TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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The heart of the canonical t structure is equivalent to the original abelian category

Statement

The degree-zero functor $A \to D (A)$ identifies $A$ with the heart of the canonical t-structure. The inverse equivalence is $H^{0}$ .

Facts & Assumptions

Given: The degree-zero functor $A \to D (A)$ identifies $A$ with the heart of the canonical t-structure. The inverse equivalence is $H^{0}$ .

[F1]

The canonical t-structure has the boundary Hom formula (The canonical pair is a t structure).

[F2]

Canonical truncations have the claimed cohomology and natural maps (Canonical truncation is a complex and has the claimed cohomology).

Proof

1.1

For objects $M, N$ in degree zero the boundary Hom formula with $a = b = 0$ gives ${Hom}_{D} (M [0], N [0]) = {Hom}_{A} (M, N)$ , and the identification takes a map to its degree-zero cohomology map. This proves full faithfulness, including zero objects and identity maps.

2.1

If $X$ is in the heart, the natural zigzag $X \leftarrow τ^{\leq 0} X \to τ^{\geq 0} τ^{\leq 0} X = H^{0} (X) [0]$ consists of quasi-isomorphisms. It is functorial, and $H^{0} (M [0]) = M$ . These natural isomorphisms give the inverse equivalence and essential surjectivity.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Proposition 12.4.1, p. 64 (standard reference, not scraped)