# On the Riemann-Roch formula without projective hypotheses

@article{Navarro2017OnTR, title={On the Riemann-Roch formula without projective hypotheses}, author={Alberto Navarro and Jos'e Navarro}, journal={arXiv: Algebraic Topology}, year={2017} }

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 hypothesis neither on the schemes nor 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 analogue result for the relative cohomology of a morphism.
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