Towards an understanding of ramified extensions of structured ring spectra

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 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.

Detecting and describing ramification for structured ring spectra

“…In [19] (see also [20] for a correction), we show that the relative topological Hochschild homology spectra and have highly nontrivial homotopy groups. Here, we extend these results to the relative -spectra of , and .…”

“…We know from [19] that shows features of a tamely ramified extension of number rings, and Sagave shows [50, Theorem 6.1] that is log-étale.…”

Ramification and descent in homotopy theory and derived algebraic geometry

“…Many authors have used THH to study ramification (as an informal concept) of extensions arising in stable homotopy theory. These include Blumberg-Mandell [2], who confirmed a philosophy of Hesselholt that ku (connective complex k-theory) is a tamely ramified extension of its Adams summand ℓ, Dundas-Lindenstrauss-Richter [3] and Höning-Richter [4] who studied ku{ko and various extensions related to tmf , and others.…”

“…(It is the desuspension of reduced THH.) Dundas-Lindenstrauss-Richter [3] and others have studied reduced THH as a measure of ramification of A{R.…”