Two remarks on integral morphisms of affine k-varieties
type: Item
instance of: scholarly article
title: Two remarks on integral morphisms of affine k-varieties
zbMATH Open document ID: 0586.14004
DOI: 10.1007/BF02848535
publication date: 1983-01-01T00:00:00Z
review text: The notion of integral morphism between affine algebraic k-varieties (k any field) has been studied by the author and \textit{M. Raimondo} [in Commutative algebra, Proc. Conf., Trento/Italy 1981, Lect. Notes Pure Appl. Math. 84, 183-196 (1983; Zbl 0508.14009)], where an affine variety defined over k is a couple (X,\({\mathcal O}_ X)\) with X homeomorphic to a Zariski closed subset of some \(k^ n\) and \({\mathcal O}_ X\) the sheaf of germs of regular functions on X. In this note we add two pinpoints on the topic. More precisely, let \({\mathcal V}_ k\) denote the category of affine k-varieties and relative morphisms. We first show that in \({\mathcal V}_ k\) any pair (\(\Phi\),\(\Psi)\) of morphisms such that \(\Phi\) is a closed immersion and \(\Psi\) is integral has a push-out. Secondly we give an example of X in \({\mathcal V}_ k\) for which the weak normalization (in the sense of Andreotti and Bombieri) of the affine scheme \((Spec \Gamma (X,{\mathcal O}_ X),\Gamma (X,{\mathcal O}_ X)^{\sim})\) is not in \({\mathcal V}_ k\).
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14A10
zbMATH DE Number: 3939487
zbMATH Keywords: integral morphism between affine algebraic k-varieties
zbMATH Keywords: weak normalization
author: Maria Grazia Marinari
MaRDI profile type: Publication
cites work: Q5576163
cites work: Q5605289
cites work: Q4161796
cites work: Some properties of weakly normal varieties
cites work: On seminormality
full work available at URL: https://doi.org/10.1007/bf02848535
OpenAlex ID: W2045962599
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 3939487