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.
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