# Motivic twisted K-theory

@article{Spitzweck2010MotivicTK, title={Motivic twisted K-theory}, author={Markus Spitzweck and Paul arne Ostvaer}, journal={arXiv: Algebraic Topology}, year={2010} }

This paper sets out basic properties of motivic twisted K-theory with respect to degree three motivic cohomology classes of weight one. Motivic twisted K-theory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BG_m-bundle for the classifying space of the multiplicative group scheme. We show a Kuenneth isomorphism for homological motivic twisted K-groups computing the latter as a tensor product of K-groups over the K-theory of BG_m. The…

## 35 Citations

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## References

SHOWING 1-10 OF 56 REFERENCES

### Twisted K-theory

- Mathematics
- 2004

Twisted complex K-theory can be defined for a space X equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C ∗ -algebras. Up to equivalence, the twisting…

### A possible new approach to the motivic spectral sequence for algebraic K-theory

- Mathematics
- 2001

Despite the considerable progress in motivic cohomology and motivic homotopy theory achieved in recent years we still do not have a simple construction of the spectral sequence relating motivic…

### Loop groups and twisted K‐theory I

- Mathematics
- 2011

This is the first in a series of papers investigating the relationship between the twisted equivariant K‐theory of a compact Lie group G and the “Verlinde ring” of its loop group. In this paper we…

### 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…

### On the Motivic Spectra Representing Algebraic Cobordism and Algebraic K-Theory

- Mathematics
- 2009

We show that the motivic spectrum representing alge- braic K-theory is a localization of the suspension spectrum of P ∞ , and similarly that the motivic spectrum representing periodic alge- braic…

### Chern classes, K-theory and Landweber exactness over nonregular base schemes

- Mathematics
- 2008

The purpose of this paper is twofold. First, we use the motivic Landweber exact functor theorem to deduce that the Bott inverted infinite projective space is homotopy algebraic $K$-theory. The…