Free Resolutions and Representation Theory

本研究计划涉及代数的两个分支：交换代数和颤振的表示。交换代数研究由多项式方程定义的集合。这部分的建议涉及到研究多项式方程来定义某些几何特征集。抖动的表示是一种将矢量数据与一些有向图的顶点相关联的方法。图的边可以看作是这些数据之间的关系。抽象代数允许系统地研究这些对象。这项研究的结果可能会导致处理线性代数问题的更好的算法，这在其他数学领域以及其他科学中有许多应用。该研究员发表的一些研究导致了这样的算法。该项目将包括研究生参与他的研究。这个项目由几个相互关联的部分组成。第一部分是研究有限自由分辨率下的一般环的结构。研究者建议发展他的发现，在长度为3的一般环和与图T（p,q,r）相关的Kac-Moody李代数之间的联系。特别地，当且仅当T（p,q,r）是Dynkin图时，泛环是noether环。PI建议在泛型环是诺etherian的情况下，对泛型环的生成器进行明确的描述。他还建议继续将这种方法推广到长度为3的完美复合体，对于余维数为4的Gorenstein理想的分辨率和更高长度的分辨率。在一个相关的项目中，PI提议研究反映图T（p,q,r）对称性的Buchsbaum-Rim连杆的作用。第二部分是局部上同的计算。PI计划发展他在行列式理想支持的一般矩阵上多项式环的局部上同调的计算。他计划在偏对称矩阵的pfaffian和对称矩阵的minor中找到类似的结果。然后，他计划开发用于在所有三种情况下找到不可约等变理想的自由分辨率的技术。第三部分涉及非交换代数。研究者提出进一步研究震颤的潜力。他打算继续研究David Berenstein关于具有势的颤振的雅可比代数的整体维数的猜想。他还计划研究雅可比代数的性质与相应的完备雅可比代数之间的关系。PI提出研究I型Vinberg表示中轨道闭包的非交换解析，特别是对于偏对称矩阵的pfaffian和对称矩阵的子矩阵。最后，他提出研究带关系颤振的表示空间的分量几何。最有趣的问题是试图决定给定代数的表示类型到其表示空间的几何形状。在这种情况下，PI感兴趣的是这些组件的正态性以及它们的MF和DO属性。

This research project is related to two branches of algebra: commutative algebra and representations of quivers. Commutative algebra studies sets defined by polynomial equations. This part of the proposal involves studying polynomial equations defining certain sets characterized geometrically. A representation of a quiver is a way to associate vector data to the vertices of some oriented graph. The edges of a graph can be viewed as relations between these data. Abstract algebra allows the study of such objects systematically. The results of this research might lead to better algorithms dealing with linear algebra problems, which have many applications in other areas of mathematics as well as in other sciences. Some of the published research of the investigator led to such algorithms. The PI will involve graduate students in his research.This project consists of several interrelated parts. The first part is to study the structure of generic rings for finite free resolutions. The investigator proposes to develop his discovery of a link between the generic ring for resolutions of length 3 and Kac-Moody Lie algebras related to graphs T(p,q,r). In particular the generic ring is Noetherian if and only if T(p,q,r) is a Dynkin diagram. The PI proposes to work towards an explicit description of the generators of the generic ring in the case when they are Noetherian. He also proposes to continue to generalize this approach to perfect complexes of length 3, for resolutions of Gorenstein ideals of codimension 4 and for resolutions of higher length. In a related project the PI proposes to study the role of Buchsbaum-Rim linkage which reflects the symmetry of the graphs T(p,q,r). The second part is related to calculating local cohomology. The PI plans to develop his calculation of local cohomology of the polynomial ring on generic matrix supported in the determinantal ideal. He plans to find similar results for Pfaffians of skew symmetric matrices and minors of symmetric matrices. He then plans to develop the techniques used to be able to find the free resolutions of irreducible equivariant ideals in all three cases. The third part involves noncommutative algebra. The investigator proposes to further study quivers with potential. He plans to pursue the conjecture of David Berenstein concerning global dimension of the Jacobian algebra of a quiver with potential. He also plans to study the relation between properties of a Jacobian algebra and the corresponding completed Jacobian algebra. The PI proposes to study the noncommutative resolutions of orbit closures in Vinberg representations of type I, in particular for Pfaffians of skew symmetric matrices and minors of symmetric matrices. Finally, he proposes to study the geometry of the components of representation spaces for quivers with relations. The most interesting problem is trying to decide the representation type of a given algebra to the geometry of its representation spaces. In this context the PI is interested in normality of these components as well as in their MF and DO properties.