Pure and Applied Mathematics Quarterly
Volume 12 (2016)
Special Issue: In Honor of Eduard Looijenga, Part 2 of 3
Guest Editor: Gerard van der Geer
Variants of normality for Noetherian schemes
Pages: 1 – 31
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 $Z \subset X$ 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.