Kahn, Bruno The Milnor conjecture after V. Voevodsky. (La conjecture de Milnor d’après V. Voevodsky.) (French) Zbl 0916.19001 Séminaire Bourbaki. Volume 1996/97. Exposés 820–834. Paris: Société Mathématique de France, Astérisque. 245, 379-418, Exp. No. 834 (1997). This paper is a very useful detailed survey on the recent proof of the famous Milnor conjecture by Voevodsky. It starts with a down-to-earth introduction giving a precise formulation of the more general Kato conjecture and a summary of Voevodsky’s proof. In section 1, the author describes what was known before Voevodsky’s work and explains some elementary reductions. Section 2 gives an axiomatic introduction to motivic cohomology after Suslin and Voevodsky [see also the survey article, “Motivic complexes of Suslin and Voevodsky”, Asterisque 245, 355–378 (1997; Zbl 0908.19005) by E. M. Friedlander] and reduces the Milnor (or Kato) conjecture to a certain generalization of Noether’s formulation of Hilbert 90.Sections 3 and 4 contain an inductive procedure which reduces the latter condition to the existence of so-called splitting varieties. Section 5 and 6 deal with the heart of the story: the homotopy theory for schemes after Morel and Voevodsky is developed [see also V. Voevodsky’s paper, “\(\mathbb A^1\)-homotopy theory”, Doc. Math., J. DMV, Extra Vol. ICM 1998, vol. I, 579–604 (1998; Zbl 0907.19002)] and Voevodsky’s construction of Steenrod operations for motivic cohomology is explained. In sections 7 and 8, these tools together with the Rost motive are applied to prove that certain projective quadrics are splitting varieties. The final section 9 briefly indicates some results by Voevodsky which reduce the more general Kato conjecture to very special problems.While the first half of the paper is comparatively elementary and more or less self-contained, the second half of the paper is more ambitious and a bit more sketchy, in particular several proofs are still missing. Another article which gives an overview of the same topic is F. Morel’s paper [“Voevodsky’s proof of Milnor’s conjecture”, Bull. Am. Math. Soc., New Ser. 35, No. 2, 123–143 (1998; Zbl 0916.19002)].For the entire collection see [Zbl 0910.00034]. Reviewer: Bernhard Köck (Karlsruhe) Cited in 3 ReviewsCited in 4 Documents MSC: 19-02 Research exposition (monographs, survey articles) pertaining to \(K\)-theory 19D45 Higher symbols, Milnor \(K\)-theory 14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry 19E20 Relations of \(K\)-theory with cohomology theories 14A20 Generalizations (algebraic spaces, stacks) 55P42 Stable homotopy theory, spectra 14F20 Étale and other Grothendieck topologies and (co)homologies 14C25 Algebraic cycles 12G05 Galois cohomology Keywords:Kato conjecture; Milnor K-theory; Galois cohomology; motivic cohomology; splitting variety; homotopy theory for schemes; Eilenberg-MacLane spectra; Steenrod operations; Rost motive; Milnor conjecture Citations:Zbl 0908.19005; Zbl 0907.19002; Zbl 0916.19002 PDFBibTeX XMLCite \textit{B. Kahn}, in: Séminaire Bourbaki. Volume 1996/97. Exposés 820--834. Paris: Société Mathé\-matique de France. 379--418, Exp. No. 834 (1997; Zbl 0916.19001) Full Text: Numdam EuDML