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Abstract

Our goal is to settle the following faded problem: The Jacobian Conjecture (JC_n): If f_1,..,f_n are elements in a polynomial ring k[X_1,..,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,..,f_n] = k[X_1,..,X_n]. For this purpose, we generalize it to the following: The Deep Jacobian Conjecture (DJC): 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 an (algebraically) simply connected normal domain. Then \varphi is an isomorphism. To settle (DJC), we show the following result on Krull domains. Theorem: Let R be a Krull domain and let Delta_1 and Delta_2 be subsets of Ht_1(R) such that Delta_1\cup Delta_2 = 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. From this theorem, we have Theorem: Let k be a field and let X be a k-affine (irreducible) variety of dimension n. Then X contains a k-affine open subvariety U which is isomorphic to a k-affine space \mathbb{A}^n_k if and only if X = U \cong \mathbb{A}^n_k. In addition, for the consistency of our discussion, we raise some serious questions and some comments concerning the examples given by the certain mathematicians. The existence of such examples would be against our original target Conjecture (DJC). Our conclusion is that they are not perfect as shown explicitly in Section 6.