Exact functor

Equivalently, $F$ is exact if and only if $F$ is both left exact and right exact.

In abelian categories, exactness can also be characterized as preservation of both kernels and cokernels (hence images and coimages).

1. Restriction of scalars. For a ring homomorphism $\varphi:R\to S$, the forgetful/restriction functor
   $$\mathrm{Res}_\varphi:S\text{-}\mathbf{Mod}\to R\text{-}\mathbf{Mod}$$
   is exact: it does not change the underlying abelian group maps, so kernels and cokernels are preserved.

1. Localization (commutative rings). If $R$ is commutative and $S\subseteq R$ is multiplicative, then
   $$S^{-1}(-):R\text{-}\mathbf{Mod}\to S^{-1}R\text{-}\mathbf{Mod}$$
   is exact because $S^{-1}(-)\cong -\otimes_R S^{-1}R$ and $S^{-1}R$ is flat over $R$.

1. Tensor with a flat module / Hom from a projective module.

• The functor $-\otimes_R M$ is exact iff $M$ is flat.
• The functor $\mathrm{Hom}_R(P,-)$ is exact iff $P$ is projective.

(These are standard sources of exact functors in module categories.)