Right exact functor - Imported Knowlpedia Content

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

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 $\mathcal B$ .

Equivalent characterizations

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

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.

Quotient by an ideal (via tensor). For a ring $R$ and ideal $I$ , the functor $M \longmapsto M/IM$ is right exact; in fact $M/IM \cong M\otimes_R (R/I)$ .

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).