TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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 the Grothendieck spectral sequence

Statement

Under the hypotheses and choice/data conventions of the Grothendieck spectral sequence there is a natural exact sequence $0 \to R^{1} G (F (A)) \to R^{1} (G F) (A) \to G (R^{1} F (A)) \overset{d_{2}^{0, 1}}{\to} R^{2} G (F (A)) \to R^{2} (G F) (A) .$ The unnamed arrows are the canonical edges. No surjectivity onto the last term is asserted.

Facts & Assumptions

Given: The hypotheses of the Grothendieck theorem.

[F1]

The second page is $R^{p} G (R^{q} F (A))$ , the target is $R^{p + q} (G F) (A)$ and its normalized filtration is finite (Grothendieck spectral sequence).

[F2]

A first-quadrant cohomological sequence with such finite abutment has the five-term exact sequence with the middle $d_{2}$ and the extremal edge arrows (Five term exact sequence of a first quadrant cohomological spectral sequence).

Proof

1.1

In F2 substitute $H^{n} = R^{n} (G F) (A)$ and the page of F1. Left exactness gives $R^{0} F (A) = F (A)$ and $R^{0} G = G$ . Thus the three page entries $(1, 0), (0, 1), (2, 0)$ are respectively $R^{1} G (F (A))$ , $G (R^{1} F (A))$ and $R^{2} G (F (A))$ . The only relevant later differential is $d_{2}^{0, 1}$ with target $(2, 0)$ .

F1F2

2.1

After the substitutions of step 1.1, the displayed sequence is exactly the five-term exact sequence exported by F2, and F2 identifies the arrows adjacent to the two abutment terms as the corresponding edges. Because it is applied to the filtered sequence of F1, these are the Grothendieck spectral sequence's canonical edges. The spectral-sequence maps supplied by F1 commute with $d_{2}$ and with those edge maps, proving naturality. If $d_{2} = 0$ or a displayed object is zero, F2's same exact sequence still applies. Its Statement has no outgoing arrow from $H^{2}$ , so no final epimorphism is claimed.

F1F2step 1.1∎

Depends on

Used by

Five-term sequence of a composite functor Example
Five-term exact sequence from LHS Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Weibel, Theorem 5.8.3, low-degree sequence, printed p.151 (standard reference, not scraped)