Applications of Homological Algebra in Algebra, Geometry, and Physics

同调代数在代数、几何和物理中的应用

• 批准号: RGPIN-2017-06572
• 负责人: Buchweitz, RagnarOlaf
• 金额: $ 1.93万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=646469
• 关键词: Applications; Homological; Algebra; Geometry; Physics

I propose to work on the following topics.***(1) Construction and Classification of (graded) Matrix Factorizations and Maximal Cohen-Macaulay Modules over Gorenstein Rings. These objects from commutative algebra are now exploited in singularity theory and string theory with tools from (noncommutative) algebraic geometry, and, more generally, homological algebra.***(a) With Iyama (Nagoya) and Yamaura (Yamanuishi) we will describe how to construct all graded maximal Cohen-Macaulay modules over reduced commutative graded Gorenstein curve singularities.***(b) With my former student Alexander Pavlov (Madison) we will write down all matrix factorizations of smooth plane cubics that have linear entries, corresponding to so-called Ulrich bundles. Based on the determination in Pavlov's thesis of all possible graded Betti numbers of indecomposable graded maximal Cohen-Macaulay modules on homogeneous coordinate rings of elliptic curves, we will then determine all graded matrix factorizations in this case.***(2) Properties and Role of Hochschild-Tate Cohomology in Algebra and Geometry.***This is the appropriate version of Hochschild cohomology for the stable category of maximal Cohen-Macaulay Modules. I collaborate with some of my current students to study the Hochschild(-Tate) cohomology of cohomology rings of flag varieties, essentially equal to the classical homology of the free loop space over these manifolds.***(3) Representation Theory of Algebras and Non-commutative Desingularizations.***With Hille (Muenster) we study higher representation-infinite algebras that arise from (weakly) Fano varieties and with him and Iyama we investigate tilting and cluster tilting for projective varieties. I will also continue to study tilting theory on determinantal and other varieties; Castelnuovo-Mumford regularity for varying t-structures; Maximal Cohen-Macaulay endomorphism rings; Rigidity of maximal Cohen-Macaulay modules.***(4) McKay Correspondence for Reflection Groups.***In collaboration with Eleonore Faber (Ann Arbor/Leeds) and Ingalls (UNB) we will extend the classical McKay correspondence for finite subgroups of SL(2,C) to finite reflection subgroups of GL(n,K), thereby obtaining noncommutative desingularizations of the highly singular discriminants of these group actions. This relates intimately to the next point:***(5) Continued Study of Free Divisors and Discriminants.***These hypersurfaces are singular in codimension one, but with highly structured singular locus. I am particularly interested in rank one maximal Cohen-Macaulay modules on such hypersurfaces as those provide compact determinantal expressions of their equations.******This research requires expertise in (Homological) Algebra, Algebraic and Complex Geometry, and in***Representation Theory, primarily applying and investigating homological methods.

我建议就以下议题开展工作。***(1) Gorenstein环上（分级）矩阵分解和极大Cohen-Macaulay模的构造与分类。这些来自交换代数的对象现在在奇点理论和弦理论中利用（非交换）代数几何的工具，更普遍的是，同调代数。***(a)利用Iyama （Nagoya）和Yamaura (Yamanuishi)，我们将描述如何在约简交换梯度Gorenstein曲线奇点上构造所有梯度极大Cohen-Macaulay模。***(b)和我以前的学生亚历山大·巴甫洛夫（麦迪逊）一起，我们将写下所有光滑平面立方体的矩阵分解，它们有线性项，对应于所谓的乌尔里希束。基于巴甫洛夫理论中对椭圆曲线齐次坐标环上不可分解的梯度极大Cohen-Macaulay模的所有可能的梯度Betti数的确定，我们将在此情况下确定所有的梯度矩阵分解。***(2) Hochschild-Tate上同调在代数和几何中的性质和作用。***这是极大Cohen-Macaulay模稳定范畴的Hochschild上同的适当版本。我和我现在的一些学生合作研究旗型上同调环的Hochschild（-Tate）上同调，本质上等于这些流形上自由环空间的经典同调。***(3)代数的表示理论与非交换非量子化。***我们与Hille （Muenster）一起研究了由（弱）Fano变异体产生的高表示无限代数，与他和Iyama一起研究了射影变异体的倾斜和簇倾斜。我还将继续研究关于行列式和其他品种的倾斜理论；变t型结构的Castelnuovo-Mumford正则性极大Cohen-Macaulay自同态环最大Cohen-Macaulay模的刚度。***(4)反思组的McKay对应。***我们与Eleonore Faber （Ann Arbor/Leeds）和Ingalls （UNB）合作，将SL（2，C）的有限子群的经典McKay对应推广到GL（n，K）的有限反射子群，从而得到了这些群作用的高度奇异判别的非交换去量子化。这与下一点密切相关：***(5)自由除数和判别式的继续研究。***这些超曲面在余维1上是奇异的，但具有高度结构化的奇异轨迹。我特别感兴趣的是在这些超曲面上的秩一极大Cohen-Macaulay模，因为它们提供了它们的方程的紧凑行列式表达式。******本研究需要（同调）代数、代数和复几何以及***表示理论方面的专业知识，主要应用和研究同调方法。