PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

Identities and composites of chain maps are chain maps

Statement

For every chain complex $C_{∙}$ , the identity family $1_{C_{n}}$ is a chain map $1_{C} : C \to C$ . If $f : C \to D$ and $g : D \to E$ are chain maps, then the componentwise composite $(g_{n} f_{n})_{n}$ is a chain map $g \circ f : C \to E$ .

Facts & Assumptions

Given: Chain complexes $C_{∙}, D_{∙}, E_{∙}$ and chain maps $f : C \to D$ , $g : D \to E$ .

[L1]

A chain map is a degree-zero family commuting with the differentials (Chain map).

Proof

technique · direct

1.1L1givenalgebra

For the identity family, $d_{n}^{C} \circ 1_{C_{n}} = 1_{C_{n - 1}} \circ d_{n}^{C}$ holds trivially in every degree, so [L1] makes $1_{C}$ a chain map.

2.1L1givenalgebra∎

Since $f$ and $g$ are chain maps, [L1] gives $d_{n}^{E} g_{n} = g_{n - 1} d_{n}^{D}$ and $d_{n}^{D} f_{n} = f_{n - 1} d_{n}^{C}$ . Therefore $d_{n}^{E} (g_{n} f_{n}) = g_{n - 1} (d_{n}^{D} f_{n}) = g_{n - 1} f_{n - 1} d_{n}^{C},$ so the composite family again satisfies [L1].

Depends on

Chain map

Used by

A chain homotopy equivalence Definition
The category of chain complexes Definition
Chain homotopy is compatible with addition and composition Lemma
A chain isomorphism is a chain homotopy equivalence Proposition
Homology respects identities and composition Proposition
Isomorphic chain maps have isomorphic cones Proposition
The cone construction commutes with shift up to the canonical sign isomorphism Proposition
The mapping cylinder factors a chain map Theorem

Dependency tree · two levels

2 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)