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

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The rth differential squares to zero

Statement

For every $r \geq 0$ the page differential satisfies $(d^{r})^{2} = 0$ .

Facts & Assumptions

Given: A filtered chain complex and its induced page differential at page r≥0.

[F1]

The page differential is induced by the chain differential on the numerator quotient (The filtered differential induces d r on the r page).

Proof

technique · direct

1.1

Let $π : A_{p, n}^{r} ↠ E_{p, q}^{r}$ be the numerator quotient (for r=0 use $F_{p} C_{n}$ ). Applying the defining identity of [F1] twice gives $(d^{r})^{2} π = π^{''} d_{n - 1} d_{n} = 0$ . The second target is $(p - 2 r, q + 2 r - 2)$ , of total degree $n - 2$ , so the composite is typed.

F1algebra

2.1

The quotient projection π is epic; hence its right cancellation in $(d^{r})^{2} π = 0 π$ gives $(d^{r})^{2} = 0$ . This also applies to r=0 and zero numerators.

step 1.1∎

Source notes

Weibel, Chapter 5, Construction 5.4.6 and Lemma 5.4.7, pp.133–134; Sharifi, Theorem 4.2.3, pp.91–92. Increasing homological indices are used here.

Depends on

The filtered differential induces d r on the r page

Used by

The next page is the homology of the current page Theorem

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

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 5 (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)