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Collapse does not in general split the abutment
Statement
Collapse does not in general split the filtered abutment.
Facts & Assumptions
Given: The degree-zero stalk of the filtered ℤ/4 in F1.
ℤ/4 with its order-two subgroup has two ℤ/2 graded pieces and is not isomorphic to their direct sum (Isomorphic associated graded objects need not give isomorphic filtered objects).
A degreewise finite filtered complex abuts to its image-filtered homology (Bounded filtered complex spectral sequence abuts to filtered homology).
Collapse means all subsequent differentials vanish (Collapse at a page).
Splitting an extension requires a section of the quotient (Extension problem of a convergent spectral sequence).
A stalk complex has a single nonzero chain object and zero differential (Zero complex and stalk complex).
The r-page is the specified filtered numerator/denominator quotient (R page of the spectral sequence of a filtered complex).
The page differential is induced by the chain differential (The filtered differential induces d r on the r page).
Proof
Put the filtered A=ℤ/4 of [F1] in chain degree zero and zero elsewhere, using [F5]. Every differential of C is zero. Hence and the page denominator is for every r, so the pages have ℤ/2 at (0,0) and (1,-1), zero elsewhere, and all page differentials vanish. Thus the sequence collapses from page zero by [F3].
Its homology is A in degree zero with exactly the given filtration; finite convergence [F2] identifies these two stable pieces with its graded homology. The quotient A→A/{0,2}≅ℤ/2 cannot have a section: the image of 1 would have to be 1 or 3 modulo 4, but either doubles to 2, while a homomorphism from ℤ/2 must send 1 to an element killed by 2. Therefore the extension does not split in the sense of [F4].
Source notes
Weibel, §5.2 pp.123–124; the nonsplitting witness is computed here.
Depends on
- Isomorphic associated graded objects need not give isomorphic filtered objects
- Bounded filtered complex spectral sequence abuts to filtered homology
- Collapse at a page
- Extension problem of a convergent spectral sequence
- Zero complex and stalk complex
- R page of the spectral sequence of a filtered complex
- The filtered differential induces d r on the r page
Used by
- Collapse with a nonsplit extension problem Counterexample
- UCT and Kunneth collapse retains an extension problem Proposition
Dependency tree · two levels
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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)