10 SERGE BOUC
2. In particular if A B C, then for any positive integer n, define inductively
the functor M ^
M®n
by
Mm=M M®n
=
M^n~^
g M if n\
Then the functor M ^ M®n is right exact.
PROOF.
The functor F C* D
Fr
is the functor from A to C defined by the compo-
sition
. A . F x F m ® m
A^ Ax A B x B ^ B
So it is exact by proposition 2.9.
If moreover A B = C, the functor M i— M®n is right exact by an easy
induction argument.
2.4. Extension of functors. The main result concerning right exact functors
is the following:
THEOREM
2.14. Let V be a full subcategory of an abelian category A with the
following properties:
1. The objects ofV are projective in A.
2. Any object of A is a quotient of an object ofV.
3. If P and Q are objects ofV, then so is P 0 Q.
Then any functor F from V to an abelian category B can be uniquely extended (up
to isomorphism of functors) to a right exact functor from A to B.
PROOF.
First uniqueness is almost obvious: let F\ and F2 be right exact func-
tors from A to B, and 0 be an isomorphism from the restriction of F\ to V to the
restriction of V. In particular, for any object P of P, there is an isomorphism Op
from F ^ P ) to P2(P)-
Now uniqueness will follow from the following
PROPOSITION
2.15. Let A be an abelian category, and V be a full subcategory
of A satisfying the hypothesis of theorem 2.14- Let i*\ and F2 be (non-necessarily
additive) right exact functors from A to an abelian category B. Then if 0 is a
natural transformation from the restriction of F\ to V to the restriction of F2 to V,
there exists a unique natural transformation 0 from F\ to F2 which coincides with
0 onV.
PROOF.
For any object M of A, choose a short exact sequence
(2.5) Q ^ P ^ M - ^ 0
with P and Q in V. Such a sequence exists by condition 2). Since F\ and F2 are
right exact, the rows of the following commutative diagram are exact
P i ( Q 0 P ) ^ 4 Fi(P) ^ ^ A F1(M) -0
A P
2
M ' P2WO
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