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Given a functor $F:A\to B$ of abelian categories we may say that $F$ is left exact if it maps exact sequences to left exact sequences, and similarily for right. For arbitrary categories, we may say that $F$ is left exact if it preserves finite limits (supposedly, this was introduced in SGAIV, but I don't have it). The question is thus: are these definitions equivalent in an abelian category? That this latter definition implies the first is clear to me, but the other gives me more trouble.



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Yes. $F$ is exact in the first sense if it maps left exact sequences to left exact sequences (this is an easy exercise). So $F$ is exact in the first sense iff it preserves kernels. But an additive functor that preserves kernels preserves all finite limits (because any finite limit can be built from a product of direct products and equalizers, and finite direct products are always preserved by an additive functor).

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Thanks a lot! I suspected it would have something to do with building the limit from products and equalizers. That fact will probably prove a good exercise :) – Eivind Dahl Apr 3 '11 at 14:56

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