DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)

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.

Differential associated to an exact couple

Definition

For a page- $r$ exact couple, define its associated differential by $d_{p, q} = j_{p - 1, q} k_{p, q} : E_{p, q} ⟶ E_{p - r, q + r - 1} .$ The bidegrees of $k$ and $j$ add to $(- 1, 0) + (1 - r, r - 1) = (- r, r - 1)$ , so total degree decreases by one. The square-zero identity is proved in The exact couple differential squares to zero ↗ before homology is formed. For an initial couple $r = 1$ this map has bidegree $(- 1, 0)$ . The zero couple has zero differential. The composite is specified uniquely; neither a section of $i$ nor a preimage selection is part of this definition.

Depends on

Exact couple

Used by

The exact couple differential squares to zero Lemma

Dependency tree · two levels

3 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

Stacks Project, Lemma 12.21.2 and Remark 12.21.5 (standard reference, not scraped)