Norm residue isomorphism theorem seminar

## Fall 2021

The goal of this seminar is to understand (some portion of) the proof of the norm residue isomorphism theorem by Voevodsky, Rost, Suslin, Weibel, Haesemeyer, and others. Since the proof is quite complicated we will sketch some parts of the proof of the existence of Rost varieties.

### Time and place

10:00-11:30 AM Tuesdays, room 622 in the math building.

### References

[1] Christian Haesemeyer and Charles A Weibel. The norm residue theorem in motivic cohomology. Princeton University Press, 2019.

[2] Carlo Mazza, Vladimir Voevodsky, Charles Weibel. Lecture notes on motivic cohomology. American Mathematical Society, 2006.

[3] Alexander Merkurjev. On the norm residue homomorphism of degree two. Translations of the American Mathematical Society-Series 2, 219:103-124, 2006.

[4] Wilberd van der Kallen. The Merkurjev-Suslin theorem. In Orders and their Applications, pages 157-168. Springer, 1985.

[5] Vladimir Voevodsky. Motivic cohomology with $$\mathbb{Z}/2$$-coefficients. Publications Mathématiques de l'IHÉS, 98:59-104, 2003.

[6] Vladimir Voevodsky. On motivic cohomology with $$\mathbb{Z}/\ell$$-coefficients. Annals of Mathematics, pages 401-438, 2011.

### Schedule

 Date Speaker Topic References Notes September 21 Avi Zeff Introduction and reductions [1, §1.1-1.4], [3] PDF September 28 Caleb Ji Motivic cohomology [2] PDF October 5 Hung Chiang $$\mathbb A^1$$-homotopy theory, the Nisnevich topology, and the reverse induction steps [1, §2.1-2.2], [2] PDF October 12 Hung Chiang, Avi Zeff Completion of reverse induction and cohomology of singular varieties/with supports [1, §2.3-2.4] PDF October 19 Baiqing Zhu Proof up to Rost varieties and equivalences [1, §3.1-3.3] October 26 Haodong Yao Equivalence of conditions [1, §2.5-2.7], [2] November 2 [everyone] Recap and planning November 9 David Marcil Proof of injectivity on cohomology assuming Rost motives [1, §4] November 16 Avi Zeff Existence of Rost motives assuming Rost varieties [1, §5] PDF November 23 [everyone] Discussion: applications of Bloch-Kato to higher local class field theory November 30 Avi et al. More higher local class field theory: the isomorphism theorem December 7 Avi Zeff K-theory of the integers