PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-05‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Natural transformations of base functors give morphisms of derived delta functors

Statement

Assume the Axiom of Dependent Choice.

Let $P$ be supplied projective resolution data, let $I$ be supplied injective resolution data, and let $α : F \Rightarrow G$ be a natural transformation between additive functors $A \to B$ .

If $F$ and $G$ are right exact, then the induced transformations $L_{n}^{P} (α) : L_{n}^{P} F \Rightarrow L_{n}^{P} G$ assemble into a morphism of homological delta functors.

If $F$ and $G$ are left exact, then the induced transformations $R_{I}^{n} (α) : R_{I}^{n} F \Rightarrow R_{I}^{n} G$ assemble into a morphism of cohomological delta functors.

Facts & Assumptions

Given: A short exact sequence $0 \to A^{'} \to A \to A^{''} \to 0$ and an integer $n \geq 0$ .

[L1]

A natural transformation induces degreewise natural transformations on left and right derived functors (A natural transformation induces natural transformations of left derived functors, A natural transformation induces natural transformations of right derived functors).

[L2]

The left and right derived families already carry delta-functor structures (Left derived functors form a homological delta functor, Right derived functors form a cohomological delta functor).

[L3]

The homology and cohomology connecting morphisms are natural with respect to morphisms of short exact sequences of complexes (Naturality of the homology connecting morphism, Naturality of the cohomology connecting morphism).

[L4]

A morphism of delta functors is a degreewise natural transformation that commutes with the connecting maps (Morphism of homological delta functors, Morphism of cohomological delta functors).

Proof

technique · direct

1.1

By [L1], the components $L_{n}^{P} (α)$ and $R_{I}^{n} (α)$ are already natural in the object variable.

L1given

2.1

Compute the connecting maps for the chosen short exact sequence from a horseshoe resolution on the projective or injective side as in [L2]. Applying $α$ degreewise gives a morphism between the two short exact sequences of complexes obtained after applying $F$ and $G$ . By [L3], the corresponding connecting squares in homology or cohomology commute.

L2L3step 1.1construct

3.1

Step 2.1 is exactly the compatibility demanded in [L4]. Therefore the induced degreewise natural transformations from step 1.1 assemble into morphisms of the corresponding derived delta functors.

L4step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

32 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)