Towards an understanding of ramified extensions of structured ring spectra

TL;DR

This paper explores how topological Hochschild homology can measure ramification in structured ring spectra, providing explicit calculations that detect tame ramification in specific extensions.

Contribution

It introduces topological Hochschild homology as a tool for understanding ramification in structured ring spectra and computes key examples demonstrating its effectiveness.

Findings

Determined second order topological Hochschild homology of p-local integers.

Showed topological Hochschild homology detects tame ramification.

Computed relative topological Hochschild homology for specific extensions.

Abstract

We propose topological Hochschild homology as a tool for measuring ramification of maps of structured ring spectra. We determine second order topological Hochschild homology of the $p$-local integers. For the tamely ramified extension of the map from the connective Adams summand to $p$-local complex topological K-theory we determine the relative topological Hochschild homology and show that it detects the tame ramification of this extension. We also determine relative topological Hochschild homology for the complexification map from connective real to complex topological K-theory and for some quotient maps with commutative quotients.