Exact functor

Let $\mathcal A,\mathcal B$ be abelian categories and let $F:\mathcal A\to\mathcal B$ be an additive functor .

Definition

The functor $F$ is exact if it preserves short exact sequences: whenever

is exact in $\mathcal A$, then

is exact in $\mathcal B$.

Equivalently,

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

Examples

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.

2. 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 $R$.

3. 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.)