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