LemmaStatement: 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.

The left derived connecting map is independent of the horseshoe resolution and lifts

Statement

Assume the Axiom of Dependent Choice.

In the situation of The connecting map for left derived functors, the map $\partial_{n}^{H} : L_{n}^{P} F (A^{''}) \to L_{n - 1}^{P} F (A^{'})$ does not depend on the chosen horseshoe middle resolution $H_{∙}$ or on the comparison lifts used to transport the homology connecting morphism to the fixed datum $P$ . More generally, a morphism of short exact sequences and fixed comparison lifts on the two end resolutions admit a compatible comparison map between chosen horseshoe middle resolutions; any two such middle maps induce the same maps on homology.

Facts & Assumptions

Given: Two choices of horseshoe middle resolution and comparison lifts for the same short exact sequence $0 \to A^{'} \to A \to A^{''} \to 0$ .

[L1]

Item 9 defines the connecting map by transporting homology's connecting morphism from a chosen horseshoe sequence (The connecting map for left derived functors).

[L2]

Two horseshoe middle comparison maps fitting the same side data are chain-homotopic (Horseshoe resolutions are compatible with morphisms of short exact sequences up to homotopy).

[L3]

A horseshoe resolution is degreewise the biproduct of the chosen end resolutions (The horseshoe lemma for projective resolutions).

[L4]

The homology connecting morphism is natural under morphisms of short exact sequences of complexes (Naturality of the homology connecting morphism).

[L5]

Chain-homotopic maps induce the same map on homology (Chain-homotopic maps induce the same map on homology).

[L6]

Any two comparison maps between projective resolutions that lift the same object morphism are chain-homotopic, whether or not they were chosen as the same side maps of a horseshoe ladder (Projective comparison maps are unique up to chain homotopy).

Proof

technique · direct

1.1

By [L1], each horseshoe choice gives a short exact sequence of complexes after applying $F$ , hence a homology connecting morphism. For a morphism of the underlying short exact sequences, fix comparison lifts on the two end resolutions. By [L3], write each horseshoe term as the biproduct of its end terms. Inductively in the degree, define the middle comparison map as the fixed diagonal pair of side maps plus an off-diagonal correction. The chain-map defect in degree $n$ lands in the kernel of the target augmentation or differential; projectivity of the corresponding right-hand resolution term lifts it, giving the next correction. This produces a middle chain map for which both side squares commute, hence a morphism of short exact sequences of complexes.

L1L3givenconstruct

2.1

Applied to the identity morphism of the original short exact sequence, step 1.1 supplies a comparison between any two chosen horseshoes. By [L2], any two compatible middle comparisons with the fixed side maps are chain-homotopic.

L2step 1.1

3.1

Applying [L4] to the ladder from step 1.1 shows that the homology connecting morphisms commute with the induced maps on the two ends. If the end comparison lifts are changed, [L6] makes the old and new lifts chain-homotopic and [L5] makes their induced end homology maps equal. With either fixed pair of side lifts, [L2] likewise makes any two compatible middle maps chain-homotopic. Thus all transport maps occurring in the connecting square are unchanged on homology. Applying this to the comparison in step 2.1 shows that the resulting map $L_{n}^{P} F (A^{''}) \to L_{n - 1}^{P} F (A^{'})$ is independent of both the horseshoe and transport choices.

L2L4L5L6step 1.1step 2.1algebra∎

Depends on

Used by

FALSE: the horseshoe connecting map is independent without a comparison proof False statement
Left derived functors form a homological delta functor Theorem

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)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)