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This note presents a uniform treatment of normality and three of itsvariants---topological, weak and seminormality---for Noetherian schemes. Thekey is to define these notions for pairs $(Z, X)$ consisting of a (notnecessarily reduced) scheme $X$ and a closed, nowhere dense subscheme $Z$. Anadvantage of the new definitions is that, unlike the usual absolute ones, theyare preserved by completions. This shortens some of the proofs and leads tomore general results. Version 2: small changes.

Variants of normality for Noetherian schemes