4
SERGE BOUC
DEFINITION 2.3. Let F : A * B be a (non-necessarily additive) functor be-
tween abelian categories. I will say that F is right exact, if for any exact sequence
(2.1) M ^N^L-^0
the associated sequence
AF(ip) F(ib)
(2.2) F(M © N) ^-4 F(N) ^ U F(L) - 0
is exact.
In particular, a right exact functor maps epimorphims to epimorphisms.
REMARK
2.4. If F is additive, then AF((p) F((p, 0), so the previous sequence
factors as
F(M © TV)
F ( 1 , Q )
) F(M)
F (
^ ) F{N)
F
^ F{L) - 0
The left morphism is a split epimorphism. So F is right exact for the modified
definition if and only if the sequence
F(ip) F(ib)
F(M) ^ - U F(N) ^ - F(L) - 0
is exact, that is if and only if F is right exact in the usual sense. So the new
definition is equivalent to the usual one for additive functors.
REMARK 2.5. Let P be any object of B. Define F{M) = P for any object M
of A, and F((p) = I dp for any map p in A. Then F is a (trivial) example of a right
exact functor, which is not additive if P is non-zero.
REMARK
2.6. A functor F is exact if and only if the sequence (2.2) is exact for
any short exact sequence
(2.3) 0 - M ^N ^L-0
Indeed this is obviously a necessary condition. Conversely, if the sequence (2.2) is
exact for any short exact sequence (2.3), then in particular, the functor F maps
epimorphisms to epimorphisms. Now if
M
' £ N X L - 0
is an exact sequence, denoting by M the cokernel of //, then // factors through
M a s ^ ' = to(7, where i is a monomorphism and a is an epimorphism. Now the
map I J is an epimorphism from M' © T V to M © TV, and so is F (
Moreover
AF
W
.*(; ; ) - F ( M ) ( j
!)M«u)( ; ?))-..
... = F ( ^ 1 ) - F ( 0 , 1 ) = A * V )
So if 0 : F(iV) - P is any map in fi, then 6 o AF(i) is zero if and only if 0 o AF(ip')
is, since F I J is an epimorphism. Thus AF(L) and AF(ip') have the same
image, and the sequence
F(Mf
© TV) ^-4 F(N) —^2_ F{L) -+ 0
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