DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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Cohomological spectral sequence

Definition

A cohomological spectral sequence in an abelian category $A$ has objects $E_{r}^{p, q}$ of $A$ for $r \geq r_{0} \geq 0$ and square-zero maps $d_{r} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1},$ with specified isomorphisms $H (E_{r}, d_{r})^{p, q} \to E_{r + 1}^{p, q}$ in $A$ . Its total degree is $p + q$ , raised by one by $d$ . Equivalently it is a Homological spectral sequence in the same abelian category under $E_{r}^{p, q} = E_{- p, - q}^{r}$ : negating the target indices $(- p - r, - q + r - 1)$ gives $(p + r, q - r + 1)$ . Negating again recovers the homological sequence, including its transition data.

Source notes

Weibel, Chapter 5, §5.2 pp.122–125 and Definitions 5.4.2–4 pp.132–133; Sharifi, §4.1 pp.87–89 and Definition 4.2.1 p.90.

Depends on

Homological spectral sequence

Used by

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Sources

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