From the theory of pro-affine algebras and ind-affine varieties to the Jacobian Problem

1996年，我们引入了一个新的关于代数闭地域K上的仿射代数和仿射变元的理论，这个理论现在已经按照Grothendieckian的代数方案理论进行了彻底的回顾和重建，尽管我们的理论仍然在域K上。我们发展了仿射代数的理想理论和局部化理论，并构造了这类代数的对偶对象，自然地称为仿射方案。在每个内仿射方案上构造一束仿射代数，适当地定义它的茎将是一个仿射局部环。这些结果已经被写在一篇论文《关于Pro-仿射代数和IND-仿射方案的一些基本定理》中，这篇论文将在不久的将来提交出版。作为副产品，我们得到了仿射空间的自同构群是内仿射的新证明，并证明了仿射簇的态射集是内仿射的。这些结果已经被写进了一篇论文《仿射簇的态射作为内仿射方案》，这篇论文将在最终编辑后发表。

In 1996 we introduced a new theory of pro-affine algebras and ind-affine varieties over an algebraically closed ground field K. This theory has now been thoroughly reviewed and rebuilt in accordance with Grothendieckian theory of algebraic schemes, albeit ours being still over a field K. We have developped, amoung other things, ideal theory and localization theory for pro-affine algebras, and have constructed the dual objects of this type of algebras, which are naturally called ind-affine schemes. A sheaf of pro-affine algebras are built over each ind-affine schemes, and its stalk appropriately defined will be a pro-affine local ring. These results have been written up in a paper, "Some Basic Theorems on Pro-Affine Algebras and Ind-Affine Schemes," and this paper will be submitted for pubication in the near future. As a by-product we have obtained a new proof of the fact that the automorphism group of an affine space is ind-affine, and we also proved that he set of morphisms of affine varieties is ind-affine. These results have been written into a paper, "Morphisms of affine varieties as ind-affine schemes," and this one will be published after the final editing.