On The Jacobian Conjecture and Open Embeddings of Affine Spaces in Affine Varieties
Susumu Oda
TL;DR
This work targets the Jacobian Conjecture $JC_n$ and its generalized form $DJC$ by developing a Krull-domain framework that controls unramified maps via subintersections and unit-group data. A key contribution is a core theorem for Krull domains ensuring equality of subintersections under flatness and unit-group constraints, from which the $DJC$ and hence all $JC_n$ are resolved under the stated hypotheses. It further derives a geometric criterion: an irreducible $k$-affine variety is a $k$-affine space $\mathbb{A}^n_k$ exactly when it contains such a space as an open subvariety, and extends JC-type statements to a base ring $A$, illustrating how these algebraic constraints interplay with open-embedding and affine-space questions. The paper also critically analyzes purported counterexamples in the literature, arguing they do not constitute valid refutations of $DJC$, and emphasizes several open questions about affineness and simple connectivity in this context.
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.