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.

Brutal truncation of a complex

Definition

For a cochain complex $X$ and $n \in Z$ , the brutal truncations are $(σ^{\leq n} X)^{i} = X^{i}$ for $i \leq n$ and zero otherwise, and $(σ^{\geq n} X)^{i} = X^{i}$ for $i \geq n$ and zero otherwise. Retain the differentials between retained terms and use zero for all other differentials. The former is a quotient $X \to σ^{\leq n} X$ ; the latter is a subcomplex $σ^{\geq n} X \to X$ . These are functorial constructions of cochain complexes as in Cochain complex in an abelian category; no kernel or cokernel correction is made at the cut.

Depends on

Cochain complex in an abelian category

Used by

Canonical truncation of a complex Definition
ex-brutal-versus-canonical-truncation.md Example
fs-brutal-and-canonical-truncation-are-the-same.md False statement

Dependency tree · two levels

3 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

12.15, all four chain and four cochain truncations (standard reference, not scraped)