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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A natural transformation induces natural transformations of left derived functors

Statement

Assume the Axiom of Dependent Choice.

Let $P$ be a supplied projective resolution datum, let $F, G : A \to B$ be additive functors between abelian categories, and let $α : F \Rightarrow G$ be a natural transformation. Then for every $n \in Z$ the maps $L_{n}^{P} (α)_{A} : = H_{n} ⁣ (α_{P (A)_{d e l}}) : L_{n}^{P} F (A) \to L_{n}^{P} G (A)$ define a natural transformation $L_{n}^{P} (α) : L_{n}^{P} F \Rightarrow L_{n}^{P} G .$

Facts & Assumptions

Given: An integer $n$ .

[L1]

A natural transformation is a family of components satisfying the naturality equation on every morphism (Natural transformation and its components).

[L2]

Additive functors apply degreewise to complexes and chain maps (An additive functor applies degreewise to complexes and chain maps).

[L3]

Every chain map induces a well-defined map on homology (A chain map induces a well-defined map on homology).

[L4]

The left derived maps are the homology maps induced from comparison lifts (The left derived map relative to supplied resolution data).

[L5]

The source and target assignments are already functors (Left derived functors relative to supplied data are additive functors).

Proof

technique · direct

1.1

For each object $A$ , the components $α_{P_{k} (A)}$ form a chain map $α_{P (A)_{d e l}} : F (P (A)_{d e l}) \to G (P (A)_{d e l}),$ because [L1] makes them commute with each differential of the chosen deleted resolution.

L1L2givenalgebra

2.1

By [L3], step 1.1 induces a morphism $L_{n}^{P} (α)_{A} : L_{n}^{P} F (A) \to L_{n}^{P} G (A)$ for each $A$ .

L3step 1.1construct

3.1

Let $u : A \to B$ , and choose a comparison lift ${\tilde{u}}_{∙} : P_{∙} (A) \to P_{∙} (B)$ . Naturality in [L1] gives $G ({\tilde{u}}_{k}) α_{P_{k} (A)} = α_{P_{k} (B)} F ({\tilde{u}}_{k})$ for every degree $k$ , so the square of chain maps commutes. Passing to homology and using [L4] gives $L_{n}^{P} G (u) \circ L_{n}^{P} (α)_{A} = L_{n}^{P} (α)_{B} \circ L_{n}^{P} F (u) .$ Thus the components from step 2.1 are natural.

L1L4L5step 2.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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)