Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12
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E infinity is the abutment object

Statement

It is false that E∞ is the abutment itself, without extension data.

Facts & Assumptions

Given: The finite filtered group ℤ/4 described in F2.

[F1]

Abutment identifies each stable term with a successive filtration quotient (Abutment to a filtered object).

[F2]

The filtered ℤ/4 has two ℤ/2 graded pieces, while every element of their direct sum is killed by 2 (Isomorphic associated graded objects need not give isomorphic filtered objects).

[F3]

An abstract sequence consists of square-zero differentials and specified homology isomorphisms (Homological spectral sequence).

Refutation

technique · direct
1.1

Take H0=Z/4 with 0⊂{0,2}⊂ℤ/4 as in [F2]. A spectral sequence constant on every page, with ℤ/2 at (0,0) and (1,-1), zero elsewhere and zero differentials, has the identity quotient identifications as its transitions. Give it the abutment isomorphisms to these two graded pieces, as allowed by [F1].

F1F2F3
2.1

The finite direct sum of its stable terms in total degree zero is (ℤ/2)², killed by 2. The abutment H0 has 2[1]₄=[2]₄≠0 by [F2]. Thus even summing the stable pieces does not give an isomorphic abutment. The stable page supplies the quotients, not the nonsplit extension.

F1F2step 1.1∎

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.

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Sources