Variants of normality for Noetherian schemes

@article{Kollar2015VariantsON,
  title={Variants of normality for Noetherian schemes},
  author={J'anos Koll'ar},
  journal={arXiv: Algebraic Geometry},
  year={2015}
}
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. Version 2: small changes. 
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