ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)verified 2026-09-10 (Codex)

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Tor of two cyclic groups from a two-term resolution

Example

Assume AC (The Axiom of Choice) for balanced Tor. Then ${Tor}_{1}^{Z} (Z / 12, Z / 18) ≅ Z / 6$ . The fixed two-term resolution calculation is choice-free; AC is inherited only for the balanced comparison in Tor one of two cyclic abelian groups is cyclic of gcd order.

Verification

Given: the two-term resolution $0 \to Z \overset{12}{\to} Z \to Z / 12 \to 0$ , and AC for balanced comparison.

[L1] The repaired two-cyclic theorem supplies the kernel identification, the explicit cyclic resolutions and the inherited AC comparison (Tor one of two cyclic abelian groups is cyclic of gcd order).

1.1givenL1

By [L1], tensoring with $Z / 18$ gives the degree-one kernel of multiplication by $12$ on $Z / 18$ . The comparison with balanced Tor uses the stated AC hypothesis; no additional resolution-existence assertion is needed.

2.1step 1.1algebra

The congruence is equivalent to $3 ∣ 2 x$ , hence to $3 ∣ x$ , since $x = 2 (2 x) - 3 x$ . The kernel residues modulo $18$ are therefore $0, 3, 6, 9, 12, 15$ .

3.1step 1.1step 2.1algebra∎

The additive map $Z / 6 \to Z / 18$ , $[t]_{6} \mapsto [3 t]_{18}$ , is well defined because $3 \cdot 6 = 18$ , has image the listed kernel, and is injective because $18 ∣ 3 t$ iff $6 ∣ t$ . This proves the isomorphism and the choice-free kernel calculation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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 (standard reference, not scraped)