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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On the Riemann-Roch formula without projective hypotheses
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by A. Navarro and J. Navarro PDF
Trans. Amer. Math. Soc. 374 (2021), 755-772


Let $S$ be a finite dimensional noetherian scheme. For any proper morphism between smooth $S$-schemes, we prove a Riemann-Roch formula relating higher algebraic $K$-theory and motivic cohomology, thus with no projective hypotheses either on the schemes or on the morphism. We also prove, without projective assumptions, an arithmetic Riemann-Roch theorem involving Arakelov’s higher $K$-theory and motivic cohomology as well as an analogous result for the relative cohomology of a morphism.

These results are obtained as corollaries of a motivic statement that is valid for morphisms between oriented absolute spectra in the stable homotopy category of $S$.

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Additional Information
  • A. Navarro
  • Affiliation: Departamento de Matemática Aplicada, E.T.S. de Arquitectura, Universidad Politéc- nica de Madrid, 28040 Madrid, Spain
  • MR Author ID: 946466
  • J. Navarro
  • Affiliation: Departamento de Matemáticas, Universidad de Extremadura, 06006 Badajoz, Spain
  • MR Author ID: 846164
  • ORCID: 0000-0001-5257-176X
  • Received by editor(s): March 1, 2018
  • Received by editor(s) in revised form: September 9, 2019
  • Published electronically: November 3, 2020
  • Additional Notes: The first author was supported by MTM2016-79400-P (MINECO)
    The second author was supported by grants GR18001 and IB18087

  • Dedicated: To our father
  • © Copyright 2020 by the authors
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 755-772
  • MSC (2010): Primary 14C40, 14F42, 19E15, 19E20, 19L10
  • DOI:
  • MathSciNet review: 4196376