On Open Embeddings of Affine Spaces in Affine Varieties and the Jacobian Conjecture

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 core 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.