Variants of normality for Noetherian schemes

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.