Towards an understanding of ramified extensions of structured ring spectra

What is it about?

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 show that the complexification map from connective topological real to complex K-theory shows features of a wildly ramified extension. We also determine relative topological Hochschild homology for some quotient maps with commutative quotients.

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Why is it important?

Commutate ring spectra are an important generalization of commutative rings and understanding their arithmetic properties is crucial. Ramification is a central notion in number theory and in our paper we make a first step to understand ramified extensions of ring spectra.