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 truncation of a complex

Definition

For a cochain complex $X$ , define the canonical truncations by

$(τ^{\leq n} X)^{i} = {\begin{cases} X^{i} & i < n, \\ \ker d_{X}^{n} & i = n, \\ 0 & i > n, \end{cases} (τ^{\geq n} X)^{i} = {\begin{cases} 0 & i < n, \\ coker d_{X}^{n - 1} & i = n, \\ X^{i} & i > n . \end{cases}$

The differential into $\ker d_{X}^{n}$ is the factorization of $d_{X}^{n - 1}$ ; the differential out of $coker d_{X}^{n - 1}$ is induced by $d_{X}^{n}$ . All other retained differentials are those of $X$ . There are natural maps $τ^{\leq n} X \to X \to τ^{\geq n} X$ . Unlike Brutal truncation of a complex, these constructions correct the boundary using the cycle and boundary objects of Cohomology object of a cochain complex.

Depends on

Dependency tree · 2 levels

Sources

12.15, all four chain and four cochain truncations