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The homotopy category of an abelian category is triangulated
Statement
If is an abelian category, then , with its shift and distinguished cone triangles, is a triangulated category.
Facts & Assumptions
Given: An abelian category .
Proof
is additive and its shift is an additive autoequivalence.
The chosen class of cone triangles satisfies TR1, TR2 with the declared sign, TR3, and TR4 by the four preceding lemmas; these data therefore meet the definition of a triangulated category.
Depends on
- The homotopy category of chain complexes
- The homotopy category is additive
- Shift is an additive autoequivalence of the complex and homotopy categories
- Distinguished cone triangle in the homotopy category
- Cone triangles satisfy TR1
- Cone triangles satisfy TR2 with the declared rotation sign
- Cone triangles satisfy TR3
- Cone triangles satisfy the octahedral axiom
Used by
- A triangle is distinguished whenever the three composites vanish False statement
- The third map in a morphism of triangles is unique False statement
- An additive functor on abelian categories induces an exact functor on homotopy categories Proposition
- Homology is a homological functor on the homotopy category Theorem
Dependency tree · two levels
25 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
- Amnon Yekutieli, A Course on Derived Categories, Theorem 9.2.2 (standard reference, not scraped)