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

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FALSE: an extension of M by N represents Ext one of N by M

Statement

Facts & Assumptions

Given: $M = Z / n Z$ , $N = Z$ , and $n \geq 2$ in the category of abelian groups.

Refutation

technique · direct

1.1givenalgebra

An extension of $M$ by $N$ has the typed form $0 \to N \to E \to M \to 0$ ; its connecting homomorphism starts at $1_{M} \in Hom (M, M)$ and lands in ${Ext}^{1} (M, N)$ .

2.1step 1.1algebra∎

Here ${Ext}^{1} (Z / n Z, Z) ≅ Z / n Z$ , whereas ${Ext}^{1} (Z, Z / n Z) = 0$ because $Z$ is projective. Interchanging the endpoints therefore changes the group and disproves the asserted order.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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, Chapters 3–4 (standard reference, not scraped)