On purely inseparable extensions $K[X,Y]/K[X’,Y’]$ and their generators
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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 $n\ge 0$ 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 $\phi \in \operatorname {qt} R$ and $n\ge 0$ such that $\mathbf {k} (u,\phi )\supseteq R^{p^n}$, then $u$ is in fact a $p$-generator of $R$.References
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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
- Received by editor(s): June 7, 1994
- Additional Notes: The author was supported by a grant from NSERC Canada
- Communicated by: Eric M. Friedlander
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 1337-1345
- MSC (1991): Primary 13F20
- DOI: https://doi.org/10.1090/S0002-9939-96-03377-1
- MathSciNet review: 1327003