Cohomology theories for algebraic varieties

代数簇的上同调理论

• 批准号: 2883661
• 金额: --
• 依托单位: University of Nottingham
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2883661
• 关键词: Cohomology; theories; algebraic; varieties

After the ground breaking works of V. Voevodsky, it became possible to work with algebraic varieties by completely topological methods. An important role in this context is played by the so-called Generalized Cohomology Theories. This includes classical algebraic K-theory, but also a rather modern (and more universal) Algebraic Cobordism theory. The study of such theories and cohomological operations on them is a fascinating subject. It has many applications to the classical questions from algebraic geometry, quadratic form theory, and other areas. One can mention, for example: the Rost degree formula, the problem of smoothing algebraic cycles, and u-invariants of fields. This is a new and rapidly developing area that offers many promising directions of research.

在 V. Voevodsky 的开创性工作之后，通过完全拓扑方法来处理代数簇成为可能。所谓的广义上同调理论在这方面发挥着重要作用。这包括经典的代数 K 理论，但也包括相当现代（且更普遍）的代数协调理论。对此类理论及其上同调运算的研究是一个令人着迷的课题。它在代数几何、二次型理论和其他领域的经典问题中有许多应用。例如，可以提到：罗斯特度公式、平滑代数环问题以及域的 u 不变量。这是一个快速发展的新领域，提供了许多有前途的研究方向。