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Bounded, bounded below, and bounded above complexes
Definition
A chain complex is:
- bounded below if for all ;
- bounded above if for all ;
- bounded if outside some finite interval of integers.
For a cochain complex the same words mean:
- for all ;
- for all ;
- outside a finite interval.
Under the reindexing convention of Cochain complex in an abelian category, bounded below and bounded above exchange roles, while boundedness is unchanged.
Depends on
Used by
- Bounded graded bimodule complexes and signed tensor totalization Definition
- Derived category of an abelian category Definition
- Euler characteristic of a finite complex of finite-rank free abelian groups Definition
- Flat abelian sheaves Definition
- Homotopically projective bounded above complex Definition
- K-flat complexes of abelian sheaves in the bounded-above setting Definition
- Perfect complexes over a ring and its graded version Definition
- Tensor product of abelian sheaves and its total complex Definition
- The category of chain complexes Definition
- Associator, symmetry and unitors of the abelian sheaf tensor product Lemma
- Derived tensor product of abelian sheaves Lemma
- Derived tensor–Hom and inverse/direct image for field sheaves Lemma
- Flat resolutions of abelian sheaves and K-flatness of bounded-above flat complexes Lemma
- K-flat sheaf complexes preserve quasi-isomorphisms Lemma
- Koszul coherence of derived sheaf tensor Lemma
- A bounded above acyclic complex of injective objects is contractible when its cycle monomorphisms split Theorem
- A bounded below acyclic complex of projective objects is contractible when its cycle epimorphisms split Theorem
Dependency tree · two levels
3 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, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
- Romyar Sharifi, Homological Algebra, Definition 2.7.6 (standard reference, not scraped)