A generalized Contou-Carrère symbol and its reciprocity laws in higher dimensions
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- by Oliver Braunling, Michael Groechenig and Jesse Wolfson HTML | PDF
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 679-753
Abstract:
We generalize Contou-Carrère symbols to higher dimensions. To an $(n+1)$-tuple $f_0,\dots ,f_n \in A((t_1))\cdots ((t_n))^{\times }$, where $A$ denotes an algebra over a field $k$, we associate an element $(f_0,\dots ,f_n) \in A^{\times }$, extending the higher tame symbol for $k = A$, and earlier constructions for $n = 1$ by Contou-Carrère, and $n = 2$ by Osipov–Zhu. It is based on the concept of higher commutators for central extensions by spectra. Using these tools, we describe the higher Contou-Carrère symbol as a composition of boundary maps in algebraic $K$-theory, and prove a version of Parshin–Kato reciprocity for higher Contou-Carrère symbols.References
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Additional Information
- Oliver Braunling
- Affiliation: Department of Mathematics, University of Freiburg, Freiburg, Germany
- MR Author ID: 1036829
- ORCID: 0000-0003-4845-7934
- Email: oliver.braeunling@math.uni-freiburg.de
- Michael Groechenig
- Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada
- MR Author ID: 1058689
- Email: michael.groechenig@utoronto.ca
- Jesse Wolfson
- Affiliation: Department of Mathematics, University of California - Irvine, Irvine, California
- MR Author ID: 1141407
- Email: wolfson@uci.edu
- Received by editor(s): November 12, 2015
- Received by editor(s) in revised form: December 17, 2016, November 19, 2018, May 25, 2019, and September 28, 2020
- Published electronically: August 2, 2021
- Additional Notes: The first author was supported by DFG SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties”, the Alexander von Humboldt Foundation, and DFG GK1821 “Cohomological Methods in Geometry” . The second author was partially supported by EPRSC Grant No. EP/G06170X/1. The third author was partially supported by an NSF Graduate Research Fellowship under Grant No. DGE-0824162, by an NSF Research Training Group in the Mathematical Sciences under Grant No. DMS-0636646, and by an NSF Post-doctoral Research Fellowship under Grant No. DMS-1400349. This research was supported in part by NSF Grant No. DMS-1303100 and EPSRC Mathematics Platform grant EP/I019111/1
- © Copyright 2021 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 679-753
- MSC (2020): Primary 19D45
- DOI: https://doi.org/10.1090/btran/81
- MathSciNet review: 4294267