Right 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 right exact if it preserves finite colimits ; equivalently (in abelian categories), if it preserves cokernels .

A standard “exact sequence” formulation is:

For every short exact sequence in $\mathcal A$,

$$> 0 \longrightarrow A' \xrightarrow{u} A \xrightarrow{v} A'' \longrightarrow 0, >$$

the sequence

$$> F(A') \xrightarrow{F(u)} F(A) \xrightarrow{F(v)} F(A'') \longrightarrow 0 >$$

is exact in $\mathcal B$.

Equivalently, $F$ preserves epimorphisms and cokernels, but need not preserve kernels or monomorphisms.

Relation to other exactness notions

• If $F$ is both left exact and right exact, then $F$ is exact .
• Any additive left adjoint functor between abelian categories is right exact (because left adjoints preserve colimits).

Examples

1. Tensor product is right exact. In $R\text{-}\mathbf{Mod}$, for a fixed right $R$-module $M$ (or in the commutative case, a fixed $R$-module),

   $$-\otimes_R M : R\text{-}\mathbf{Mod}\to \mathbf{Ab}$$

   is right exact. It is exact iff $M$ is flat.

2. Quotient by an ideal (via tensor). For a ring $R$ and ideal $I$, the functor

   $$M \longmapsto M/IM$$

   is right exact; in fact $M/IM \cong M\otimes_R (R/I)$.

3. Extension of scalars is right exact. For a ring map $\varphi:R\to S$, the functor

   $$-\otimes_R S : R\text{-}\mathbf{Mod}\to S\text{-}\mathbf{Mod}$$

   is left adjoint to restriction of scalars, hence right exact (and exact iff $S$ is flat as an $R$-module).