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Induced filtration on homology
Definition
For a filtered chain complex, the induced filtration on homology is The arrow is induced by the inclusion chain map from Filtered chain complex, using A chain map induces a well-defined map on homology. Since factors through , functoriality in Homology respects identities and composition shows that its homology image is contained in . Thus this is an increasing family of subobjects of . The domain of the arrow need not embed into .
Source notes
Stacks §12.24, Definition 12.24.5; Weibel §5.5 p.135.
Depends on
Used by
- Quasi isomorphism criterion from a filtered map Corollary
- An exhaustive nonseparated filtration with the wrong naive abutment Counterexample
- The filtration on homology is h n of the filtered subcomplex False statement
- The associated graded of filtered homology is a subquotient of chain level data Lemma
- Failure of separatedness or completeness can destroy the claimed abutment Proposition
- A first quadrant filtered complex spectral sequence converges to filtered homology Theorem
- Complete exhaustive filtered complex convergence criterion Theorem
- Homological Serre spectral sequence Theorem
- The column filtration spectral sequence of a first quadrant double complex Theorem
- The row filtration spectral sequence of a first quadrant double complex Theorem
Dependency tree · two levels
9 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)