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Date: Monday, 28 November 2022 Time: 3.30 pm - 4.30 pm Venue: Lecture Hall 4 On a residual coordinate which is a non-trivial line Amartya Kumar Dutta ISI Kolkata. 28-11-22 Abstract Let $R$ be a Noetherian integral domain. %The notation $A=R^{[n]}$ will denote that $A$ is a polynomial ring in $n$ indeterminates over $R$. A polynomial $F$ in the polynomial ring $R[X,Y]$ is said to be a {\it coordinate} if $R[X,Y]=R[F,G]$ for some $G \in R[X,Y]$. An obvious necessary condition for a polynomial $F$ to be a coordinate is that it should be a {\it line} in $R[X,Y]$, i.e., $R[X,Y]/(F)=R^{[1]}$. When $R$ is a field of characteristic zero, the famous Epimorphism Theorem of Abhyankar-Moh shows that any line in $R[X,Y]$ is necessarily a coordinate. The Generalized Epimorphism Theorem of S.M. Bhatwadekar shows that the result holds when $R$ is a seminormal domain of characteristic zero or if $R$ contains $\mathbb Q$. Examples of B. Segre and M. Nagata show that when $R$ is a field of positive characteristic, a line need not be a coordinate in $R[X,Y]$. Another necessary condition for a polynomial $F$ in $R[X,Y]$ to be a {\it coordinate} is that $F$ should be a {\it residual coordinate}, i.e., for every prime ideal $P$ of $R$, the image of $F$ in $k(P)[X,Y]$ should be a coordinate in $k(P)[X,Y]$, where $k(P)$ denotes the residue field $R_P/PR_P$. Again, it has been shown by Bhatadekar-Dutta that if $R$ contains $\mathbb Q$ or if $R$ is seminormal, then a residual coordinate in $R[X,Y]$ is necessarily a coordinate. Bhatwadekar has also shown that if the characteristic of $R$ is zero, then a line in $R[X,Y]$ is necessarily a residual coordinate. However, an example of Bhatwadekar-Dutta shows that, in general, a residual coordinate in $R[X,Y]$ need not be a line; in particular, it need not be a coordinate. To summarize, when $R$ is neither seminormal nor contains $\mathbb Q$, then a line need not be a coordinate and a residual coordinate need not be a line. This naturally raises the question whether a residual coordinate in $R[X,Y]$ which is also a line in $R[X,Y]$ is necessarily a coordinate in $R[X,Y]$. We shall demonstrate a negative answer to the above question by presenting a residual coordinate $F$ in $R[X,Y]$ over the one-dimensional Noetherian local domain $R=k[[t^2, t^3]]$, where $k$ is a field of characteristic $>2$, such that $F$ is a line in $R[X,Y]$ but $F-1$ is not. Let $\widetilde{R}$ denote the ring $k[[t]]$ and $\tilde{k}$ the ring $\widetilde{R}/t^2\widetilde{R}$. The main tool in the construction of the example is the technique of lifting of a certain $\tilde{k}$-automorphism of $\tilde{k}[X,Y]$ to an $\tilde{R}$-automorphism of $\tilde{R}[X,Y]$ introduced by T. Asanuma. A modification of the above example shows that the Generalized Epimorphism Theorem of S.M. Bhatwadekar cannot be extended to an arbitrary Noetherian domain of characteristic zero. This is a joint work with Prof. T. Asanuma.
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