On the Riemann-Roch formula without projective hypotheses

Navarro, A.; Navarro, J.

doi:10.1090/tran/8107

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
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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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