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

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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.