Alphabeta MathHow statement and proof provenance work
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The triangulated five lemma
Statement
If a morphism of distinguished triangles has two adjacent object components isomorphisms, then its remaining component is an isomorphism.
Facts & Assumptions
Given: A morphism of distinguished triangles with two adjacent isomorphism components.
Proof
Apply to the morphism and use the long exact Hom sequences; the ordinary five lemma makes the map on induced by the third component an isomorphism for every .
Taking to be each source and evaluating inverse natural maps at identities yields a two-sided inverse for the third component, hence it is an isomorphism.
Depends on
Used by
- Euler class of a bounded projective complex is derived invariant and triangle additive Lemma
- Perfect complexes form an essentially small triangulated subcategory Lemma
- Truncation triangles are essentially unique and give adjoint truncation functors Lemma
- Two isomorphism components of a morphism of triangles force the third Proposition
Dependency tree · two levels
7 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 10, Exercise 10.2.2 (standard reference, not scraped)