Complex orientations for THH of some perfectoid fields

TL;DR

This paper explores the extension of topological Hochschild homology to perfectoid fields, establishing complex orientations and potential applications in algebraic K-theory and chromatic homotopy theory.

Contribution

It introduces a new framework for topological Hochschild homology of perfectoid fields, leveraging recent advances to define complex orientations and connect to K-theory.

Findings

Extension of THH to perfectoid fields using Scholze's work

Construction of spectra with complex orientations

Potential for new Chern character analogs in chromatic homotopy theory

Abstract

This sketch argues that work of Hesselholt on the topological Hochschild homology of $\Cp$ extends, using work of Scholze and others, to define complex orientations for a version of topological Hochschild homology for rings of integers in a natural class of generalized cyclotomic perfectoid fields; and that the resulting spectra provide geometrically interesting targets for analogs of the Chern character, defined for certain integral lifts of the extraordinary $K$-functors of chromatic homotopy theory.