Left 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 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 $\mathcal B$.

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

1. $\mathrm{Hom}$ 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 $\mathrm{Ext}^1_R(M,-)$).

2. 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.

3. 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.