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. 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 $S$.

中文翻译：

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关于没有投影假设的黎曼-罗赫公式

令 $S$ 是一个有限维诺特方案。对于光滑$S$-schemes 之间的任何真态射，我们证明了一个关于高代数$K$-理论和动机上同调的Riemann-Roch 公式，因此在方案和态射上都没有投影假设。我们还证明了在没有投影假设的情况下，一个算术 Riemann-Roch 定理，该定理涉及 Arakelov 的更高 $K$ 理论和动机上同调，以及一个态射的相对上同调的类似结果。这些结果是作为动机陈述的推论获得的，该陈述适用于稳定同伦范畴 $S$ 中定向绝对谱之间的态射。