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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On purely inseparable extensions $k[X,Y]/k[X',Y']$
and their generators


Author: D. Daigle
Journal: Proc. Amer. Math. Soc. 124 (1996), 1337-1345
MSC (1991): Primary 13F20
MathSciNet review: 1327003
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Abstract: Let ${\mathbf{k}} $ be a field of characteristic $p>0$ and $R={\mathbf{k}} [X,Y]$ a polynomial algebra in two variables. By a $p$-generator of $R$ we mean an element $u$ of $R$ for which there exist $v\in R$ and $n\ge0$ such that ${\mathbf{k}} [u,v]\supseteq R^{p^n}$. We also define a $p$-line of $R$ to mean any element $u$ of $R$ whose coordinate ring $R/uR$ is that of a $p$-generator. Then we prove that if $u\in R$ is such that $u-T$ is a $p$-line of ${\mathbf{k}} (T)[X,Y]$ (where $T$ is an indeterminate over $R$), then $u$ is a $p$-generator of $R$. This is analogous to the well-known fact that if $u\in R$ is such that $u-T$ is a line of ${\mathbf{k}} (T)[X,Y]$, then $u$ is a variable of $R$. We also prove that if $u$ is a $p$-line of $R$ for which there exist $\varphi\in\operatorname{qt} R$ and $n\ge0$ such that ${\mathbf{k}} (u,\varphi)\supseteq R^{p^n}$, then $u$ is in fact a $p$-generator of $R$.


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Additional Information

D. Daigle
Affiliation: Department of Mathematics, University of Ottawa, Ottawa, Canada K1N 6N5
Email: daniel@zenon.mathstat.uottawa.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03377-1
PII: S 0002-9939(96)03377-1
Received by editor(s): June 7, 1994
Additional Notes: The author was supported by a grant from NSERC Canada
Communicated by: Eric M. Friedlander
Article copyright: © Copyright 1996 American Mathematical Society