Le principe de scindage et l'inexistence d'une \(K\)-théorie de Milnor globale. (The splitting principle and the non-existence of global Milnor \(K\)-theory) (Q1208252)

The main result of this paper is a minimality result for the Quillen \(K\)- theory of smooth quasi-projective schemes over a field \(k\). Let \(F\) be a functor on such schemes which satisfies some natural axioms; in particular there is to be a natural transformation \(\eta:F\to K\), where \(K\) is the Quillen \(K\)-theory. The author shows that \(\eta:F_ *(X)\to K_ *(X)\) is surjective for each \(X\). As a consequence, there is no good extension of the Milnor \(K\)-theory \(K_ *^ M(k)\) to the category of schemes.