ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.

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ex-brutal-versus-canonical-truncation.md

Example

For $X = (Z \overset{2}{\to} Z)$ in degrees $0, 1$ , its four truncations at zero are $σ^{\leq 0} X = Z [0]$ , $τ^{\leq 0} X = 0$ , $σ^{\geq 0} X = X$ , and $τ^{\geq 0} X = X$ .

Facts & Assumptions

Given: For $X = (Z \overset{2}{\to} Z)$ in degrees $0, 1$ , its four truncations at zero are $σ^{\leq 0} X = Z [0]$ , $τ^{\leq 0} X = 0$ , $σ^{\geq 0} X = X$ , and $τ^{\geq 0} X = X$ .

[F1]

Brutal truncations simply delete terms (Brutal truncation of a complex).

[F2]

Canonical truncations use a kernel at an upper cut and a cokernel at a lower cut (Canonical truncation of a complex).

[F3]

Canonical truncations preserve the cohomology on the retained side (Canonical truncation is a complex and has the claimed cohomology).

Verification

1.1

Upper brutal truncation retains the degree-zero $Z$ and deletes the next term. Upper canonical truncation replaces it by $\ker (2) = 0$ . All lower degrees are already zero.

F1F2algebra

2.1

The lower brutal truncation deletes only already-zero terms. The lower canonical boundary is $coker (0 \to Z) = Z$ , so its differential remains multiplication by two. Thus both lower truncations equal $X$ . Its $H^{0}$ is zero, agreeing with the canonical upper truncation and disagreeing with the brutal upper truncation. The four constructions need not be four distinct complexes.

F1F2F3step 1.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

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