Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-01‡ 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.

Naturality of the homology connecting morphism

Statement

A morphism of short exact sequences of complexes induces a commutative square Hn(C)→∂nHn−1(A)↓↓Hn(C′)→∂n′Hn−1(A′) for every n∈Z.

Facts & Assumptions

Given: A morphism of short exact sequences of complexes.

[L1]

A morphism of short exact sequences of complexes is a commutative three-column ladder of chain maps (A morphism of short exact sequences of complexes).

[L2]

The arrow category of an abelian category is abelian, with kernels, cokernels, and the relevant diagrams computed componentwise (The arrow category of an abelian category).

[L3]

The homology connecting morphism is the connecting morphism attached to the quotient-kernel snake diagram of a short exact sequence of complexes (The connecting morphism in homology).

[L4]

The weaker Stacks snake construction applies in every abelian category (Snake lemma under the weaker Stacks hypotheses).

Proof

technique · direct
1.1

By [L1], each degree of the given ladder is a morphism of short exact sequences. Passing to the quotient-kernel diagrams used on this page therefore gives a morphism between two weaker Stacks snake diagrams. Regard every comparison arrow as an object of the arrow category. By [L2], the resulting diagram is itself weaker Stacks snake data in that abelian category.

L1L2L3givenconstruct
2.1

Apply [L4] in the arrow category to the diagram from step 1.1. Its connecting morphism is an arrow object whose two components are the connecting morphisms of the original weaker snake diagrams; being a morphism in the arrow category says exactly that the square between those components commutes. Under the identifications in [L3], this is the displayed homology connecting square.

L2L3L4step 1.1construct∎

Depends on

Used by

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