Homological Properties of Commutative Rings and Applications

在这个项目中，交换代数中的几个问题将被研究。它们是关于环和模的同调性质的具体陈述，但与其他数学领域有着意想不到的有趣联系。该项目的一部分提出研究诺特环上Ext和Tor函子的消失行为。更好地理解这种行为可以用来攻击一组不同的问题，如加伯猜想皮卡德群的穿孔谱的完整的交叉，或存在的非交换决议的奇异性。其他主题包括分类的子类别的最大科恩-麦考利模块，解决方程的半环向量丛代数品种和规律性的斯坦利-Reisner理想与单纯复形。交换代数涉及数学中一些最普遍的对象，例如多变量方程组。它与代数和算术几何、数论、表示论和组合学等领域有着自然而深刻的联系。这样的特点使人们能够在现代研究的前沿提出许多有趣的问题，即使是早期的研究生也能理解。该项目提出的许多课题都是关于这种联系的，它们的调查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.