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

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The cohomological filtered complex construction

Statement

Let K be a cochain complex with decreasing filtration by subcomplexes. Set $C_{n} = K^{- n}$ , $F_{p} C_{n} = F^{- p} K^{- n}$ and $d_{n} = d_{K}^{- n}$ . Then $E_{r}^{p, q} (K) = E_{- p, - q}^{r} (C), d_{r} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1} .$ The construction gives $E_{0}^{p, q} = {gr}^{p} K^{p + q}$ and $E_{1}^{p, q} ≅ H^{p + q} ({gr}^{p} K)$ . With degreewise finite filtration it naturally abuts to the decreasing image filtration on $H^{p + q} (K)$ , with $E_{\infty}^{p, q} ≅ {gr}^{p} H^{p + q} (K)$ .

Facts & Assumptions

Given: A decreasing filtered cochain complex K.

[F1]

Negating both indices converts homological into cohomological spectral sequences (Cohomological spectral sequence).

[F2]

Finite filtered homological construction has natural graded-homology abutment (Bounded filtered complex spectral sequence abuts to filtered homology).

[F3]

Filtered chain maps induce compatible maps on all pages (A filtered chain map induces a morphism of spectral sequences).

[F4]

A cochain complex has d raising degree and consecutive differential composite zero (Cochain complex in an abelian category).

[F5]

The initial homological page is the associated graded complex (E zero is the associated graded complex).

[F6]

The first homological page is homology of that graded complex (E one is homology of the associated graded complex).

Proof

technique · direct

1.1F1F4

Since $F^{- (p + 1)} K^{- n} \supseteq F^{- p} K^{- n}$ , the specified $F_{p}$ is increasing. The cochain differential maps $K^{- n} \to K^{- n + 1} = C_{n - 1}$ and preserves that piece; its square is zero by [F4]. Thus this is a filtered chain complex. The index rule in [F1] turns the target $(- p - r, - q + r - 1)$ into $(p + r, q - r + 1)$ .

2.1F1F2step 1.1F5F6

At r=0, the chain quotient at (-p,-q) is $F^{p} K^{p + q} / F^{p + 1} K^{p + q}$ . Its differential is induced by $d_{K}$ , so its homology is $H^{p + q} ({gr}^{p} K)$ . The first-page identification in [F6] supplies the next-page isomorphism, giving the displayed $E_{1}$ . Its square-zero property and every subsequent transition are preserved by the index substitution.

3.1F2F3step 1.1step 2.1∎

Degreewise finite decreasing bounds become finite increasing bounds after negating their indices. Apply [F2] with chain degree −(p+q): its image filtration $F_{- p} H_{- (p + q)} (C)$ is precisely the image $H^{p + q} (F^{p} K) \to H^{p + q} (K)$ . The next lower chain piece corresponds to $F^{p + 1}$ , giving the stated graded quotient. A filtered cochain map reindexes to a filtered chain map by the same formula; [F3] and the naturality in [F2] prove naturality of all identifications.

The cohomological filtered complex construction

Statement

Facts & Assumptions

Proof

Source notes

Depends on

Dependency tree · 2 levels

Sources