Dualizing modules in algebra and geometry

代数和几何中的对偶模块

• 批准号: 1606479
• 负责人: Vesna Stojanoska
• 金额: $ 9.26万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-12-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1606479&HistoricalAwards=false
• 关键词: Dualizing; modules; algebra; geometry

The PI proposes to investigate the existence of homotopical dualizing modules in number theory and algebraic geometry and the implications thereof. Specifically, there will be two valuable instances of such dualizing modules. One will be studied in joint work with Tomer Schlank; it will give a homotopical extension of the arithmetic duality theorems of Tate-Poitou, which state that the cohomology of certain absolute Galois groups has a twisted form of self-duality. The PI and Tomer Schlank will also investigate applications of such duality results to problems of existence of rational points on algebraic varieties. A different example is related to the spectra of topological modular forms with or without level structures. Previous work of the PI suggests that the spectrally derived moduli stack of generalized elliptic curves has a simply describable dualizing sheaf of commutative ring spectra; an objective of the proposed project is to prove that result. Topological modular forms are crucial for understanding v2 periodic homotopy in the sphere spectrum; though somewhat removed from the theme of duality, the PI will collaborate with Mark Behrens, Kyle Ormsby, and Nathaniel Stapleton to compute the cooperations in the homology based on connective topological modular forms. Duality is a pervasive concept in mathematics; the proposed project will study different types of duality in a unified framework, thereby arriving at novel and otherwise inaccessible information. In particular, the project is motivated by the idea that introducing a homotopical viewpoint can shed new light on our understanding of geometry and arithmetic. One of the objects with such curious self-duality properties, namely topological modular forms, lends itself to vast applications, some already explored and others not, as it mirrors itself in homotopy, algebraic geometry, number theory, and even quantum field theory as the receptacle of the Witten genus.

PI 提议研究数论和代数几何中同伦对偶模的存在及其含义。具体来说，这种二元化模块将有两个有价值的实例。其中一项将与托默·施兰克 (Tomer Schlank) 联合研究；它将给出泰特-普瓦图算术对偶定理的同伦扩展，该定理指出某些绝对伽罗瓦群的上同调具有扭曲的自对偶形式。 PI 和 Tomer Schlank 还将研究这种对偶性结果在代数簇上有理点存在性问题中的应用。一个不同的例子与具有或不具有水平结构的拓扑模形式的谱有关。 PI 之前的工作表明，广义椭圆曲线的谱导出模堆栈具有可简单描述的交换环谱的对偶束；拟议项目的目标是证明这一结果。拓扑模形式对于理解球谱中的 v2 周期同伦至关重要；虽然有点偏离二元性的主题，但 PI 将与 Mark Behrens、Kyle Ormsby 和 Nathaniel Stapleton 合作，计算基于连接拓扑模块形式的同调中的合作。对偶性是数学中普遍存在的概念。拟议的项目将在一个统一的框架中研究不同类型的二元性，从而获得新颖的和其他方式无法获得的信息。特别是，该项目的动机是引入同伦观点可以为我们对几何和算术的理解提供新的启示。具有如此奇特的自对偶性质的对象之一，即拓扑模形式，使其具有广泛的应用，其中一些已经被探索过，而另一些则没有，因为它在同伦、代数几何、数论，甚至量子场论中反映了自己作为维滕属的花托。