PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 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.

Homology respects identities and composition

Statement

For every $n \in Z$ :

$H_{n} (1_{C}) = 1_{H_{n} (C)}$ for every chain complex $C$ .
If $f : C \to D$ and $g : D \to E$ are chain maps, then $H_{n} (g \circ f) = H_{n} (g) \circ H_{n} (f) .$

Facts & Assumptions

Given: Chain maps $f : C \to D$ and $g : D \to E$ .

[L1]

Identities and composites of chain maps are chain maps (Identities and composites of chain maps are chain maps).

[L2]

A chain map induces a unique map on homology compatible with the quotient from cycles (A chain map induces a well-defined map on homology).

Proof

technique · direct

1.1

By [L1], $1_{C}$ is a chain map. The identity on $H_{n} (C)$ satisfies the same compatibility with the quotient from cycles as the map $H_{n} (1_{C})$ from [L2], so uniqueness in [L2] gives $H_{n} (1_{C}) = 1_{H_{n} (C)}$ .

L1L2

2.1

Again by [L1], $g \circ f$ is a chain map. Both $H_{n} (g \circ f)$ and $H_{n} (g) \circ H_{n} (f)$ compose with the cycle quotient to the map induced by the composite on cycles, so [L2] forces them to agree.

L1L2algebra∎

Depends on

Used by

Isomorphisms of complexes are quasi-isomorphisms Proposition
Homology is an additive functor Theorem

Dependency tree · two levels

6 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)