False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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The two spectral sequences of a double complex have identical e one pages

Statement

False: The two spectral sequences of every double complex have identical $E^{1}$ pages.

Facts & Assumptions

[F1]

The row filtration spectral sequence of a first quadrant double complex computes horizontal-first row pages; The column filtration spectral sequence of a first quadrant double complex computes vertical-first column pages.

[F2]

Abelian-group model for spectral-sequence computations supplies the nonzero group $k = Z / 2$ . The two double complex spectral sequences have the same abutment but not the same pages distinguishes common targets from page equality.

Refutation

Given: $C_{1, 0} = C_{0, 0} = k$ , $h_{1, 0} = 1_{k}$ , and every other component and map zero. All double-complex identities hold because every possible double composite is zero.

1.1

The only nonzero horizontal complex is $k \overset{1}{\to} k$ . Its kernel at the source and cokernel at the target are zero. Thus every row $E^{1}$ term is zero. The two nonzero vertical complexes each consist of a single $k$ with zero differential; therefore column $E_{1, 0}^{1} = E_{0, 0}^{1} = k$ . The row and column formulas have exactly the hypotheses in [F1], since the witness is first quadrant and finitely supported.

F1F2

2.1

In particular the row term at $(0, 0)$ is zero while the column term there is the nonzero group $k$ , so the pages are not even isomorphic as bigraded objects. Both sequences nevertheless abut to the zero homology of the total identity complex. The zero double complex would have equal pages, but cannot rescue the universal assertion. The witness uses only two components and zero or identity maps, with no choice assumption.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

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