Pure and Applied Mathematics Quarterly

Volume 12 (2016)

Number 1

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

DOI: https://dx.doi.org/10.4310/PAMQ.2016.v12.n1.a1


János Kollár (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)


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.

Published 15 February 2017