Motives in May 2025

Organized by Jake Huryn and William C. Newman.

The goal of this minicourse was to learn something about Voevodsky's triangulated category of mixed motives, with the hope of seeing some ideas from the proof of the norm residue isomorphism theorem. Also, it served as a bit of preparation for the conference Arithmetic, K-Theory, and Algebraic Cycles.

References:
[HW]  Haesemeyer–Weibel, The Norm Residue Theorem in Motivic Cohomology
[MVW] Mazza–Voevodsky–Weibel, Lecture Notes on Motivic Cohomology
[Voe] Voevodsky, "Motivic cohomology with Z/2-coefficients"

Notes from pretalks (in progress; version of 4/26)

Pretalks

• Advertisement (Jake): April 11, in coordination with OSU TAGSS
• Chow groups (Jake): April 15. Exercises
• Sites (Jake): April 22. Exercises
• Cohomology theories (Will): April 25. Exercises
• Triangulated categories (Will): April 29. Exercises
• Pure motives (Jake): April 30. Exercises

Talks

• Presheaves with transfers [MVW, Lectures 1–2] (Will): May 2.
• Homotopy invariant presheaves, motivic cohomology [MVW, Lectures 2–3] (Jake): May 5.
• Milnor K-theory and motivic cohomology [MVW, Lecture 5] (Will): May 7.
• Sheaves with transfers [MVW, Lectures 6 and 13] (Will): May 9.
• Triangulated categories of mixed motives [MVW, Lectures 14, 8, and 9] (Jake): May 12, May 14.
• Mayer–Vietoris, vector bundles [MVW, Lecture 14] (Will): May 14.
• Norm residue isomorphism theorem I (Jake): May 16.
• Realizations, motive of Pⁿ [MVW, Lecture 15] (Will): May 19.
• Norm residue isomorphism theorem II [HW, Chapter 2] (Jake): May 21.
• Motives with compact support and the cdh topology [MVW, Lectures 12–13 and 16] (Will): May 23.
• Norm residue isomorphism theorem III [HW, Chapter 1], [Voe, §§4 and 7] (Jake): May 26.