LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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.

Extending a partial comparison homotopy by one degree

Statement

Let $f_{∙}, g_{∙} : P_{∙} \to Q_{∙}$ be augmentation-preserving maps of projective resolutions lifting the same object morphism. Suppose $h_{0}, \dots, h_{n - 1}$ have already been chosen so that $f_{k} - g_{k} = d_{k + 1}^{Q} h_{k} + h_{k - 1} d_{k}^{P}$ holds for every $k < n$ (with $h_{- 1} = 0$ ). Then there exists $h_{n} : P_{n} \to Q_{n + 1}$ extending the homotopy identity to degree $n$ .

Facts & Assumptions

Given: Projective resolutions $P_{∙} \to A$ , $Q_{∙} \to B$ , two comparison maps $f_{∙}, g_{∙}$ lifting the same object morphism, and a partial chain homotopy through degree $n - 1$ .

[L1]

A chain homotopy is given by the equation $f_{n} - g_{n} = d_{n + 1}^{Q} h_{n} + h_{n - 1} d_{n}^{P}$ (A chain homotopy).

[L2]

Cycle objects are kernels of the differentials (Cycle and boundary subobjects of a complex).

[L3]

Projective objects lift across epimorphisms (Projective object).

Proof

technique · direct

1.1L1L2givenalgebra

Put $c_{n} : = f_{n} - g_{n} - h_{n - 1} d_{n}^{P},$ with $h_{- 1} = 0$ when $n = 0$ . Using the chain-map identities for $f_{∙}$ and $g_{∙}$ and the already verified lower-degree homotopy equations, one gets $d_{n}^{Q} c_{n} = 0$ . Thus $c_{n}$ lands in $Z_{n} (Q)$ by [L2]; when $n = 0$ , the common augmentation condition on $f_{0}$ and $g_{0}$ says exactly that $c_{0}$ lands in $\ker (ε_{Q}) = Z_{0} (Q)$ .

2.1L3step 1.1construct∎

Exactness of $Q_{∙}$ makes $Q_{n + 1} ↠ Z_{n} (Q)$ epic, and $P_{n}$ is projective. By [L3], lift $c_{n}$ to a map $h_{n} : P_{n} \to Q_{n + 1}$ . Then $d_{n + 1}^{Q} h_{n} = c_{n}$ , which is precisely the degree- $n$ homotopy equation from [L1].

Depends on

Used by

Projective comparison maps are unique up to chain homotopy Theorem

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, An Introduction to Homological Algebra (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)