LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Ext one of Z modulo n by Z is Z modulo n

Statement

Assume the Axiom of Dependent Choice and fix the supplied resolution data used for balanced Ext in $A b$ . For $n > 1$ , ${Ext}_{Z}^{1} (Z / n, Z) ≅ Z / n$ ; in particular it is nonzero.

Facts & Assumptions

Given: The projective resolution $0 \to Z \overset{n}{\to} Z \to Z / n \to 0$ .

Proof

technique · direct

1.1givenconstruct

Applying ${Hom}_{Z} (-, Z)$ gives $0 \to Z \overset{n}{\to} Z \to 0$ , in cohomological degrees $0, 1$ .

2.1step 1.1algebra∎

Its first cohomology is $coker (n : Z \to Z) = Z / n$ . Ext can be computed from any projective resolution of the first variable identifies this cohomology with ${Ext}_{Z}^{1} (Z / n, Z)$ . Since $n > 1$ , the quotient is nonzero.

Depends on

Used by

Dependency tree · two levels

17 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)