Analysis of the universal compactifying space

2014 Fiscal Year Final Research Report

• Project/Area Number: 23740017
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Osaka City University (2012-2014); Kyoto University (2011)
• Principal Investigator: TAKAGI Satoshi 大阪市立大学, 大学院理学研究科, 研究員 (20456841)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: 圏論 / 代数幾何学 / 数論

Outline of Final Research Achievements

We characterized the Zariski-Riemann space (which is treated in Algebraic geometry) as an analog of Stone-Cech compactification, which is treated in Topology. We also re-defined proper morphisms to reach this goal. Also, we succeeded in formularizing convex geometry in pure algebraic manner; this means that there exists an algebraic type \tau such that convex polytopes can be regarded as a \tau-algebra. Furthermore, applying this technique to arithmetics, we succeeded in formularizing arithmetic compactification of the spectrum of algebraic integer ring in a pure-algebraic manner. This is accomplished by considering the above \tau-algebras and the construction of the Zariski-Riemann space simultaneously.

Free Research Field

代数幾何学