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Extension problem of a convergent spectral sequence
Definition
For the abutment data of Abutment to a filtered object, the extension problem is reconstructing the filtered from its successive pieces . Each step has a subobject and quotient Specifying the two outside objects does not specify the middle one or the maps. A splitting would supply a section of the quotient map and an identification with the direct sum compatible with the subobject. Neither collapse nor a list of graded pieces supplies such sections. The one-jump case has no nontrivial extension between two nonzero pieces.
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.
Depends on
Used by
- Collapse solves all extension problems False statement
- Collapse does not in general split the abutment Proposition
Dependency tree · two levels
4 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
- Charles A. Weibel, An Introduction to Homological Algebra, Chapter 5 (standard reference, not scraped)
- Romyar Sharifi, Homological Algebra (standard reference, not scraped)