# 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.
These…

#### One Citation

On the higher Riemann-Roch without denominators.

- Mathematics
- 2019

We prove two refinements of the higher Riemann-Roch without denominators: a statement for regular closed immersions between arbitrary finite dimensional noetherian schemes, with no smoothness…

#### References

SHOWING 1-10 OF 31 REFERENCES

Riemann-Roch for homotopy invariant K-theory and Gysin morphisms

- Mathematics
- 2016

We prove the Riemann-Roch theorem for homotopy invariant $K$-theory and projective local complete intersection morphisms between finite dimensional noetherian schemes, without smoothness assumptions.…

On Grothendieck's Riemann-Roch Theorem

- Mathematics
- 2016

We prove that, for smooth quasi-projective varieties over a field, the $K$-theory $K(X)$ of vector bundles is the universal cohomology theory where $c_1(L\otimes \bar L)=c_1(L)+c_1(\bar…

Mixed Weil cohomologies

- Mathematics
- 2012

Abstract We define, for a regular scheme S and a given field of characteristic zero K, the notion of K-linear mixed Weil cohomology on smooth S-schemes by a simple set of properties, mainly:…

Orientation theory in arithmetic geometry

- Mathematics
- 2011

This work is devoted to study orientation theory in arithmetic geometric within the motivic homotopy theory of Morel and Voevodsky. The main tool is a formulation of the absolute purity property for…

Riemann-Roch Theorems for Oriented Cohomology

- Mathematics
- 2004

The notion of an oriented cohomology pretheory on algebraic varieties is introduced and a Riemann-Roch theorem for ring morphisms between oriented pretheories is proved. An explicit formula for the…

Bivariant theories in motivic stable homotopy

- Mathematics
- 2017

The purpose of this work is to study the notion of bivariant theory introduced by Fulton and MacPherson in the context of motivic stable homotopy theory, and more generally in the broader framework…

Riemann-Roch for general algebraic varieties

- Mathematics
- 1983

Grothcndicck proved the Ricmann-Roch theorem for a morphism between smooth projcctive varieties, and Baum-Fulton-MacPherson extended it to proper morphisms between quasi-projective, possibly…

A UNIVERSALITY THEOREM FOR VOEVODSKY'S ALGEBRAIC COBORDISM SPECTRUM

- Mathematics
- 2007

An algebraic version of a theorem of Quillen is proved. More precisely, for a regular Noetherian scheme S of finite Krull dimension, we consider the motivic stable homotopy category SH(S) of P 1…

Oriented cohomology theories of algebraic varieties II

- Mathematics
- 2009

This article contains proofs of the results announced in (21) in the part concerning general properties of oriented cohomology theories of algebraic varieties. It is constructed one-to-one corres-…

Around the Gysin triangle II

- Mathematics
- 2007

The notions of orientation and duality are well understood in algebraic topology in the framework of the stable homotopy category. In this work, we follow these lines in algebraic geometry, in the…