Moduli Spaces for Rings and Ideals

环和理想的模空间

• 批准号: 1147782
• 负责人: Melanie Wood
• 金额: $ 12.51万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-05-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1147782&HistoricalAwards=false
• 关键词: Moduli; Spaces; Rings; Ideals

The investigator studies moduli spaces of finite, flat covers and line bundles of those covers. Important basic examples include orders in number fields (finite flat covers of the integers) and finite covers of the complex projective line. The methods involve working over an arbitrary base scheme, so for example one gets results about both the number theoretic and geometric examples above. The project is to find moduli spaces that have explicit descriptions and reasonable geometry so that they can be worked with concretely. Since the work of Gauss in 1801, polynomials have been use to study number systems that are bigger than the usual counting numbers 1,2,3.., For example, a larger number system might also include the square root of 2, which cannot be found among 1,2,3... When we include the square root of 2, it is a quadratic extension of the usual numbers, and if we had included the cube root of 2 it would have been a cubic extension of the usual numbers. This work tries to understand what the possible low degree extensions of the usual numbers are by working explicitly with polynomials that are related to the extensions. This then allows one to make computations regarding the larger number systems by reducing them to easier computations about polynomials.

研究者研究了有限平坦覆盖的模空间和这些覆盖的线丛。重要的基本例子包括数域中的序(整数的有限平坦覆盖)和复射影直线的有限覆盖。这些方法涉及处理任意基方案，因此，例如，可以得到关于上述数论和几何示例的结果。这个项目是寻找具有明确描述和合理几何的模空间，以便它们可以被具体地处理。自从1801年Gauss的工作以来，多项式被用来研究比通常的计数数1，2，3更大的数系。例如，更大的数系可能还包括2的平方根，这是在1，2，3之间找不到的。当我们包括2的平方根时，它是通常数字的二次扩展，如果我们包括2的立方根，它将是通常数字的三次扩展。这项工作试图通过显式地使用与扩展相关的多项式来理解通常数的可能的低次扩展是什么。这使得人们可以通过将较大的数系简化为关于多项式的更容易的计算来进行关于较大数系的计算。