Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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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.
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FALSE: Ext is defined before choosing or supplying resolutions

Statement

FALSE: Ext is defined before choosing or supplying resolutions

Facts & Assumptions

Given: An abelian category, objects M,N, and the two resolution-based Ext constructions of this page.

Refutation

technique · direct
1.1

The injective construction is the cohomology of Hom⁡(M,I(N)) and requires a supplied injective resolution N↪I(N); the projective construction similarly requires a supplied resolution P(M)↠M. Neither complex is specified before that datum is supplied.

givenalgebra
2.1

Thus the raw resolution constructions are not definitions of a choice-free object merely from M and N. Choice-independence is a later comparison result, so it cannot make the claim “defined before supplying resolutions” true.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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