LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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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.

1.1F1given

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.

2.1F2step 1.1algebra∎

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.

Depends on

Dependency tree · 2 levels

Sources

Stacks Project, Lemma 12.21.2; indexed calculation supplied locally