Variants of normality for Noetherian schemes

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This note presents a uniform treatment of normality and three of its variants—topological, weak and seminormality— for Noetherian schemes. The key is to define these notions for pairs Z ⊂ X consisting of a (not necessarily reduced) scheme X and a closed, nowhere dense subscheme Z. An advantage of the new definitions is that, unlike the usual absolute ones, they are preserved by completions. This shortens some of the proofs and leads to more general results.

Original languageEnglish (US)
Pages (from-to)1-31
Number of pages31
JournalPure and Applied Mathematics Quarterly
Issue number1
StatePublished - 2016

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Completion
  • Nagata scheme
  • Noetherian scheme
  • Normalization


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