POWRE: Excellent Local Rings

POWRE：优秀的本地环

• 批准号: 9973069
• 负责人: Susan Loepp
• 金额: $ 4.88万
• 依托单位: Williams College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-09-01 至 2000-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9973069&HistoricalAwards=false
• 关键词: POWRE; Excellent; Local; Rings

The PI will study the relationship between excellent local rings and their completions. It has long been known that the formal fibers of a ring encode important information about the ring. Because of this, understanding the formal fiber over the zero ideal (called the generic formal fiber) of a local integral domain is critical. The PI will work to explore the relationship between a ring and its generic formal fiber. She will study the relationship between a ring R and R[X] (where X is an indeterminate) by examining the generic formal fiberof R and R[X]. She will work on the problem of characterizing when a ring satisfies the complete homomorphic images property, and will also study a question relating to the localization problem for tight closure. In addition, she will explore several rigidity questions.The project will take place at Michigan State University during the 1999-2000 academic year. The PI will spend the year collaborating with Christel Rotthaus, one of the world's leading experts in excellent rings. The PI will attend commutative algebra seminars and discuss overlapping research interests with the resident algebraists and visiting algebraists at Michigan State. This will aid the PI in broadening her research focus and learning new research skills. In addition, the PI will brainstorm with these algebraists to discover new problems in the field that will be suitable for her undergraduate students to study. The ideas coming fromthis collaboration will fuel undergraduate research projects for years to come.The PI for this project is currently in her third year as an Assistant Professor at Williams College. She has previously advised the research of four undergraduate students and is currently advising the research of another. She is dedicated to contributing to the advancement of undergraduate research in mathematics by continuing along and broadening her own research path and advising the original research of undergraduates.The PI will devote a major portion of this project to broadening the focus of her research and the focus of the problems on which her undergraduate students can work. A one-time input of funds at this critical stage will serve to greatly advance her career as a mathematician at a highly selective liberal arts college.

PI将研究优秀局部环与它们的完备性之间的关系。人们早就知道，环的形式纤维编码了关于环的重要信息。正因为如此，理解局部积分域的零理想上的形式纤维(称为一般形式纤维)是至关重要的。PI将致力于探索环与其普通形式纤维之间的关系。她将通过考察R和R[X]的一般形式纤维来研究环R和R[X](其中X是不定的)之间的关系。她将致力于刻画环何时满足完全同态映象性质的问题，并将研究与紧闭集的局部化问题有关的问题。此外，她还将探讨几个僵化问题。该项目将于1999-2000学年在密歇根州立大学进行。PI将在今年与克里斯特尔·罗特豪斯合作，克里斯特尔·罗特乌斯是世界领先的优秀吊环专家之一。PI将参加交换代数研讨会，并与密歇根州立大学的常驻代数学家和访问代数学家讨论重叠的研究兴趣。这将有助于PI拓宽她的研究范围和学习新的研究技能。此外，PI将与这些代数学家集思广益，发现该领域的新问题，适合她的本科生学习。来自这一合作的想法将在未来几年推动本科生的研究项目。这个项目的PI目前是她在威廉姆斯学院担任助理教授的第三年。她之前曾为四名本科生的研究提供建议，目前正在为另一名本科生的研究提供建议。她致力于通过继续和拓宽自己的研究道路，并为本科生的原创研究提供建议，为促进本科生的数学研究做出贡献。PI将把这个项目的主要部分投入到拓宽她的研究重点和她的本科生可以研究的问题的重点上。在这个关键阶段一次性投入资金，将极大地促进她在一所高度挑剔的文理学院担任数学家的职业生涯。