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On the Riemann-Roch formula without projective hypotheses. (English) Zbl 1468.14015

Summary: 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 \(S\).

MSC:

14C40 Riemann-Roch theorems
14F42 Motivic cohomology; motivic homotopy theory
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
19E20 Relations of \(K\)-theory with cohomology theories
19L10 Riemann-Roch theorems, Chern characters
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