A Purely Algebraic Approach to The Generalized Jacobian Conjecture
Our goal is to settle the following faded problem, {\sf The Jacobian Conjecture $(JC_n)$~:} {\it 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 $ \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: {\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 factorial and that $S$ is a simply connected normal domain. Then $\varphi$ is an isomorphism.} We settle Conjecture $(GJC)$, which resolves $(JC_n)$ as a corollary. To settle $(GJC)$, we show the following result on Krull domains with some conditions. {\sf Theorem :} {\it Let $R$ be a Krull domain domain and let $\Delta_1$ and $\Delta_2$ be subsets of ${\rm Ht}_1(R)$ such that $\Delta_1\cup \Delta_2 = {\rm Ht}_1(R)$ and $\Delta_1\cap \Delta_2 = \emptyset$. Put $R_i := \bigcap_{Q\in \Delta_i}R_Q\ (i=1,2)$, subintersections of $R$. Assume that $\Delta_2$ is a finite set, that $R_1$ is factorial and that $R\hookrightarrow R_1$ is flat. If $R^\times = (R_1)^\times$, then $\Delta_2 = \emptyset$ and $R = R_1$. }