what is a faithfully exact functor?

Question

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

Answer 1

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.