DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How 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.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Canonical t structure on a derived category

Definition

For a triangulated category $T$ with shift $[1]$ , a t-structure is a pair of strictly full subcategories $(T^{\leq 0}, T^{\geq 0})$ such that $T^{\leq 0} [1] \subseteq T^{\leq 0}$ , $T^{\geq 0} [- 1] \subseteq T^{\geq 0}$ , ${Hom}_{T} (T^{\leq 0}, T^{\geq 1}) = 0$ , and every $X$ has a distinguished triangle $A \to X \to B \to A [1]$ with $A \in T^{\leq 0}$ and $B \in T^{\geq 1}$ . Here $T^{\geq 1} = T^{\geq 0} [- 1]$ and $T^{\leq n} = T^{\leq 0} [- n]$ , $T^{\geq n} = T^{\geq 0} [- n]$ . Its heart is $T^{\leq 0} \cap T^{\geq 0}$ .

For the triangulation of The derived category inherits a triangulated structure, the canonical candidate is $D^{\leq 0} = {X : H^{i} (X) = 0 for i > 0}$ and $D^{\geq 0} = {X : H^{i} (X) = 0 for i < 0}$ , using Cohomology factors through the derived category.

Depends on

Used by

The canonical pair is a t structure Theorem

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

Lecture 3, section 3, definition and Main example, pp. 28–29 (standard reference, not scraped)