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

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A universal-coefficient splitting cannot in general be chosen naturally

Statement

The noncanonical splitting in the homological UCT cannot in general be chosen naturally in the complex and coefficient group.

Counterexample

Given: the free complex $C_{1} = Z e \oplus Z f$ , $C_{0} = Z g$ , with $d (e) = 2 g$ and $d (f) = 0$ , together with $G = Z / 2$ .

1.1givenalgebra

Since $H_{1} (C) = Z f$ , $H_{0} (C) = Z / 2$ , and $d \otimes G = 0$ , direct kernel and cokernel calculations give the exact degree-one sequence $0 ⟶ (Z / 2) f ⟶ (Z / 2) e \oplus (Z / 2) f \overset{q}{\to} Z / 2 ⟶ 0$ , with $q (a e + b f) = a$ . This is the UCT sequence in this finite example: its left map tensors the cycle $f$ , and its right map records the coefficient of $e$ modulo $2$ .

2.1step 1.1algebra

The chain automorphism $u_{1} (e) = e + f$ , $u_{1} (f) = f$ , $u_{0} (g) = g$ induces the identity on both integral homology groups and hence on both outer UCT terms, while its action on the middle term is $(a, b) \mapsto (a, a + b)$ .

3.1step 2.1contradiction∎

A section must send $1$ to $(1, t)$ for some $t \in Z / 2$ . Naturality with respect to $u$ would force $(1, t + 1) = (1, t)$ , which is impossible. Therefore a UCT splitting cannot be chosen naturally for all complexes and coefficient groups.

Depends on

Used by

A nonnatural choice of universal-coefficient splitting Counterexample
The universal coefficient theorem does not always give a natural direct-sum decomposition False statement

Dependency tree · two levels

7 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

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