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The full subcategory of acyclic complexes is thick in the homotopy category
Statement
For an abelian category , the full subcategory of consisting of acyclic complexes is thick.
Facts & Assumptions
Given: An abelian category and the class of acyclic complexes in its homotopy category.
Proof
A cone long exact sequence and the shift identification show that acyclicity is stable under shifts and satisfies two-out-of-three for cone triangles.
Homology takes a retract of a complex to a retract of its homology, and degreewise finite biproducts preserve acyclicity; hence the class is closed under direct summands and is thick.
Depends on
Used by
Dependency tree · two levels
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Sources
- The Stacks Project, Derived Categories, Section 13.10 (standard reference, not scraped)