this will be the case in particular if it is exact, and if M, N and L are

projective in *4). Then the sequence

AF(cp) F(tb)

F(M © N) ^-4 F(JV) — ^ - F(L) -• 0

is also exact and split: indeed, set

A = -F f ~

a

J : F(7V) - F( M 0 TV)

£ - F(/3) : F(L) -+ F(N)

Then it is clear that F(i/;) oB = F(ip o /3) = F(l) - 1, and that

AF(y)oA + BoFtyO = - ( F ( ^ , 1) -F(0,1) ) ° ^ ( ~ J +F(0oil) = ...

...

=

_ F ( - ^ o a + 1) + F(l) + F(/3 oip)= F(l) = 1

So the condition of the definition of a right exact functor is void on the split exact

sequences. In particular, if every exact sequence in A is split, then every functor

from A to an abelian category is right exact.

REMARK 2.8. Let p = F(0,1) : F(M 0 N) - F(N), and i = F(°) : F(N) -

F(M®N). Then iop = F ( ) is an idempotent endomorphism of F(M®N).

Moreover

F{}p, 1) (l - F ( J J ) ) = i ^ , 1) - F(0,1) = AFM

So the functor F is right exact if and only if for any short exact sequence (2.1), the

sequence

( 1 - F ( g ; ) ) F ( M © A T ) ^ ^ F ( i V ) ^ l

F ( L )

^ o

is exact.