Homological and positive characteristic questions in commutative algebra

这个项目涉及四个不同的方向。第一个方向是研究正特征中的弱Lefschetz性质，从PI的观点来看，这与寻找某些多项式上的最小可能的关联度有关。在正特征中，Frobenius自同态在生成出乎意料的小度关系中起着重要作用。本课题的第二个方向是研究某些不存在非自由全自反模的环类。在考虑的环中，我们提到了几乎是Gorenstein的环(在Huneke和PI在2003年定义的意义上)，以及行列式环。PI希望研究由于非自由全自反模的存在而对环施加的结构限制(例如希尔伯特函数)。第三个方向是使用我们在第一部分中发展的关系度的知识来计算希尔伯特-昆兹重数。这个项目的第四部分是试图解决Conca、Krtenthaler和Watanabe关于何时是由变量的幂和组成的某一集的猜想。PI工作在交换代数中，这是一个主要研究环中的模和理想的代数分支。所考虑的大多数对象来自代数几何，对应于多项式方程的解的集合。同调交换代数涉及找到这些解之间的关系，并通过找到关系上的关系来迭代这个过程，从而产生可能无限的解。PI在她的大部分作品中也使用了积极的特色方法。在正特征设置下，环被赋予一种称为Frobenius同态的附加结构。PI的工作利用了这种额外的结构，以表明它产生了有趣的新现象，这些现象在它不存在的情况下不会发生。

This project involves four separate directions. The first direction is an investigation of the weak Lefschetz property in positive characteristic, which from the PI's point of view is related to finding the smallest possible degrees of relations on certain polynomials. In positive characteristic, the Frobenius endomorphism plays an important role in producing relations of unexpectedly small degree.The second direction of the project is the study of certain classes of rings for which there are no non-free totally reflexive modules. Among the rings under consideration we mention rings which are almost Gorenstein (in the sense defined by Huneke and the PI in 2003), and determinantal rings. The PI wishes to investigate structural restrictions that are imposed on the ring (such as the Hilbert function) by the existence of a non-free totally reflexive module. The third direction is towards using our knowledge of degrees of relations developed in the first part in order to compute Hilbert-Kunz multiplicities. The fourth part of the project is an attempt to solve a conjecture of Conca, Krattenthaler, and Watanabe regarding when is a certain set consisting of sums of powers of the variables a system of parameters in the polynomial ring with three variables.The PI works in commutative algebra, a branch of algebra mainly concerned with the study of modules and ideals in a ring. Most of the objects under consideration arise from algebraic geometry, corresponding to sets of solutions of polynomial equations. Homological commutative algebra has to do with finding relations between these solutions, and iterating the process by finding relations on relations, etc., thus giving rise to possibly infinite resolutions. The PI also uses positive characteristic methods in most of her work. In a positive characteristic setting, the ring is endowed with an additional structure called the Frobenius homomorphism. The PI's work takes advantage of this additional structure to show that it gives rise to interesting new phenomena that do not occur in its absence.