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