Fun example: empty colimit does not commute with empty limit
2026ЗаметкиBogdan Zavyalov
Why the definition of a filtered category insists on non-emptiness: the empty colimit fails to commute with the empty limit, and that is the only failure.
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A non-Noetherian local ring with finitely generated maximal ideal
2026ЗаметкиBogdan Zavyalov
A complete local ring with a finitely generated maximal ideal is Noetherian. Drop completeness and it fails: an ultraproduct construction gives a non-Noetherian local ring whose maximal ideal is principal.
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