Some Comments around The Examples against The Generalized Jacobian Conjecture

Abstract

We have studied a faded problem, the Jacobian Conjecture ~: \noindent {\sf The Jacobian Conjecture $(JC_n)$}~: If $f_1, \cdots, f_n$ are elements in a polynomial ring $k[X_1, \cdots, X_n]$ over a field $k$ of characteristic $0$ such that the Jacobian $\det(\partial f_i/ \partial X_j) $ is a nonzero constant, then $k[f_1, \cdots, f_n] = k[X_1, \cdots, X_n]$. For this purpose, we generalize it to the following form~: \noindent {\sf The Generalized Jacobian Conjecture $(GJC)$}~: {\it Let $\varphi : S \rightarrow T$ be an unramified homomorphism of Noetherian domains with $T^\times = \varphi(S^\times)$. Assume that $T$ is a factorial domain and that $S$ is a simply connected normal domain. Then $\varphi$ is an isomorphism. } For the consistency of our discussion, we raise some serious (or idiot) questions and some comments concerning the examples appeared in the papers published by the certain excellent mathematicians (though we are unwilling to deal with them). Since the existence of such examples would be against our original target Conjecture$(GJC)$, we have to dispute their arguments about the existence of their respective (so called) counter-examples. Our conclusion is that they are not perfect counter-examples as are shown explicitly.