### Citations

145 |
Algebraic K-theory and quadratic forms,
- Milnor
- 1970
(Show Context)
Citation Context ...)) and for general X ∈ Smk, we set S(X) = ∏ i∈X(0) S(Xi). 3.2. Residue and specialization morphisms. In this section, we recall the residue morphisms on Grothendieck-Witt rings. We will mostly follow =-=[Mil69]-=- and [Mor12]. The residue morphisms provide us with the data (D1)-(D3) for unramified Fk-data. The first such datum is obvious - (D1) just requires that we have induced morphisms for field extensions.... |

86 | A1-homotopy theory of schemes. - Morel, Voevodsky - 2001 |

12 | On the Bass-Quillen conjecture concerning projective modules over polynomial rings - Lindel |

10 | The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes,
- Schlichting
- 2010
(Show Context)
Citation Context ...ture of Grothendieck-Witt rings. The reference used will be [KK82]. The reader should be aware that there are more modern and much more high-tech approaches to Grothendieck-Witt groups available, cf. =-=[Sch10]-=-. In the present work, we will not need more than the structure of units and the existence of residue maps. Throughout the paper we assume that the base field k has characteristic 6= 2. 2.1. Basic def... |

7 | The Gersten conjecture for Witt groups in the equicharacteristic case,
- Balmer, Gille, et al.
- 2002
(Show Context)
Citation Context ...lutions are Date: February 2013. 2010 Mathematics Subject Classification. 55R25,19G12,14F42. Key words and phrases. Grothendieck-Witt rings, A1-homotopy theory. 1 2 MATTHIAS WENDT already established =-=[BGPW]-=- resp. [Mor12, Chapters 4,5]. We describe a Gerstentype resolution of the units in the Grothendieck-Witt rings, based on the explicit computation of contractions. It is noteworthy and strange that thi... |

5 | Classifying spaces and fibrations of simplicial sheaves
- Wendt
- 2009
(Show Context)
Citation Context ...rations. Corollary 1.1. Let S2n,n denote an A1-local fibrant model of the motivic sphere, and denote by B hAut•(S 2n,n) the corresponding classifying space of Nisnevich locally trivial torsors, as in =-=[Wen11]-=-. Then this space is in fact A1-local and classifies Nisnevich locally trivial spherical fibrations. The relation between strong A1-invariance of the units and the existence of a classifying space of ... |

2 |
A1-algebraic topology over a field. Lecture notes in mathematics 2052
- Morel
- 2012
(Show Context)
Citation Context ...eneral X ∈ Smk, we set S(X) = ∏ i∈X(0) S(Xi). 3.2. Residue and specialization morphisms. In this section, we recall the residue morphisms on Grothendieck-Witt rings. We will mostly follow [Mil69] and =-=[Mor12]-=-. The residue morphisms provide us with the data (D1)-(D3) for unramified Fk-data. The first such datum is obvious - (D1) just requires that we have induced morphisms for field extensions. The obvious... |

2 | Fibre sequences and localizations of simplicial sheaves
- Wendt
- 2010
(Show Context)
Citation Context ...oup of homotopy self-equivalences of a sphere with the units in the Grothendieck-Witt ring. On the other hand, the general theory of localization of fibre sequences of simplicial sheaves developed in =-=[Wen13]-=- implies the existence of an A1-local classifying space of fibrations with fibre X provided the sheaf of homotopy self-equivalences is strongly A1-local. These two results are the starting point for t... |

1 |
Classes d’homotopie de fractions rationnelles
- Cazanave
(Show Context)
Citation Context ...iated to vector bundles. By Proposition 5.8, these are classified exactly by H1Nis(P 1,NQ) ∼= H0Nis(Spec k, (NQ)−1) ∼= W (k) (1) tor/S. Example 6.14. Cazanave has described the automorphisms of P1 in =-=[Caz08]-=-. As a consequence of the results of [Caz08], the rational function X3 − ( a3 a2 + a2 a1 ) X a1X2 − a1a3 a2 UNITS IN GROTHENDIECK-WITT RINGS AND A1-SPHERICAL FIBRATIONS 19 is the endomorphism associat... |