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Localization for $THH(ku)$ and the Topological Hochschild and Cyclic Homology of Waldhausen Categories
About this Title
Andrew J. Blumberg, Department of Mathematics, The University of Texas, Austin, Texas 78712 and Michael A. Mandell, Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 265, Number 1286
ISBNs: 978-1-4704-4178-4 (print); 978-1-4704-6140-9 (online)
DOI: https://doi.org/10.1090/memo/1286
Published electronically: May 11, 2020
Keywords: Topological Hochschild homology,
topological cyclic homology,
Waldhausen category,
localization sequence,
$K$-theory,
dévissage theorem
MSC: Primary 19D55; Secondary 55P43, 19L41, 19D10
Table of Contents
Chapters
- Introduction
- 1. Review of $THH$, $TR$, and $TC$
- 2. $THH$ and $TC$ of simplicially enriched Waldhausen categories
- 3. $K$-theory theorems in $THH$ and $TC$
- 4. Localization sequences for $THH$ and $TC$
- 5. Generalization to Waldhausen categories with factorization
Abstract
We develop a theory of $THH$ and $TC$ of Waldhausen categories and prove the analogues of Waldhausen’s theorems for $K$-theory. We resolve the longstanding confusion about localization sequences in $THH$ and $TC$, and establish a specialized dévissage theorem. As applications, we prove conjectures of Hesselholt and Ausoni-Rognes about localization cofiber sequences surrounding $THH(ku)$, and more generally establish a framework for advancing the Rognes program for studying Waldhausen’s chromatic filtration on $A(*)$.- Vigleik Angeltveit, Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill, Tyler Lawson, and Michael A. Mandell, Topological cyclic homology via the norm, Doc. Math. 23 (2018), 2101–2163. MR 3933034
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