Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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fs-two-roofs-are-equal-whenever-their-right-hand-arrows-are-equal.md

Statement

Two roofs between the same objects are equal in the derived category whenever their right-hand arrows are equal.

Facts & Assumptions

Given: Two roofs between the same objects are equal in the derived category whenever their right-hand arrows are equal.

[F1]

A roof represents numerator composed with the inverse of its denominator (The calculus of fractions constructs the localization).

[F2]

Cohomology factors through localization (Cohomology factors through the derived category).

Refutation

1.1

On X=Z[0], compare (X←1X→1X) and (X←−1X→1X). Both denominators are quasi-isomorphisms and the numerator in each is the identity. The roof formula gives the morphisms 1 and −1 respectively. All other degrees are zero.

F1algebra
2.1

The functor H0 sends the two morphisms to 1 and −1 on Z, unequal at the element 1. Hence the roofs are unequal despite identical numerators. The denominator is essential data.

F2step 1.1algebra∎

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