TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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.

Five term exact sequence of a first quadrant cohomological spectral sequence

Statement

Let $E_{r}^{p, q}$ be a cohomological spectral sequence in an abelian category, first quadrant from page $2$ , with specified finite abutment to $H^{n}$ and decreasing filtration normalized by $F^{0} H^{n} = H^{n}$ , $F^{n + 1} H^{n} = 0$ for $n \geq 0$ . There is an exact sequence $0 ⟶ E_{2}^{1, 0} ⟶ H^{1} ⟶ E_{2}^{0, 1} \overset{d_{2}}{\to} E_{2}^{2, 0} ⟶ H^{2},$ whose maps adjacent to $H^{1}, H^{2}$ are the corresponding edge maps. No terminal surjectivity onto $H^{2}$ is asserted.

Facts & Assumptions

[F1]

Cohomological spectral sequence gives degree $(r, 1 - r)$ and the specified homology transitions.

[F2]

Edge homomorphisms of a first quadrant spectral sequence specifies the cohomological edge maps through the finite filtration's extreme graded pieces.

[F3]

Spectral sequence subquotient and local lifting calculus gives canonical kernel, image and quotient comparisons in an abelian category.

Proof

Given: The sequence and normalized abutment in the statement. All support claims refer to pages $r \geq 2$ ; a zero term remains zero under a homology transition.

1.1

At $(1, 0)$ the outgoing target $(1 + r, 1 - r)$ has negative second coordinate and the incoming source $(1 - r, r - 1)$ has negative first coordinate. Both are zero. Thus $E_{2}^{1, 0}$ is canonically the stable term $E_{\infty}^{1, 0} = F^{1} H^{1} / F^{2} H^{1} = F^{1} H^{1}$ . Its edge map is the monic filtration inclusion into $H^{1}$ .

F1F2

1.2

At $(0, 1)$ every incoming source $(- r, r)$ is zero. The outgoing target is $(r, 2 - r)$ , which is in the first quadrant only for $r = 2$ . Consequently the transition identifies $E_{3}^{0, 1}$ with $\ker (d_{2} : E_{2}^{0, 1} \to E_{2}^{2, 0})$ , and all later transitions there are stationary. By abutment this kernel is $E_{\infty}^{0, 1} = H^{1} / F^{1} H^{1}$ . Therefore the edge $H^{1} \to E_{2}^{0, 1}$ is the quotient onto this kernel followed by its inclusion.

F1F2F3

1.3

At $(2, 0)$ all outgoing targets $(2 + r, 1 - r)$ are zero. The incoming source $(2 - r, r - 1)$ is in the quadrant only for $r = 2$ , when it is $(0, 1)$ . Hence $E_{\infty}^{2, 0} = coker d_{2}$ . Abutment identifies this with $F^{2} H^{2} / F^{3} H^{2} = F^{2} H^{2}$ . Its edge into $H^{2}$ is the cokernel projection followed by that filtration inclusion.

F1F2F3

2.1

Step 1.1 proves exactness at $E_{2}^{1, 0}$ including the initial zero. The kernel at $H^{1}$ in step 1.2 is $F^{1} H^{1}$ , the preceding image. Its image at $E_{2}^{0, 1}$ is exactly $\ker d_{2}$ . Step 1.3 says that the kernel of the next edge at $E_{2}^{2, 0}$ is exactly $im d_{2}$ , because its second factor is monic. These are every asserted exactness position; $H^{2}$ has no outgoing arrow in the statement. The formulas remain valid when any term or $d_{2}$ is zero; when $d_{2} = 0$ , its kernel and cokernel are their whole source and target. All identifications use specified transitions and abutment maps, not chosen splittings, and require no AC.

F3step 1.1step 1.2step 1.3∎

Used by

A first quadrant five term exact sequence Example

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Weibel, An Introduction to Homological Algebra, Chapter 5 (standard reference, not scraped)
The Stacks Project, Homological Algebra (standard reference, not scraped)