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

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.

E one is homology of the associated graded complex

Statement

There is a canonical isomorphism $E_{p, q}^{1} ≅ H_{p + q} ({gr}_{p}^{F} C)$ , inverse to the specified initial next-page map. If $d (F_{p} C_{n}) \subseteq F_{p - 1} C_{n - 1}$ for all $p, n$ , then $E_{p, q}^{1} ≅ {gr}_{p} C_{p + q}$ and $d^{1}$ is the map between these graded objects induced by $d$ .

Facts & Assumptions

Given: A filtered chain complex and integers p,q.

[F1]

$α_{0} : H (E^{0}) \to E^{1}$ is a natural isomorphism (The next page is the homology of the current page).

[F2]

$E^{0}$ is the associated graded complex with its differential (E zero is the associated graded complex).

[F3]

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

Proof

technique · direct

1.1F2

Fix p. By [F2], the terms at total degrees n are ${gr}_{p} C_{n}$ , and the differential lowers n by one while fixing p. Consequently the kernel and boundary quotient at $(p, q)$ is exactly $H_{p + q} ({gr}_{p} C)$ .

2.1F1step 1.1

Apply the inverse of $α_{0}$ from [F1] and then the identification in step 1.1. The composite is the asserted isomorphism; each factor is canonical and natural, so their composite is too.

3.1F2F3step 1.1step 2.1∎

Under the additional lowering hypothesis, the induced differential on $F_{p} C_{n} / F_{p - 1} C_{n}$ is zero, because the image of $d$ lies in the denominator. Its homology is therefore the same graded object. Steps 1.1 and 2.1 identify this with $E^{1}$ . At the next filtration level, $d (F_{p - 1} C_{n}) \subseteq F_{p - 2} C_{n - 1}$ , so $[x] \mapsto [d x]$ is well defined from ${gr}_{p} C_{n}$ to ${gr}_{p - 1} C_{n - 1}$ . By [F3] this is precisely $d^{1}$ , including when either graded object is zero.

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.

The lowering-hypothesis clause is the increasing homological translation of Stacks §12.24, Lemma 12.24.3(2). The proof here is componentwise and requires no countable direct sum.

E one is homology of the associated graded complex

Statement

Facts & Assumptions

Proof

Source notes

Depends on

Dependency tree · 2 levels

Sources