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

在1996年，我们引入了代数闭域K上的仿射代数和仿射簇的新理论。这个理论现在已经彻底审查和重建根据格罗滕迪克理论的代数计划，虽然我们仍然是在一个域K。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，自然地称之为ind-affine概型。在每一个ind-仿射概型上建立一个亲仿射代数层，适当定义其茎是一个亲仿射局部环。这些结果已被写在“关于Pro-Affine代数和Ind-Affine概型的一些基本定理”一文中，这篇文章将在不久的将来发表。作为副产品，我们得到了仿射空间的自同构群是ind-affine的一个新的证明，并证明了仿射簇的态射集是ind-affine的。这些结果已经被写进了一篇论文，“作为ind-affine schemes的仿射变体的态射”，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.