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
MathSciNet review:
4196376
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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
Article copyright:
© Copyright 2020
by the authors