TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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An additive category with kernels is idempotent complete

Statement

Every additive category in which every morphism has a kernel is idempotent complete.

Facts & Assumptions

Given: An additive category $C$ and an idempotent $e : A \to A$ .

[L1]

In an additive category, hom-sets admit subtraction and there is a zero object (Additive category).

[L2]

On a biproduct, the injection-projection maps satisfy the identity-sum relation (On a biproduct, the injections and projections satisfy the identity-sum relation).

[L3]

The criterion "monic iff kernel zero" holds in a preadditive category with a zero object (In a preadditive category with a zero object, a morphism is monic exactly when its kernel is zero).

[L4]

Idempotent completeness means that every idempotent splits (Idempotent complete category, Idempotent and split idempotent).

Proof

technique · direct

1.1

Let $k : K \to A$ be the kernel of $1_{A} - e$ , which exists by hypothesis. Since $(1_{A} - e) e = 0$ , the kernel universal property gives a morphism $p : A \to K$ with $k p = e$ .

L1L4construct

2.1

Because $(1_{A} - e) k = 0$ , one has $e k = k$ . Hence $k p k = e k = k$ . The morphisms $p k$ and $1_{K}$ both satisfy $k (-) = k$ , so the kernel universal property forces $p k = 1_{K}$ . Therefore $k p = e$ and $p k = 1_{K}$ , so $e$ splits through $K$ .

L1step 1.1

3.1

Thus every idempotent in $C$ splits. By [L4], the category is idempotent complete. The additive setting is what makes $1_{A} - e$ meaningful; [L2] and [L3] record the surrounding zero-object calculus on which that subtraction sits.

L2L3L4step 2.1∎

Depends on

Used by

FALSE: every idempotent splits False statement

Dependency tree · two levels

16 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 12.3, Lemma 12.3.16 (standard reference, not scraped)
Dixy Msapato, The Karoubi envelope and weak idempotent completion of an extriangulated category, Proposition 2.2 (standard reference, not scraped)