2. NON ADDITIVE EXACT FUNCTORS

7

••-'-:: SMS?

So 5 o A = 1, and in particular 5 is an epimorphism. So if # : G o F(N) — P is

any map in B, then

So D and D' have the same image, and the sequence (2.4) is exact. This completes

the proof of the proposition. •

2.3.2. Products and sums. If A and A! are abelian categories, then their prod-

uct A x A' is also abelian. If / : M — T V and / ' : M' — • TV' are maps in „4 and

*4', I will denote by

[M,Mf]

and [TV,

TV7]

the associated couples in A x A', and

[/, /'] : [M, M'] - • [TV,

TV7]

the associated morphism. The image of [M, M'} under

a functor F will be denoted by F[M,

Mf)

instead of F([M, M']).

PROPOSITION 2.10. Let F : A -* B and F' : A! — B' be right exact functors

between abelian categories. Then

FxF' :AxA! -BxB'

is right exact.

PROOF.

This is obvious, since a product of exact sequences is exact. •

COROLLARY

2.11. Let F,F' : A — B be right exact functors between abelian

categories. Then F 0 F is right exact.

PROOF. The functor F e F factors as

A^AxAFxF'BxB^B

where A is the diagonal functor, mapping the object M to [M, M] and the morphism

/ to [/,/], and E is the direct sum functor mapping the object [P, Q] to P © Q,

and the morphism [/, g] to / 0 g. Those two functors are obviously additive and

exact, so the corollary follows from proposition 2.9. •

2.3.3. Pairings. If A, A', and B are abelian categories, a pairing F : Ax A' — B

is just a biadditive functor: for any object M of A, the functor F[M, — ] : A! — B

is additive, and for any object M' of A', the functor F[—, M'] : A — B is additive.

Note that F itself is not additive in general.

PROPOSITION

2.12. Let A, A!, andB be abelian categories, and F : Ax A! —• B

be a pairing. The following are equivalent:

1. The functor F is right exact.

2. For any objects M of A and M' of A!', the (additive) functors F[M, —] and

F[—,M'\ are right exact.

3. For any exact sequences

M

^NXL-,O M'^N'^L'-,O

the sequence

(F[p,i\,F[l,p']) Fu^i

F[M, N'] © F[N, M'} 4 F[N, N'} -J11Z4 p[L, L'} - 0

is exact.