Homological Properties of Commutative Rings and Applications

在这个项目中，将研究交换代数中的几个问题。它们是关于环和模的同调性质的具体陈述，但与数学的其他领域有着意想不到和有趣的联系。该项目的一部分建议研究Notherian环上Ext和Tor函子的消失行为。对这种行为的更好的理解可以用来解决一系列不同的问题，例如加伯的S关于完全交叉点的穿孔谱的Picard群的猜想，或者奇点的非对易分解的存在。其他主题包括极大Cohen-Macaulay模的子范畴的分类，代数簇上向量丛半环上的方程的求解，以及与单纯复形相关的Stanley-Reisner理想的正则性。交换代数处理数学中一些最普遍的对象，例如具有多个变量的方程式组。它与代数和算术几何、数论、表示论和组合学等领域有着天然和深刻的联系。这些特点使人们能够在现代研究的前沿，以即使是早期研究生也能理解的形式，提出许多有趣的问题。本项目提出的许多课题都是关于这种联系的，他们的研究特别适合年轻研究人员之间的培训和合作。

In this project, several questions in commutative algebra will be studied. They are concrete statements about homological properties of rings and modules, but have unexpected and intriguing connections to other areas of mathematics. One part of the project proposes to investigate the vanishing behavior of Ext and Tor functors over Noetherian rings. Better understanding of such behavior can be used to attack a diverse set of problems such as Gabber 's conjecture about Picard groups of punctured spectra of complete intersections, or the existence of non-commutative resolution of singularities. Other topics include classification of subcategories of maximal Cohen-Macaulay modules, solving equations in the semi-ring of vector bundles on algebraic varieties and regularity of Stanley-Reisner ideals associated to simplicial complexes. Commutative algebra deals with some of the most ubiquitous objects in mathematics, such as sets of equations in several variables. It has both natural and deep connections to fields such as algebraic and arithmetic geometry, number theory, representation theory and combinatorics. Such characteristics allow one to phrase many interesting questions at the front of modern research in the form accessible to even early graduate students. Many topics proposed in this project are about such connections, and their investigation are especially suitable for training and collaboration between young researchers.