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

1999 Fiscal Year Final Research Report Summary

• Project/Area Number: 09640067
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo Denki University
• Principal Investigator: KAMBAYASHI Tatsuji Tokyo Denki University (Department of Mathematical Sciences, Professor), 理工学部, 教授 (70169803)
• Co-Investigator(Kenkyū-buntansha): NAKANO Tetsuo Tokyo Denki University (Department of Mathematical Sciences, Assistant Professor), 理工学部, 助教授 (00217796)
• Project Period (FY): 1997 – 1999
• Keywords: pro-affine algebra / ind-affine scheme / Jacobian Problem / morphism set / automorphism

Research Abstract

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.