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Localization theorem for higher arithmetic K-theory

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Abstract

Quillen’s localization theorem is well known as a fundamental theorem in the study of algebraic K-theory. In this paper, we present its arithmetic analogue for the equivariant K-theory of arithmetic schemes, which are endowed with an action of certain diagonalisable group scheme. This equivariant arithmetic K-theory is defined by means of a natural extension of Burgos–Wang’s simplicial description of Beilinson’s regulator map to the equivariant case. As a byproduct of this work, we give an analytic refinement of the Riemann–Roch theorem for higher equivariant algebraic K-theory. And as an application, we prove a higher arithmetic concentration theorem which generalizes Thomason’s corresponding result in purely algebraic case to the context of Arakelov geometry.

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Acknowledgements

The author would like to thank Damian Roessler for his careful reading of an early version of this paper and for his valuable comments. He would also like to thank the referee for his/her hard work and helpful suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, West 3rd Ring North Road 105, Beijing, 100048, People’s Republic of China

    Shun Tang

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  1. Shun Tang
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Correspondence to Shun Tang.

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This work was partially supported by NSFC (no. 11631009).

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Tang, S. Localization theorem for higher arithmetic K-theory. Math. Z. 290, 307–346 (2018). https://doi.org/10.1007/s00209-017-2019-4

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