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.
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