This paper interprets Hesselholt and Madsen’s real topological Hochschild homology functor THR in terms of the multiplicative norm construction. We show that THR satisfies cofinality and Morita invariance, and that it is suitably multiplicative. We then calculate its geometric fixed points and its Mackey functor of components, and show a decomposition result for groupalgebras. Using these structural results we determine the homotopy type of THR(Fp) and show that its bigraded homotopy groups are polynomial on one generator over the bigraded homotopy groups of H Fp. We then calculate the homotopy type of THR(Z) away from the prime 2, and the homotopy ring of the geometric fixed-points spectrum Φ
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Acknowledgments. The authors would like to thank the Hausdorff Research Institute for Mathematics in Bonn for their hospitality during the Junior Trimester Program in Topology in 2016. Much of the work on this paper was carried out during that program. The authors also acknowledge the support of the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92). The second author was supported by the Max Planck institute for Mathematics and thanks the Mittag-Leffler Institute for their hospitality. The third author was supported by the German Research Foundation Schwerpunktprogramm 1786. The fourth author was supported by Independent Research Fund Denmark’s Sapere Aude program (DFF–4002-00224) and by the Max Planck Institute for Mathematics.
© European Mathematical Society 2021
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- Hochschild homology
- ring spectra
- Ring spectra
- MODEL CATEGORIES
- ALGEBRAIC K-THEORY
- WITT VECTORS