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An additive functor is left exact exactly when it preserves kernels

Statement

Let $F : C \to D$ be an additive functor between additive categories. Then $F$ is left exact if and only if it preserves kernels.

Facts & Assumptions

Given: An additive functor $F : C \to D$ .

[L1]

Additive categories are preadditive with finite biproducts (Additive category).

[L2]

In a preadditive category, equalizers are kernels of differences (In a preadditive category, the equalizer of a parallel pair is the kernel of their difference).

[L3]

An additive functor preserves finite biproducts, hence finite products (An additive functor preserves finite biproducts).

[L4]

A functor preserves finite limits exactly when it preserves finite products and equalizers (Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals).

Proof

technique · direct

1.1L4

If $F$ is left exact, then it preserves all finite limits by definition, so in particular it preserves kernels because a kernel is a finite limit.

1.2L1L2L3

Conversely, assume $F$ preserves kernels. By [L1] the source and target are preadditive, and by [L3] the functor preserves finite products. If $e : E \to A$ equalizes $f, g : A ⇉ B$ , then [L2] identifies $e$ as a kernel of $f - g$ . Since $F$ is additive, $F (f - g) = F f - F g$ , so the image of $e$ is a kernel of $F f - F g$ , hence again an equalizer of $F f$ and $F g$ by [L2]. Therefore $F$ preserves equalizers.

2.1L4step 1.2

Now [L4] applied to step 1.2 shows that $F$ preserves all finite limits. So $F$ is left exact.

3.1step 1.1step 2.1∎

Thus left exactness and kernel preservation are equivalent for additive functors.

Depends on

Used by

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Section 12.7, Lemma 12.7.2 (standard reference, not scraped)