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TOWARDS A NON-ARCHIMEDEAN ANALYTIC ANALOG OF THE BASS–QUILLEN CONJECTURE

Part of: Grothendieck groups and $K_0$ Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  01 February 2019

Moritz Kerz ,
Shuji Saito  and
Georg Tamme
Moritz Kerz
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040Regensburg, Germany (moritz.kerz@ur.de)
Shuji Saito
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914, Japan (sshuji@msb.biglobe.ne.jp)
Georg Tamme
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040Regensburg, Germany (georg.tamme@ur.de)
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Abstract

We suggest an analog of the Bass–Quillen conjecture for smooth affinoid algebras over a complete non-archimedean field. We prove this in the rank-1 case, i.e. for the Picard group. For complete discretely valued fields and regular affinoid algebras that admit a regular model (automatic if the residue characteristic is zero) we prove a similar statement for the Grothendieck group of vector bundles $K_{0}$.

Keywords

affinoid algebraPicard grouphomotopy invariance

MSC classification

Primary: 14G22: Rigid analytic geometry
Secondary: 19A49: $K_0$ of other rings
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 6 , November 2020 , pp. 1931 - 1946
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Copyright
© Cambridge University Press 2019

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Footnotes

The authors are supported by the DFG through CRC 1085 Higher Invariants (Universität Regensburg).

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