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: http://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.

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