LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-12

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The associated graded of filtered homology is a subquotient of chain level data

Statement

Put $Z_{n} = \ker d_{n}$ , $D_{n} = im d_{n + 1}$ and $P_{p} = F_{p} C_{n} \cap Z_{n}$ . Then the induced homology filtration satisfies ${gr}_{p}^{F} H_{n} (C) ≅ \frac{P_{p}}{P_{p - 1} + (F_{p} C_{n} \cap D_{n})} .$ The isomorphism is induced by the cycle quotient $Z_{n} \to H_{n} (C)$ and is natural for filtered chain maps.

Facts & Assumptions

Given: A filtered chain complex, an integer n, and an integer filtration index p.

[F1]

$F_{p} H_{n} (C)$ is the image of the inclusion on homology (Induced filtration on homology).

[F2]

Coimage equals image, quotients descend, and the sum/intersection and modular identities hold (Spectral sequence subquotient and local lifting calculus).

Proof

technique · direct

1.1F1F2

The identity $d_{n} d_{n + 1} = 0$ factors the boundary image $D_{n}$ through $Z_{n}$ : composing with the epic map onto $D_{n}$ gives zero, so cancellation gives the factorization. Cycles of $F_{p} C$ are $P_{p}$ by the pullback property. Its boundaries map into $D_{n}$ , so the image on homology in [F1] is precisely the image of $P_{p} \to Z_{n} / D_{n}$ . By [F2] this is $(P_{p} + D_{n}) / D_{n}$ .

2.1F2step 1.1

The kernel of $P_{p} \to (P_{p} + D_{n}) / (P_{p - 1} + D_{n})$ is $P_{p} \cap (P_{p - 1} + D_{n})$ . Modularity [F2], using $P_{p - 1} \subseteq P_{p}$ , gives $P_{p - 1} + (P_{p} \cap D_{n}) = P_{p - 1} + (F_{p} C_{n} \cap D_{n})$ . The map is epic since $D_{n}$ is killed and $P_{p}$ supplies the remaining summand.

3.1F2step 1.1step 2.1∎

Nested quotients [F2] identify $(P_{p} + D_{n}) / (P_{p - 1} + D_{n})$ with $F_{p} H_{n} (C) / F_{p - 1} H_{n} (C)$ . Coimage-to-image applied to the epic map in step 2.1 gives the stated formula. A filtered chain map carries each $P_{p}$ and $D_{n}$ into the corresponding target subobject; all arrows just used are uniquely induced by these restrictions, so their squares commute.

Source notes

Stacks §12.24, equations 12.24.5.1–12.24.5.2; the omitted intermediate quotient calculation is proved here.

The associated graded of filtered homology is a subquotient of chain level data

Statement

Facts & Assumptions

Proof

Source notes

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