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

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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.
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The rth differential has bidegree minus r r minus one

Statement

In the homological convention the $r$ th differential has bidegree $(- r, r - 1)$ and total degree $- 1$ . The stable item ID is retained; its verbal suffix does not specify the sign convention.

Facts & Assumptions

Given: The induced page differential of a filtered chain complex, at page r≥0 and position (p,q).

[F1]

$d^{r}$ is induced from $d_{n}$ with target filtration index $p - r$ and chain degree $n - 1$ (The filtered differential induces d r on the r page).

Proof

technique · direct

1.1

At bidegree $(p, q)$ the total degree is $n = p + q$ . By [F1] the target total degree is $n - 1$ and its first index is $p - r$ . Hence its second index is $(n - 1) - (p - r) = q + r - 1$ .

F1algebra

2.1

The changes in the two indices are therefore $- r$ and $r - 1$ , whose sum is $- 1$ . At r=0 they are $(0, - 1)$ and at r=1 they are $(- 1, 0)$ , as asserted.

step 1.1algebra∎

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

Nothing in the library uses this result yet.

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)