LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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 homological universal-coefficient edge map is well defined

Statement

For a right $R$ -complex $C$ and a left $R$ -module $G$ , the formula $[z] \otimes g \mapsto [z \otimes g]$ defines an $R$ -balanced map $H_{n} (C) \otimes_{R} G \to H_{n} (C \otimes_{R} G)$ .

Proof

Given: $z \in Z_{n} C$ , $g \in G$ , and the tensor-complex differential.

1.1

Since $d z = 0$ , $d (z \otimes g) = d z \otimes g = 0$ , so $z \otimes g$ represents a homology class.

given

2.1

Replacing $z$ by $z + d w$ changes $z \otimes g$ by $d (w \otimes g)$ , and $(z r) \otimes g = z \otimes (r g)$ ; hence the formula descends through both quotients.

step 1.1algebra∎

Depends on

Used by

The universal coefficient theorem for homology over a PID Theorem

Dependency tree · two levels

7 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

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