PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Derived functors commute with finite biproducts

Statement

Assume the Axiom of Dependent Choice.

Let $P$ be a supplied projective resolution datum on a class $D_{P}$ , let $I$ be a supplied injective resolution datum on a class $D_{I}$ , and let $F : A \to B$ be an additive functor between abelian categories. For every $n \in Z$ , the additive functor $L_{n}^{P} F$ preserves finite biproducts that exist in the domain of $P$ , and the additive functor $R_{I}^{n} F$ preserves finite biproducts that exist in the domain of $I$ .

Facts & Assumptions

Given: A finite biproduct in one of the two relevant supplied-data domains.

[L1]

The left derived functor $L_{n}^{P} F$ is additive (Left derived functors relative to supplied data are additive functors).

[L2]

The right derived functor $R_{I}^{n} F$ is additive (Right derived functors relative to supplied data are additive functors).

[L3]

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

Proof

technique · direct

1.1L1L3

Apply [L3] to the additive functor $L_{n}^{P} F$ from [L1]. This gives preservation of finite biproducts on the left-derived side.

1.2L2L3

Apply [L3] to the additive functor $R_{I}^{n} F$ from [L2]. This gives preservation of finite biproducts on the right-derived side.

2.1step 1.1step 1.2∎

Therefore both derived constructions commute with finite biproducts.

Depends on

Used by

Cohomology of a finite disjoint union Theorem

Dependency tree · two levels

20 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

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)