False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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FALSE: positive Ext vanishes whenever either variable is injective

Statement

Facts & Assumptions

Given: The category of abelian groups and $0 \to Z \to Q \to Q / Z \to 0$ .

Refutation

technique · direct

1.1

The group $Q / Z$ is divisible and therefore injective, but it occurs as the quotient, hence as the first argument of the extension. If the sequence split, a retraction $Q \to Z$ would restrict to the identity on $Z$ .

givenalgebra

2.1

Every homomorphism $Q \to Z$ is zero, so the extension is nonsplit and yields $0 \neq {Ext}^{1} (Q / Z, Z)$ . Injectivity instead forces positive Ext to vanish in the second variable, and projectivity does so in the first.

step 1.1algebra∎

Depends on

Used by

Positive Ext need not vanish for an injective first variable Counterexample

Dependency tree · two levels

5 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

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 (standard reference, not scraped)