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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Algebraic $K$-theory of $\operatorname {THH}(\mathbb {F}_p)$
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by Haldun Özgür Bayındır and Tasos Moulinos PDF
Trans. Amer. Math. Soc. 375 (2022), 4177-4207 Request permission


In this work we study the $E_{\infty }$-ring $\operatorname {THH}(\mathbb {F}_p)$ as a graded spectrum. Following an identification at the level of $E_2$-algebras with $\mathbb {F}_p[\Omega S^3]$, the group ring of the $E_1$-group $\Omega S^3$ over $\mathbb {F}_p$, we show that the grading on $\operatorname {THH}(\mathbb {F}_p)$ arises from decomposition on the cyclic bar construction of the pointed monoid $\Omega S^3$. This allows us to use trace methods to compute the algebraic $K$-theory of $\operatorname {THH}(\mathbb {F}_p)$. We also show that as an $E_2$ $H\mathbb {F}_p$-ring, $\operatorname {THH}(\mathbb {F}_p)$ is uniquely determined by its homotopy groups. These results hold in fact for $\operatorname {THH}(k)$, where $k$ is any perfect field of characteristic $p$. Along the way we expand on some of the methods used by Hesselholt-Madsen and later by Speirs to develop certain tools to study the THH of graded ring spectra and the algebraic $K$-theory of formal DGAs.
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Additional Information
  • Haldun Özgür Bayındır
  • Affiliation: Department of Mathematics, City, University of London, London EC1V 0HB, United Kingdom
  • Tasos Moulinos
  • Affiliation: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne 31062 Toulouse Cedex 9, France
  • MR Author ID: 1137307
  • Received by editor(s): May 20, 2021
  • Received by editor(s) in revised form: October 7, 2021
  • Published electronically: March 4, 2022
  • Additional Notes: The first author acknowledges support from the project ANR-16-CE40-0003 ChroK. The second author was supported by grant NEDAG ERC-2016-ADG-741501 during the writing of this work
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 4177-4207
  • MSC (2020): Primary 55P99, 19D99
  • DOI:
  • MathSciNet review: 4419056