ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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A nonzero Tor correction in universal coefficients

Statement

For the complex $0 \to Z \overset{2}{\to} Z \to 0$ and coefficients $Z / 2$ , the degree-one UCT correction is ${Tor}_{1}^{Z} (Z / 2, Z / 2) ≅ Z / 2$ .

Verification

Given: $H_{0} C ≅ Z / 2$ , $H_{1} C = 0$ , and coefficients $Z / 2$ .

1.1given

Tensoring produces $0 \to Z / 2 \overset{0}{\to} Z / 2 \to 0$ , so $H_{1} (C \otimes Z / 2) ≅ Z / 2$ .

2.1step 1.1∎

The displayed integral complex is itself a length-one free resolution of $H_{0} C = Z / 2$ . Tensoring this resolution with $Z / 2$ makes its differential zero, so the definition of Tor (equivalently, Tor one of two cyclic abelian groups is cyclic of gcd order) gives ${Tor}_{1}^{Z} (H_{0} C, Z / 2) = Z / 2$ . Since $H_{1} C \otimes Z / 2 = 0$ , the degree-one correction is exactly the nonzero group computed in step 1.1. This finite free-resolution calculation needs no choice; under AC it is also the correction term in The universal coefficient theorem for homology over a PID.

Depends on

Used by

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Sources

Weibel, An Introduction to Homological Algebra (standard reference, not scraped)