Topics in commutative algebra

任何分级的，有限生成的模块提供最小的分级分辨率，这些分辨率提供了关于原始模块的丰富信息。其中一条信息是模块的贝蒂数；这些是最小自由分辨率下的自由发生器的度数。交换代数中最近最令人兴奋的发展之一是Boij-Soderberg理论的出现，该理论描述了由多项式环上Cohen-Macaulay模的所有Betti数生成的凸锥，特别是它描述了该锥的极值射线以及将任意给定的Betti图表示为这些射线上点的凸组合的方法。David Carey的项目研究了由方形自由单项式理想的Betti图生成的子锥，特别是边缘理想。该项目的目的是利用这些理想的组合结构来获得对betti锥的理解，例如，根据底层图的性质对边缘理想的betti锥中的极值射线进行分类。

BETTI NUMBERS STANLEY REISNER IDEALSAny graded, finitely generated module affords a minimal graded resolution, and these resolutions provide a wealth of information about the original module. One such piece of information are the Betti numbers of the module; these are the degrees of the free generators in a minimal free resolution.One of the most exciting recent developments in commutative algebra is the emergence of the Boij-Soderberg Theory which describes the convex cone generated by all the Betti numbers of Cohen-Macaulay modules over a polynomial ring, in particular it describes the extremal rays of this cone and ways to express any given betti diagram as a convex combination of points on these rays.David Carey's project studies the subcone generated by Betti diagrams of square free monomial ideals in general, and edge ideals in particular. The aim of the project is to exploit the combinatorial structure of these ideals to gain an understanding of the betti cones, e.g., the classification of extremal rays in the Betti cones of edge ideals in terms of the properties of the underlying graphs.