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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Distinguished triangles are closed under shifts and both rotations
Statement
Every integral translate, every left rotation, and every inverse rotation of a distinguished triangle is distinguished, with the signs determined by Rotation of a triangle.
Facts & Assumptions
Given: A distinguished triangle in a triangulated category.
Proof
TR2 gives the signed left rotation, and conversely says that its being distinguished is equivalent to that of the original triangle.
Three successive left rotations give the translate of the original triangle by , up to the sign automorphisms prescribed by Rotation of a triangle. Hence is distinguished if and only if is distinguished.
Iterating this equivalence in both directions gives for every . A right rotation is the inverse of a left rotation up to the same sign isomorphisms, so it too preserves distinguished triangles.
Depends on
Used by
Dependency tree · two levels
9 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, Definition 13.3.2 (standard reference, not scraped)