LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04

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.

The simplicial augmentation is a chain map

Statement

For every simplicial complex $K$ , one has $ε \partial_{1} = 0,$ so the augmented simplicial chain groups form a chain complex.

Proof

Given: An oriented edge $[v_{0}, v_{1}]$ and the augmented simplicial chain complex of $K$ .

1.1

By the boundary formula, $\partial_{1} [v_{0}, v_{1}] = [v_{1}] - [v_{0}]$ . Applying the augmentation gives $ε ([v_{1}] - [v_{0}]) = 1 - 1 = 0$ .

given

2.1

In degrees $n \geq 1$ , the ordinary simplicial differentials already satisfy $\partial_{n - 1} \partial_{n} = 0$ , so adding $ε$ at degree $0$ preserves the chain-complex condition.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Allen Hatcher, Algebraic Topology (standard reference, not scraped)
Vidit Nanda, Computational Algebraic Topology, Lecture 03: Homology (standard reference, not scraped)