Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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fs-brutal-and-canonical-truncation-are-the-same.md

Statement

Brutal and canonical truncation of a complex at the same degree always coincide, even up to derived isomorphism.

Facts & Assumptions

Given: Brutal and canonical truncation of a complex at the same degree always coincide, even up to derived isomorphism.

[F1]

Brutal truncation retains the original boundary term (Brutal truncation of a complex).

[F2]

Canonical upper truncation replaces its boundary term by the kernel of the outgoing differential (Canonical truncation of a complex).

Refutation

1.1

Take X=(Z→2Z) in degrees 0,1, zero elsewhere. Brutal upper truncation at zero is σ≤0X=Z[0]. Canonical upper truncation has degree-zero term ker⁡(2:Z→Z)=0, hence τ≤0X=0.

F1F2algebra
2.1

Their degree-zero cohomology groups are Z and zero, so they are neither equal complexes nor isomorphic derived objects. The endpoint kernel correction changes the answer.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources