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$ \operatorname{H}(\mathbb{Z}/p^k)$ as a Thom spectrum and topological Hochschild homology


Author: Nitu Kitchloo
Journal: Proc. Amer. Math. Soc. 148 (2020), 3647-3651
MSC (2010): Primary 55P42
DOI: https://doi.org/10.1090/proc/14968
Published electronically: March 4, 2020
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Abstract: In this short note we study the topological Hochschild homology of Eilenberg-MacLane spectra for finite cyclic groups. In particular, we show that the Eilenberg-MacLane spectrum $ \operatorname {H}(\mathbb{Z}/p^k)$ is a Thom spectrum for any prime $ p$ (except, possibly, when $ p=k=2$) and we also compute its topological Hoschshild homology. This yields a short proof of the results obtained by Brun [J. Pure Appl. Algebra 148 (2000), pp. 29-76], and Pirashvili [Comm. Algebra 23 (1995), pp. 1545-1549] except for the anomalous case $ p=k=2$.


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Nitu Kitchloo
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Email: nitu@math.jhu.edu

DOI: https://doi.org/10.1090/proc/14968
Received by editor(s): November 30, 2018
Received by editor(s) in revised form: December 9, 2019, and December 11, 2019
Published electronically: March 4, 2020
Additional Notes: The author was supported in part by the Simons Fellowship and the Max Planck Institute of Mathematics
Communicated by: Mark Behrens
Article copyright: © Copyright 2020 American Mathematical Society