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Axioms for the norm residue isomorphismCharles Weibel IntroductionThe purpose of this paper is to give an axiomatic framework for proving (by inductionon n) that the norm residue map Kn M .k/=` ! H ét n .k; ˝n/ is an isomorphism, for any`prime `>2and for any field k such that ` is invertible in k.Fix ` and n. ByaRost variety for a sequence a D .a 1 ;:::;a n / of units in k, weshall mean a smooth projective variety X of dimension d D `n 1 1 satisfying theconditions of [8, 6.3] or [6, (0.1)]; the exact definition is given in Definition 1.1 below.One key requirement is that fa 1 ;:::;a n g vanishes in Kn M .k.X//=`. Rost constructedsuch a variety in his 1998 preprint [3]; the proof that it satisfies these properties waspublished in [1] and [6].Theorem 0.1. Let n and ` be such that the norm residues maps are isomorphisms forall i