LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-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.

The exact couple differential squares to zero

Statement

The associated differential of a page- $r$ exact couple satisfies $d_{p - r, q + r - 1} d_{p, q} = 0$ for every $p, q \in Z$ and every $r \geq 1$ .

Facts & Assumptions

[F1]

Exact couple gives exactness at $E_{p - r, q + r - 1}$ , in particular $k_{p - r, q + r - 1} j_{p - 1, q} = 0$ .

[F2]

Differential associated to an exact couple specifies $d_{p, q} = j_{p - 1, q} k_{p, q}$ .

Proof

Given: A page- $r$ exact couple with the stated indexed maps.

1.1

At the component $E_{p - r, q + r - 1}$ , exactness says that the image of $j_{p - 1, q}$ lies in the kernel of $k_{p - r, q + r - 1}$ . Therefore $k_{p - r, q + r - 1} j_{p - 1, q} = 0$ as an arrow from $D_{p - 1, q}$ to $D_{p - r - 1, q + r - 1}$ . This is valid even when any component or arrow is zero.

F1given

2.1

Substituting both differential formulas and using associativity gives $d_{p - r, q + r - 1} d_{p, q} = j_{p - r - 1, q + r - 1} (k_{p - r, q + r - 1} j_{p - 1, q}) k_{p, q} = 0$ . Its target is $E_{p - 2 r, q + 2 r - 2}$ , as required for the square of a map of bidegree $(- r, r - 1)$ . This works at $r = 1$ and all larger integers, with no choice or convergence assumption.

F2step 1.1algebra∎

Depends on

Used by

Derived exact couple Definition

Cited to discharge well-definedness by Differential associated to an exact couple.

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

Stacks Project, Lemma 12.21.2; indexed calculation supplied locally (standard reference, not scraped)