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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].

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
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