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

@article{Navarro2016RiemannRochFH, title={Riemann-Roch for homotopy invariant K-theory and Gysin morphisms}, author={Alberto Navarro}, journal={arXiv: K-Theory and Homology}, year={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. We also prove a new Riemann-Roch theorem for the relative cohomology of a morphism.
In order to do so, we construct and characterize Gysin morphisms for regular immersions between cohomologies represented by spectra (examples include homotopy invariant $K$-theory, motivic cohomology, their…

## 10 Citations

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## References

SHOWING 1-10 OF 47 REFERENCES

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…

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…

Around the Gysin triangle I

- Mathematics
- 2008

We study the construction and properties of the Gysin triangle in an axiomatic framework which covers triangulated mixed motives and MGl-modules over an arbitrary base S. This allows to define the…

Motivic Landweber Exactness

- Mathematics
- 2008

We prove a motivic Landweber exact functor theorem. The main result shows the assignment given by a Landweber-type formula involving the MGL-homology of a motivic spectrum defines a homology theory…

THE RIGID SYNTOMIC RING SPECTRUM

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2014

The aim of this paper is to show that rigid syntomic cohomology – defined by Besser – is representable by a rational ring spectrum in the motivic homotopical sense. In fact, extending previous…

Higher Bott-Chern forms and Beilinson's regulator

- Mathematics
- 1996

Abstract. In this paper, we prove a Gauss-Bonnet theorem for the higher algebraic K-theory of smooth complex algebraic varieties. To each exact n-cube of hermitian vector bundles, we associate a…

Orientable homotopy modules

- Mathematics
- 2010

We prove a conjecture of Morel identifying Voevodsky’s homotopy invariant sheaves with transfers with spectra in the stable homotopy category which are concentrated in degree zero for the homotopy…

Algebraic K‐theory, A1‐homotopy and Riemann–Roch theorems

- Mathematics
- 2010

In this article, we show that the combination of the constructions done in SGA 6 and the A^1-homotopy theory naturally leads to results on higher algebraic K-theory. This applies to the operations on…