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: 1522828
• 负责人: Bhargav Bhatt
• 金额: $ 3.84万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2016-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522828&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建议研究算术几何（特别是p-adic方面），代数几何和交换代数的共享边界问题。例如，梅尔·霍克斯特的直和项猜想，它假定了正则环的一些基本性质的存在，在过去的四十年里一直是交换代数中一个重要的开放问题;相等特征情形的解决方案（由于霍克斯特从70年代开始）负责大量的现代交换代数，而p-adic情形仍然是诱人的开放。PI最近发现，p-adic Hodge理论（由于Faltings）的一些想法可以用来解决这个猜想的某些未知情况。PI打算进一步追求这一方向，通过使用强大的最近的技术-主要是Scholze的美丽理论perfectoid空间-从快速发展的主题p进霍奇理论，以接近直接和项猜想和其他纯粹的代数问题。相反，以前的工作PI的直接和项猜想，再加上一些派生的代数几何，最近已被证明有助于达到一个显着的简化某些几何方面的p进霍奇理论; PI计划发展派生方面更彻底地概念化图片更好。研究解决方案的多项式与整数系数可以追溯到古代。这里一个非常有用的技术是首先研究“近似”解，即，解模素数，然后模素数的幂。这个想法是，随着素数的幂的增加，近似变得更好。Grothendieck在后半个世纪的数学革命不仅使人们不仅可以为前面的陈述赋予精确的含义，而且还提供了一个美丽的几何背景--p-adic几何的世界--来研究这种近似解。这一背景一直是数学中许多最新进展的核心（例如怀尔斯对费马大定理的证明和朗兰兹纲领中的其他里程碑）。PI计划进一步为基础几何理论做出贡献，并开发纯代数问题的应用。