Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Collapse at a page
Definition
A spectral sequence collapses at if every differential at every bidegree is zero for every r≥s. The specified transitions then identify with the stable page of Spectral sequence stabilization at a bidegree. One zero differential does not meet this definition. Pointwise stabilization with no uniform bound R(p,q) also does not imply collapse at any fixed page. We use the same definition with subscripts in the cohomological convention.
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. The all-later-differentials convention follows Sharifi Definition 4.1.13; Weibel Definition 5.2.7 uses the narrower one-row/column terminology.
Depends on
Used by
- Grothendieck collapse when one functor is exact Corollary
- Spectral-sequence computation record Definition
- A spectral sequence collapses when one differential is zero False statement
- Collapse solves all extension problems False statement
- Collapse does not in general split the abutment Proposition
- Collapse from one column at page s ≥ 1 or one row at page s ≥ 2 Proposition
Dependency tree · two levels
2 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)