Pro-affine Algebras, Ind-affine Groups and the Jacobian Problem

Abstract

The purpose of the paper at hand is first to develop a theory of pro-affine algebras and their dual objects, ind-affine varieties and groups, and second to apply the theory to the well-known Jacobian problem, which asks whether or not a polynomial endomorphism of the complex affinen-space with nowhere vanishing Jacobian determinant is in fact an automorphism (cf. [3], for instance). Our main result, Theorem 5.3, says the affirmative answer obtains if the natural map from the automorphisms to the endomorphisms as above is a closed immersion of ind-affine varieties.Much earlier, I. R. Shafarevich in [7, 8] introduced the notion of “infinite-dimensional varieties and groups” and investigated the structure of the automorphism group of the affinen-space. Those are the ind-affine varieties and groups in our terminology. While we owe much to Shafarevich's two papers just cited, we have found some minor flaws in his theory as discussed in the main text below (also cf. [4]). More importantly, since Shafarevich's starting point is ind-affine varieties, his rings of functions are without nonzero nilpotent elements to begin with. We have found this treatment inadequate for our purpose of application to the Jacobian problem, because certain ideals naturally associated with endomorphisms with Jacobian determinant 1 and with automorphisms arenotknown to agree with their respective radicals. Indeed, the premises of our main result, Theorem 5.3, are satisfied and the affirmative answer to the Jacobian problem follows,oncethe ideals associated with automorphisms are known to be radical ideals. (See (5.4) below, as well as [2].)So, we have undertaken to build our theory starting from the pro-affine algebras. This was, in a sense, already taken up by Abe and Takeuchi [1] recently, but we had to redo it our way because their purpose and viewpoints much differ from ours. The theory we present herein is still rudimentary, and it is hoped that a further development of the theory will ultimately settle the Jacobian problem.The notion of linearly compact rings and modules, as in the 1953 paper [9], is used in essential ways in our paper. For apparent lack of recent references, we have added a short Appendix dealing with the subject.