Interactions between p-adic arithmetic geometry and commutative algebra

p进算术几何与交换代数之间的相互作用

• 批准号: 1340424
• 负责人: Bhargav Bhatt
• 金额: $ 7.43万
• 依托单位: Institute For Advanced Study
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-03-01 至 2015-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1340424&HistoricalAwards=false
• 关键词: Interactions; between; adic; arithmetic; geometry

The PI proposes to investigate problems lying at the shared boundaries of arithmetic geometry (especially p-adic aspects), algebraic geometry, and commutative algebra. For example, Mel Hochster's direct summand conjecture, which posits the existence of some fundamental properties of regular rings, has been an important open problem in commutative algebra for over four decades; the solution to the equal characteristic case (due to Hochster from the 70s) is responsible for large swathes of modern commutative algebra, while the p-adic case remains tantalizingly open. The PI recently discovered that some ideas from p-adic Hodge theory (due to Faltings) can be used solve certain unknown cases of this conjecture. The PI intends to pursue this direction further by using powerful recent techniques --- chiefly Scholze's beautiful theory of perfectoid spaces --- from the fast evolving subject of p-adic Hodge theory to approach the direct summand conjecture and other purely algebraic problems. Conversely, previous work of the PI on the direct summand conjecture, coupled with some derived algebraic geometry, has recently proven instrumental in arriving at a significant simplification of certain geometric aspects of p-adic Hodge theory; the PI plans to develop the derived aspects more thoroughly to conceptualize the picture better.The study of solutions of polynomials with integer coefficients dates back to antiquity. An extremely useful technique here is to study "approximate" solutions first, i.e., solutions modulo primes, and then modulo powers of primes. The idea is that as the power of prime increases, the approximation becomes better. Grothendieck's revolutionization of mathematics in the last half century not only allows one to not only attach a precise meaning to the previous statement, but also provides a beautiful geometric context --- the world of p-adic geometry ---- to study such approximate solutions. This context has been at the heart of numerous recent advances in mathematics (such as Wiles' proof of Fermat's last theorem and other recent milestones in the Langlands program). The PI plans to contribute further to underlying geometric theory as well as develop applications to purely algebraic problems.

PI提议调查位于算术几何（尤其是P-Adic方面），代数几何形状和交换代数的共同边界的问题。例如，梅尔·霍奇斯特（Mel Hochster）的直接求和和猜想认为存在常规环的某些基本属性，在交换代数中一直是一个重要的开放问题。相等特征案例的解决方案（由于70年代的Hochster）负责大量现代交换代数，而P-ADIC案例仍然诱人地开放。 PI最近发现，可以使用P-Adic Hodge理论（由于疲惫）的某些想法解决了该猜想的某些未知案例。 PI打算通过使用强大的最新技术来进一步追求这一方向 - 主要是Scholze的完美体形空间的美丽理论 - 从P-Adic Hodge理论的快速发展的主题来处理直接的总结和其他纯粹的代数问题。相反，PI在直接求和猜想中的先前工作，再加上一些衍生的代数几何形状，最近证明了对P-Adic Hodge理论的某些几何方面的显着简化的作用。 PI计划更彻底地开发出派生的方面，以更好地概念化图片。对整数系数的多项式解决方案的研究可以追溯到古代。这里非常有用的技术是首先研究“近似”解决方案，即溶液模型素，然后是素数的prime。这个想法是，随着素数的增加，近似值变得更好。 Grothendieck在过去的半个世纪中对数学的革命性不仅允许人们不仅在上一个陈述中附上精确的含义，而且还提供了美丽的几何环境，即P-Adic几何学的世界 - 研究这种近似解决方案。这种背景是数学最新进展的核心（例如，威尔斯的《 Fermat的最后一个定理》和Langlands计划中其他最新里程碑的证明）。 PI计划进一步为基本的几何理论做出贡献，并为纯粹的代数问题开发应用。