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Degeneration of the spectral sequence of a split tower
Statement
Under the hypotheses of Boundary maps vanish for a split tower, the tower spectral sequence of Spectral sequence of a tower has zero differentials on every page for . This assertion alone does not assert convergence or resolve extensions.
Facts & Assumptions
Given: A tower with the specified equivalence to a direct-sum tail tower.
Goal: Show that all differentials in its tower spectral sequence vanish.
Boundary maps vanish for a split tower makes every stage connecting homomorphism zero.
Spectral sequence of a tower constructs the exact couple from stage long exact sequences and derives its pages; its initial map from a graded term to a stage term is the connecting homomorphism.
Proof
In the initial exact couple the map from a graded term to a stage term is zero by [F1]. Its initial differential is therefore zero.
On derivation, the new connecting map sends a surviving graded class to the image in the derived stage group of its former connecting image; because that image is zero, the new connecting map is zero. Induction gives zero connecting map and hence zero differential at every derived page. The reindexing in [F2] gives the asserted pages . No convergence claim is used.
Depends on
Used by · 1
Dependency tree · 3 levels
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