Detecting and describing ramification for structured ring spectra

Abstract

Ramification for commutative ring spectra can be detected by relative topological Hochschild homology and by topological Andr\'e-Quillen homology. In the classical algebraic context it is important to distinguish between tame and wild ramification. Noether's theorem characterizes tame ramification in terms of a normal basis and tame ramification can also be detected via the surjectivity of the trace map. We transfer the latter fact to ring spectra and use the Tate cohomology spectrum to detect wild ramification in the context of commutative ring spectra. We study ramification in examples in the context of topological K-theory and topological modular forms.