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Exactness of a complex at a degree and acyclic complexes
Definition
Let be a chain complex and let be the canonical map from boundaries to cycles.
The complex is exact at degree when is an isomorphism. Equivalently, the pair is exact at in the sense of Exactness at a node, so that and represent the same subobject of .
The complex is acyclic if it is exact at every degree.
Depends on
Used by
- A chain map between acyclic complexes has an acyclic cone Corollary
- A short exact sequence with acyclic middle complex identifies neighbouring homology Corollary
- Relative homology vanishes exactly for quasi-isomorphisms Corollary
- The cone criterion from the general long exact sequence Corollary
- Two-out-of-three for acyclicity in a short exact sequence of complexes Corollary
- Injective resolutions in an abelian category Definition
- K-flat complexes of abelian sheaves in the bounded-above setting Definition
- Projective resolutions in an abelian category Definition
- A degreewise split exact complex with compatible splittings is contractible Lemma
- Derived tensor product of abelian sheaves Lemma
- Flat resolutions of abelian sheaves and K-flatness of bounded-above flat complexes Lemma
- K-flat sheaf complexes preserve quasi-isomorphisms Lemma
- One-step extension of a partial injective resolution Lemma
- One-step extension of a partial projective resolution Lemma
- A quasi-isomorphism has an acyclic cone in the triangulated language Proposition
- An exact sequence is a complex, and its exactness agrees with the earlier notion Proposition
- The full subcategory of acyclic complexes is thick in the homotopy category Proposition
- 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
- A chain map is a quasi-isomorphism exactly when its cone is acyclic Theorem
- A complex is exact at n exactly when its nth homology is zero Theorem
Dependency tree · two levels
10 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)
- The Stacks Project, Section 12.13: Complexes (standard reference, not scraped)