Ramified extensions of commutative ring spectra

数域上整数环的分支扩张理论是代数数论中的一个经典问题。在稳定同伦理论中，结构环谱的分支扩张作为交换环谱的Galois扩张的连通覆盖而出现。研究稳定同伦理论中的分支的系统方法是缺失的。到目前为止，对驯服和狂野分支的区分是相当临时的。到目前为止所研究的例子大多是小于或等于一的色型，即这些是涉及奇异同调和拓扑K-理论及其变体的推广。在这个项目中，我将研究色类型2或更高类型的分支扩张，并且使用这些例子，我的目标是开发一个合适的概念，即驯服分支扩张。这种映射的重要例子来自于函数谱和拓扑模形式的谱。除了在素数处的分支外，还可能在更高的色素数处分支。研究分枝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。

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.