# what is a faithfully exact functor?

Could any of you give me a definition of faithfully exact functor, please?

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 I think it's a functor which both preserves and reflects short exact sequences, but I can't find a reference which actually says this. – Qiaochu Yuan Jul 4 '11 at 11:22

Indeed a functor $F:\mathcal{A}\to \mathcal{B}$ of abelian categories is called faithfully exact if the following holds: A sequence $A\to B\to C$ in $\mathcal{A}$ is exact if and only if the induced sequence $F(A)\to F(B)\to F(C)$ in $\mathcal{B}$ is exact.

See for example Ishikawa, Faithfully exact functors and their applications to projective modules and injective modules, available on Project Euclid. There you can also find equivalent reformulations.

The motivation for this terminology is probably the following: Let $R$ be a ring and $M$ an $R$-module. Then the functor $M\otimes_R -$ is exact if and only if $M$ is flat and it is faithfully exact if and only if $M$ is faithfully flat.

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 I like "reflects exactness" better :) – Mariano Suárez-Alvarez♦ Jul 4 '11 at 14:54 @Philipp: thank you for your answer. Could you tell me, please, also the definition of faithfully flat? – Jacob Fox Jul 4 '11 at 15:09 Actually what I've written in the last paragraph is usually used as the definition of faithfully flat. Theorem 1.1 of Ishikawa's paper applied to this situation provides some equivalent definitions. See also the reference to [Bourbaki, Algèbre commutative] given in the footnote to the proof of Theorem 1.1 in Ishikawa's paper. – Philipp Hartwig Jul 4 '11 at 16:03