LemmaStatement: 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.

The cycle-boundary diagram associated to a short exact sequence of complexes

Statement

Let $0 \to A_{∙} \overset{i}{\to} B_{∙} \overset{p}{\to} C_{∙} \to 0$ be a short exact sequence of complexes in an abelian category, and fix $n \in Z$ . Write ${\overline{A}}_{n} : = A_{n} / B_{n} (A), {\overline{B}}_{n} : = B_{n} / B_{n} (B), {\overline{C}}_{n} : = C_{n} / B_{n} (C) .$ Then the differentials induce a commutative diagram $\begin{array}{ccccccc} {\overline{A}}_{n} & \to & {\overline{B}}_{n} & \to & {\overline{C}}_{n} & \to & 0 \\ ↓ & ↓ & ↓ \\ 0 & \to & Z_{n - 1} (A) & \to & Z_{n - 1} (B) & \to & Z_{n - 1} (C) \end{array}$ in which the top row is exact at ${\overline{B}}_{n}$ and ${\overline{C}}_{n}$ , and the bottom row is exact at $Z_{n - 1} (A)$ and $Z_{n - 1} (B)$ .

Facts & Assumptions

Given: The short exact sequence of complexes in the statement and an integer $n$ .

[L1]

A short exact sequence of complexes is exact in each degree in the ambient abelian category (Short exact sequence of complexes).

[L2]

Cycles are kernels of outgoing differentials and boundaries are images of incoming differentials (Cycle and boundary subobjects of a complex).

[L3]

For a morphism of short exact sequences, the induced kernel row is exact at its first two nodes and the induced cokernel row is exact at its last two nodes (The kernel row and cokernel row of a morphism of short exact sequences are exact at two nodes each).

Proof

technique · direct

1.1

Apply [L3] to the commutative square in degree $n + 1$ . Using [L2], its cokernel row is exactly ${\overline{A}}_{n} \to {\overline{B}}_{n} \to {\overline{C}}_{n} \to 0,$ so the top row is exact at ${\overline{B}}_{n}$ and ${\overline{C}}_{n}$ .

L1L2L3givenconstruct

2.1

Apply [L3] to the commutative square in degree $n - 1$ . By [L2], the resulting kernel row is $0 \to Z_{n - 1} (A) \to Z_{n - 1} (B) \to Z_{n - 1} (C),$ exact at $Z_{n - 1} (A)$ and $Z_{n - 1} (B)$ . The two rows are connected by the differentials, and the chain-map equalities from [L1] make the diagram commutative.

L1L2L3step 1.1construct∎

Depends on

Used by

The preconnecting arrow on cycles Definition
Exactness at the homology of the left complex Lemma
Exactness at the homology of the middle complex Lemma
Exactness at the homology of the right complex Lemma
Exactness at the target of the connecting map Lemma

Dependency tree · two levels

12 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, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.13: Complexes (standard reference, not scraped)