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

TL;DR

This paper addresses the Jacobian Conjecture by generalizing it to the Deep Jacobian Conjecture, proving a core theorem on Krull domains, and characterizing affine spaces within affine varieties.

Contribution

It introduces the Deep Jacobian Conjecture and proves a key theorem on Krull domains that leads to a characterization of affine spaces in affine varieties.

Findings

Proved the Deep Jacobian Conjecture under certain conditions.

Characterized when an affine variety contains an open subvariety isomorphic to affine space.

Established conditions under which a Krull domain equals a factorial subintersection.

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.