Commutative algebra: homological and homotopical aspects

交换代数：同调和同伦方面

• 批准号: 1503044
• 负责人: Srikanth Iyengar
• 金额: $ 25.87万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-31 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1503044&HistoricalAwards=false
• 关键词: Commutative; algebra; homological; homotopical; aspects

This is a proposal in commutative algebra from the point of view of cohomology and homotopy theory, with strong connections to representation theory of finite dimensional algebras. Many of the proposed topics of research are inspired by homotopy theory, which also serves as a source of intuition, and of techniques, for dealing with them. Broadly speaking, the recurrent theme in the project is the interplay of three topics: commutative algebra; differential graded algebras and modules; and triangulated categories. While differential graded algebras and triangulated category methods have long been used successfully in commutative algebra, the PI and his collaborators, among others, have been adapting classical constructions and methods from commutative algebra (for example, local cohomology, and derived completions) to the context of differential graded algebras and triangulated categories, with striking returns in the representation theory of finitely dimensional algebra, notably, group algebras, and in commutative algebra itself. This proposal seeks to further develop these, and in new directions.A number of rather diverse algebraic structures---this means, loosely speaking, not directly involving any notions from calculus like continuity or rate of change---have been developed in mathematics to model phenomenon in the (physical) world. Two examples particularly relevant to this proposal are groups and their representations, that are remarkably well-adapted to capture phenomenon involving symmetry, and rings, especially those that arise as rings of functions on various geometric objects like manifolds or solution sets of polynomial equations. However, it was only a few years ago that it was realized that if we enhance a ring by the most primitive structure from calculus, namely, a derivative, then it becomes possible to uniformly capture much of the information encoded in these various algebraic structures. Interestingly, these still rather mysterious hybrid structures, called differential graded algebras, emerged as important tools in algebraic topology already in the early 1950s. Differential graded algebras can be seen as bridges that relate various algebraic and geometric contexts in mathematics and mathematical physics. This has had the effect that methods developed in one field have profoundly influenced a host of others, and new connections among them have been discovered. Differential graded algebras are, in general, rather complicated, but there are interesting classes that appear to be amenable to methods developed in commutative ring theory, a classical topic with a large and well-developed body of tools and techniques. The broad aim of this proposal is to investigate differential graded algebras from this perspective. Besides deepening our knowledge of them, the proposed research is expected to have impact on commutative algebra, representation theory, and related fields.

这是上同调和同伦理论在交换代数中的一个建议，它与有限维代数的表示理论有很强的联系。许多被提出的研究主题都受到同伦理论的启发，同伦理论也是直觉和处理这些问题的技术的来源。一般说来，项目中反复出现的主题是三个主题的相互作用：交换代数；微分分次代数和模；以及三角范畴。虽然微分分次代数和三角范畴方法早已成功地应用于交换代数中，但PI和他的合作者们一直在将交换代数(例如局部上同调和导出补全)的经典构造和方法应用于微分分次代数和三角范畴的背景下，在有限维代数的表示理论中取得了惊人的成果，特别是在群代数中，以及在交换代数本身中。这一建议寻求进一步发展这些，并朝着新的方向。许多相当不同的代数结构-宽泛地说，这意味着，不直接涉及微积分中的任何概念，如连续性或变化率--已经在数学中发展起来，以模拟(物理)世界中的现象。与这一建议特别相关的两个例子是群及其表示，它们非常适合于捕捉涉及对称的现象，以及环，特别是那些出现在各种几何对象上的函数环，如流形或多项式方程的解集。然而，直到几年前，人们才意识到，如果我们用微积分中最原始的结构，即导数来增强环，那么就有可能统一地捕获在这些不同的代数结构中编码的大部分信息。有趣的是，这些仍然相当神秘的混合结构，称为微分分次代数，早在20世纪50年代初就作为代数拓扑学的重要工具出现了。微分分次代数可以看作是联系数学和数学物理中各种代数和几何背景的桥梁。这使得在一个领域开发的方法深刻地影响了许多其他领域的方法，并发现了它们之间的新联系。一般来说，微分分次代数是相当复杂的，但有一些有趣的类似乎服从于交换环理论中发展的方法，这是一个拥有大量且发展良好的工具和技术的经典主题。这个建议的主要目的就是从这个角度来研究可微分次代数。除了加深我们对它们的认识外，这项研究还有望对交换代数、表示理论及相关领域产生影响。

项目成果

Absolutely Koszul Algebras and the Backelin-Roos Property

• DOI: 10.1007/s40306-015-0125-0
• 发表时间: 2015-03
• 期刊: Acta Mathematica Vietnamica
• 影响因子: 0.5
• 作者: A. Conca;S. Iyengar;H. D. Nguyen;Tim Römer

Homology over trivial extensions of commutative DG algebras

交换 DG 代数的平凡扩展上的同调

• DOI: 10.1080/00927872.2018.1472272
• 发表时间: 2019
• 期刊: Communications in Algebra
• 影响因子: 0.7
• 作者: Avramov, Luchezar L.;Iyengar, Srikanth B.;Nasseh, Saeed;Sather-Wagstaff, Sean
• 通讯作者: Sather-Wagstaff, Sean

Torsion in tensor powers of modules

• DOI: 10.1215/00277630-3140827
• 发表时间: 2014-12
• 期刊: Nagoya Mathematical Journal
• 影响因子: 0.8
• 作者: Olgur Celikbas;S. Iyengar;Greg Piepmeyer;R. Wiegand

Stratification for module categories of finite group schemes

有限群方案模类别的分层

• DOI: 10.1090/jams/887
• 发表时间: 2018
• 期刊: Journal of the American Mathematical Society
• 影响因子: 3.9
• 作者: Benson, Dave;Iyengar, Srikanth B.;Krause, Henning;Pevtsova, Julia
• 通讯作者: Pevtsova, Julia

Criteria for vanishing of Tor over Complete Intersections

• DOI: 10.2140/pjm.2015.276.93
• 发表时间: 2014-12
• 期刊: arXiv: Commutative Algebra
• 作者: Olgur Celikbas;S. Iyengar;Greg Piepmeyer;R. Wiegand