Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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FALSE: balance of Ext requires spectral-sequence pages

Statement

FALSE: balance of Ext requires spectral-sequence pages

Facts & Assumptions

Given: The first-quadrant double complex Kp,q=Hom⁡(Pp,Iq) associated with supplied projective and injective resolutions.

Refutation

technique · direct
1.1givenalgebra

For each p, Hom⁡(Pp,−) is exact because Pp is projective, so the augmented q-columns are exact; the finite-diagonal acyclic-assembly lemma gives a quasi-isomorphism from the projective edge complex to Tot⁡K.

2.1step 1.1algebra∎

Dually, each Hom⁡(−,Iq) is exact, so exact rows give a quasi-isomorphism from the injective edge complex to the same total complex. The resulting cohomology isomorphism balances Ext without constructing any spectral-sequence page.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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