Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A distinguished triangle is split up to rotation exactly when one map vanishes
Statement
Call a triangle split up to rotation when it is isomorphic to a rotation of a canonical biproduct triangle. A distinguished triangle is split up to rotation if and only if at least one of its three maps vanishes.
Facts & Assumptions
Given: A distinguished triangle.
Proof
If one map vanishes, rotate until it is first and apply A distinguished triangle with zero first map is split; undoing that rotation makes the original triangle split up to rotation.
A canonical biproduct triangle has zero final map. Each of its rotations therefore has one zero map, and this property is preserved by an isomorphism of triangles; this proves the converse.
Depends on
Used by
- The split distinguished triangle Example
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
- The Stacks Project, Lemma 13.4.11 (standard reference, not scraped)