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Transactions of the American Mathematical Society

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On the Riemann-Roch formula without projective hypotheses


Authors: A. Navarro and J. Navarro
Journal: Trans. Amer. Math. Soc. 374 (2021), 755-772
MSC (2010): Primary 14C40, 14F42, 19E15, 19E20, 19L10
DOI: https://doi.org/10.1090/tran/8107
Published electronically: November 3, 2020
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Abstract | References | Similar Articles | Additional Information

Abstract: 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

J. Navarro
Affiliation: Departamento de Matemáticas, Universidad de Extremadura, 06006 Badajoz, Spain

DOI: https://doi.org/10.1090/tran/8107
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
Article copyright: © Copyright 2020 by the authors