The goal of this seminar is to understand the motivic cohomology theory of Suslin and Voevodsky. For a field $k$ and a ring $R$, we will define a triangulated category $$ \mathsf{DM}_ \mathrm{et}^{\mathrm{eff},-}(k, R), $$ of derived motives. One of the main results we will prove is that if $R = \mathbb{Z}/m\mathbb{Z}$ where $m$ is invertible in $k$, then we have an equivalence of triangulated categories $$ \mathsf{D}^-(\Gal(k^\mathrm{sep}/k), \mathbb{Z}/m\mathbb{Z}) \simeq \mathsf{DM}_ \mathrm{et}^{\mathrm{eff},-}(k, \mathbb{Z}/m\mathbb{Z}). $$ This is one ingredient in the proof of the Bloch–Kato conjecture, which relates Milnor K-theory to Galois cohomology.
Schedule
- Presheaves with transfer (Jan 15th, Stepan): [MVW, Lectures 1 and 2]
- Motivic cohomology of a smooth variety (Jan 22th, Jungtao): [MVW, Lectures 3 and 4]
- Milnor K-theory (Jan 29th, Stepan): [MVW, Lecture 5]
- Étale sheaves with transfer (Feb 5th, Daniel): [MVW, Lecture 6]
- Suslin’s rigidity theorem (Feb 12th, Jiahao): [MVW, Lecture 7]
- $\mathbb{A}^1$-weak equivalence (Feb 19th, Xinyu): [MVW, Lecture 9]
- Étale motivic cohomology (Feb 26th, Spencer): [MVW, Lecture 10]
- Introduction to Bloch–Kato (Mar 5th, Vaughan): [HW, Chapter 1]