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The universal coefficient theorem does not always give a natural direct-sum decomposition
Statement
The assertion that the natural UCT short exact sequence has a natural direct-sum decomposition is false.
Refutation
Given: the natural UCT exact sequence and its nonnatural splitting construction.
UCT supplies a natural short exact sequence, while a section can be constructed only after auxiliary choices. The cited counterexample uses a specific free complex, coefficient group , and a chain automorphism acting trivially on the two outer UCT terms but nontrivially on every possible section.
By A universal-coefficient splitting cannot in general be chosen naturally, naturality for that automorphism would force in , which is impossible. Therefore no alternative natural choice of section can exist in general, and the asserted natural direct-sum decomposition is false.
Depends on
Used by
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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)