Ramified extensions of commutative ring spectra

交换环谱的分支扩展

• 批准号: 405031884
• 负责人: Professorin Dr. Birgit Richter
• 金额: --
• 依托单位: Bereich Algebra und Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2018
• 资助国家: 德国
• 起止时间: 2017-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/405031884?language=en
• 关键词: Ramified; extensions; commutative; ring; spectra

The theory of ramified extensions of rings of integers in number fields is a classical topic in algebraic number theory. In stable homotopy theory, ramified extensions of structured ring spectra occur for instance as connective covers of Galois extensions of commutative ring spectra. A systematic approach for studying ramification in stable homotopy theory is missing. The distinction of tame and wild ramification is rather ad hoc so far. The examples that were studied until now are mostly of chromatic type less or equal to one, i.e. these are extensions that concern singular homology and topological K-theory and variations of these. In this project I will study ramified extensions of chromatic type two or higher and using these examples I aim to develop a suitable notion of tamely ramified extensions. Important examples of such maps come from function spectra and spectra of topological modular forms. Besides ramification at prime numbers there might be ramification at higher chromatic primes. Technical means for studying ramified maps are homology theories such as topological Hochschild homology, topological Andre-Quillen homology together with their logarithmic versions.

数域上整数环的分歧扩张理论是代数数论中的一个经典课题。在稳定同伦理论中，结构环谱的分歧扩张例如作为交换环谱的伽罗瓦扩张的连接覆盖出现。一个系统的方法来研究分歧稳定同伦理论是缺失的。到目前为止，驯服和野性分支的区别是相当特殊的。到目前为止研究的例子大多是小于或等于1的色型，即这些是关于奇异同调和拓扑K-理论及其变体的扩展。在这个项目中，我将研究分歧扩大色类型2或更高，并使用这些例子，我的目标是发展一个合适的概念，驯服分歧扩大。这类映射的重要例子来自函数谱和拓扑模形式的谱。除了在素数上的分歧之外，还可能有在高色素数上的分歧。研究分歧映射的技术手段是同调理论，如拓扑Hochschild同调，拓扑Andre-Quillen同调及其对数形式。