PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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An additive functor preserves split biproduct diagrams

Statement

If $F : C \to D$ is additive and

$A \overset{i}{\to} A \oplus C \overset{p}{\to} C$

is part of a biproduct diagram with complementary maps $r : A \oplus C \to A$ and $j : C \to A \oplus C$ , then

$F A \overset{F (i)}{\to} F (A \oplus C) \overset{F (p)}{\to} F C$

is again part of a biproduct diagram with complementary maps $F (r)$ and $F (j)$ .

Facts & Assumptions

Given: An additive functor $F$ and a split biproduct diagram in its source.

[L1]

An additive functor preserves finite biproducts (An additive functor preserves finite biproducts).

[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).

Proof

technique · direct

1.1L1

By [L1], the object $F (A \oplus C)$ is a biproduct of $F A$ and $F C$ with structure maps $F (i), F (j), F (r), F (p)$ .

2.1L2step 1.1

The displayed identities $r i = 1_{A}$ , $p j = 1_{C}$ , $r j = 0$ , $p i = 0$ , and $i r + j p = 1_{A \oplus C}$ are exactly the biproduct equations, so applying $F$ preserves them; [L2] identifies the last one as the identity-sum relation in the target.

3.1step 1.1step 2.1∎

Therefore the image diagram is again a split biproduct diagram.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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 12.3.7 (standard reference, not scraped)