RUI: Representation theory and homological algebra over local rings

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI将把她的注意力集中在局部noether环的特征上，根据模块的某些子类别的行为，将她的研究扩展到同调代数和表示理论的方法。这个想法源于对有限Cohen-Macaulay型环的成功描述，这种描述在20世纪80年代中期非常流行。有限维代数的结果推广到高维环。这种描述最近被推广到具有有限Gorenstein型的环。π吗?S的主要目标是了解当有限条件可用时，哪些类别的模块具有提供环描述的正确属性。为此，作者提出进一步研究全自反模范畴的性质与基环范畴的性质之间的关系。例如，PI建议理解长期存在的开放问题，以表征环力的哪些属性每个全自反模块是自由的。由于全自反模来自无限复合体，研究自然包括同调代数的方法，其中模的性质通过可能的无限近似线性化。该项目致力于研究交换代数中的问题，主要目标是理解多变量多项式方程的解集。为了描述这样的一组解，代数学家研究了当视觉直觉不可用时在解集上定义的函数集。由于函数可以相加和相乘，所以函数集具有交换环的结构，其中最常见的例子就是多项式环。众所周知，理解环就等于理解环的模的范畴，模是域上向量空间的泛化。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI will concentrate her attention on characterizing local Noetherian rings in terms of the behavior of certain subcategories of modules, extending her research to methods from homological algebra and representation theory. The idea stems from a successful description of rings of finite Cohen-Macaulay type, which flourished in the mid 1980s. Results about finite-dimensional algebras are generalized to rings of higher dimension. Such a description was recently extended to rings with a finite Gorenstein type. The PI?s main goal is to understand which categories of modules have the right properties to provide a description of the ring when finiteness conditions are available. To do so, the PI proposes to investigate further the relationship between the properties of the category of totally reflexive modules and those of the base ring. For example, the PI proposes to understand the long standing open problem to characterize which properties of the ring force every totally reflexive module to be free. As totally reflexive modules come from infinite complexes, the investigation naturally includes methods from homological algebra, where the properties of modules are linearized by a possibly infinite approximation.The project is devoted to investigating problems in commutative algebra with the main goal being to understand the set of solutions of polynomial equations in many variables. In the effort to describe such a set of solutions, algebraists study the set of functions defined on the set of solution when a visual intuition is not available. As functions can be added and multiplied, the set of functions has the structure of a commutative ring, of which the most familiar example is the ring of polynomials. It is well established that understanding a ring is tantamount to understanding the category of its modules, which are generalization of vector spaces over fields.