On Unramified Morphisms of Affine Varieties into Simply Connected Non-singular Affine Varieties

The Jacobian Conjecture is the following : If φ ∈ Endk(Ank ) with a field k of characteristic zero is unramified, then φ is an automorphism. In this paper, This conjecture is proved affirmatively in the abstract way instead of treating variables in a polynomial ring. Let k be an algebraically closed field, let Ak = Max(k[X1, . . . , Xn]) be an affine space of dimension n over k and let f : Ak −→ A n k be a morphism of affine spaces over k of dimension n. Then f is given by A n k ∋ (x1, . . . , xn) 7→ (f1(x1, . . . , xn), . . . , fn(x1, . . . , xn)) ∈ A n k . where fi(X1, . . . , Xn) ∈ k[X1, . . . , Xn]. If f has an inverse morphism, then the Jacobian det(∂fi/∂Xj) is a nonzero constant. This follows from the easy chain rule. The Jacobian Conjecture asserts the converse. If k is of characteristic p > 0 and f(X) = X +X, then df/dX = f (X) = 1 but X can not be expressed as a polynomial in f . Thus we must assume the characteristic of k is zero. The algebraic form of the Jacobian Conjecture is the following : 2000 Mathematics Subject Classification : Primary 13B25, Secondary 13M10