Ramified extensions of commutative ring spectra

整数环在数域上的分支扩展理论是代数数论中的一个经典课题。在稳定同伦理论中，结构环谱的分支扩展出现在交换环谱的伽罗瓦扩展的连接覆盖上。目前还没有一种系统的方法来研究稳定同伦理论中的分枝。到目前为止，驯化和野生分支的区分是相当特别的。到目前为止所研究的例子大多是小于等于1的色型，也就是说，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，我将研究色型二或更高的分支扩展，并使用这些例子，我的目标是发展一个合适的分支扩展的概念。这种映射的重要例子来自于函数谱和拓扑模形式的谱。除了质数上的分支外，在更高色质数上也可能有分支。研究分支映射的技术手段是同调理论，如拓扑Hochschild同调、拓扑Andre-Quillen同调及其对数版本。

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.