Left 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 left exact if it preserves finite limits; equivalently (in abelian categories), if it preserves kernels.

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 >

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

> is exact in $\mathcal B$ .

Equivalent characterizations

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

Relation to other exactness notions

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

Examples

$\mathrm{Hom}$ is left exact. In $R\text{-}\mathbf{Mod}$ , for a fixed $R$ -module $M$ , the functor $\mathrm{Hom}_R(M,-):R\text{-}\mathbf{Mod}\to \mathbf{Ab}$ is left exact: applying $\mathrm{Hom}_R(M,-)$ to a short exact sequence yields an exact sequence starting with $0$ . It is not generally right exact (failure of surjectivity at the right is measured by $\mathrm{Ext}^1_R(M,-)$ ).

Global sections of sheaves. For a topological space $X$ , the global sections functor $\Gamma(X,-):\mathrm{Sh}_{\mathbf{Ab}}(X)\to \mathbf{Ab}$ is left exact, but not right exact in general.

Restriction of scalars (actually exact). If $\varphi:R\to S$ is a ring homomorphism, the restriction-of-scalars functor $\mathrm{Res}_\varphi:S\text{-}\mathbf{Mod}\to R\text{-}\mathbf{Mod}$ is exact, hence left exact.