PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-05

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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

1.1L1given

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

2.1L2L3step 1.1construct

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.

3.1L4step 2.1∎

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.

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)