Purely inseparable coverings of exponent one of the affine plane
type: Item
instance of: scholarly article
title: Purely inseparable coverings of exponent one of the affine plane
zbMATH Open document ID: 0581.14007
DOI: 10.1016/0022-4049(83)90098-1
author: Peter Russell
author: M. Miyanishi
published in: Journal of Pure and Applied Algebra
publication date: 1983-01-01T00:00:00Z
review text: Let k be an algebraically closed field of characteristic p\(>0\). A purely inseparable covering of exponent 1 of the affine plane \({\mathbb{A}}^ 2:=Spec k[u,v]\), or affine Zariski surface, is a normal affine surface \(X:=Spec {\mathbb{A}}\) such that \(k[u,v]\subsetneqq A\subsetneqq k[x,y]\) with \(u=x^ p,\) \(v=y^ p.\) \textit{J. Lang} [Mich. Math. J. 28, 375-380 (1981; Zbl 0495.14021)] has shown that if \(char(k)=2\), then \(A:=k[u,v(uv+1)x+vy]\) is a regular, factorial, irrational k-algebra, and we will show that this is true, in fact, for arbitrary \(p>0\). These examples suggest that affine Zariski surfaces have a rich geometry to be studied. One should contrast this with the situation when \(A\subset k[x,y]\) is an affine, regular k-algebra such that k[x,y] is generically separable over A. It is known that then A is a polynomial ring over k if A is factorial.
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 3929150
zbMATH Keywords: characteristic p
zbMATH Keywords: purely inseparable covering of exponent 1
zbMATH Keywords: affine Zariski surfaces
MaRDI profile type: Publication
full work available at URL: 0022-4049(83)90098-1
OpenAlex ID: W2005451849
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description: scientific article; zbMATH DE number 3929150