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The K-Theory of Toric Schemes Over Regular Rings of Mixed Characteristic

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Abstract

We show that if X is a toric scheme over a regular commutative ring k then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for regular commutative rings containing a field. The affine case of our result was conjectured by Gubeladze. We prove analogous results when k is replaced by an appropriate K-regular, not necessarily commutative k-algebra.

Cortiñas’ research was supported by Conicet and partially supported by grants UBACyT 20021030100481BA, PIP 112-201101-00800CO, PICT 2013-0454, and MTM2015-65764-C3-1-P (Feder funds).

Walker’s research was partially supported by a grant from the Simons Foundation (#318705).

Walker’s research was partially supported by a grant from the Simons Foundation (#318705).

Weibel’s research was supported by NSA and NSF grants.

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Acknowledgements

This article is part of a collection of papers published in honour of Antonio Campillo’s 65th birthday. The first named author is very grateful to him for all his help over many years; the authors dedicate this paper to him.

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Authors and Affiliations

  1. Department of Matemática-Inst. Santaló, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina

    G. Cortiñas

  2. School of Mathematics and Statistics, University of Melbourne, Melbourne, VIC, Australia

    C. Haesemeyer

  3. Department of Mathematics, University of Nebraska – Lincoln, Lincoln, NE, USA

    M. E. Walker

  4. Department of Mathematics, Rutgers University, New Brunswick, NJ, USA

    C. A. Weibel

Authors
  1. G. Cortiñas
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  2. C. Haesemeyer
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  3. M. E. Walker
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  4. C. A. Weibel
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Corresponding author

Correspondence to G. Cortiñas .

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Editors and Affiliations

  1. Fachbereich Mathematik, Universität Kaiserslautern, Kaiserslautern, Germany

    Gert-Martin Greuel

  2. Departamento de Álgebra, Instituto de Matemáticas (IMUS), Universidad de Sevilla, Sevilla, Spain

    Luis Narváez Macarro

  3. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

    Sebastià Xambó-Descamps

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Cortiñas, G., Haesemeyer, C., Walker, M.E., Weibel, C.A. (2018). The K-Theory of Toric Schemes Over Regular Rings of Mixed Characteristic. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_19

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