Motives Seminar-WS 2024/25
Motivic and real étale stable homotopy theory
This semester we will go through the paper [B] of Tom Bachmann, which establishes an equivalence between a certain localisation of the motivic stable homotopy catogory over a base-scheme $S$, and the "classical" stable homotopy category of sheaves of spectra on the small real étale site of $S$. The localisation is obtained by inverting the map $\rho:S^0_S\to \mathbb{G}_{m,S}$ sending the base-point in $S^0_S=S\amalg S$ to 1 and the "non-basepoint" to -1. In the most basic case, $S=\text{Spec}(\mathbb{R})$, this comes down to an equivalence of $\text{SH}(\mathbb{R})[\rho^{-1}]$ with the classical stable homotopy category $\text{SH}$. There are many beautiful applications of this result, for instance, the identification of the Witt-sheaf cohomology of a smooth $\mathbb{R}$-scheme $X$ with the singular cohomology $H^*(X(\mathbb{R}), \mathbb{Z})$ after inverting 2.
The main source is Bachmann's paper [B] which is mostly self-contained, assuming some results about motivic stable homotopy theory and about the real étale topos, this latter coming from the book of Scheiderer [S]. Other references can be found in [B]. See also the workshop notes [GRMA] below
We will be meeting (most) Tuesdays, 16-18 Uhr (c.t.) in WSC-N-U-4.03
Program
Lecture 1. Oct. 15: Marc Levine : An overview [B] §1
Lecture 1 Notes
Lecture 1 Video.
Lecture 2. Oct. 22: Andrei Konovalov : Recollections on local homotopy theory [B] §2
Lecture 2 Video
Lecture 3: Oct. 29: Marc Levine : Recollections on real étale cohomology [B] §3
Lecture 3 Notes
Lecture 3 Video.
Lecture 4: Nov. 12: Clémentine Lemarié--Rieusset : Recollections on motivic homotopy theory [B] §4
Lecture 4 Notes
Lecture 4 Video.
Lecture 5. Nov. 19: Chirantan Chowdhury : Recollections on pre-motivic categories and monoidal Bousfield localisation [B] §5-6
Lecture 5 Video.
Lecture 6. Nov. 26: Jan Hennig : The theorem of Jacobsen and $\rho$-stable homotopy modules [B] §7
Lecture 6 Video
Lecture 7. Dec. 3: Linda Carnevale : Preliminary observations [B] §8
Lecture 8. Dec. 10: Clémentine Lemarie-Rieusset : Main theorems [B] §9
Lecture 9. Dec. 17: N.N. : Real realization [B] §10
Lecture 10. Jan. 7, 2025: N.N. : Application 1: the $\eta$-inverted sphere [B] §11
Lecture 11. Jan. 21: N.N. : Application 2: Some rigidity results [B] §12
???. Jan 28
References
[B] Bachmann, Tom, Motivic and real étale stable homotopy theory. Compos. Math.154(2018), no.5, 883–917.
[S] Scheiderer, Claus, Real and étale cohomology Lecture Notes in Math., 1588 Springer-Verlag, Berlin, 1994, xxiv+273 pp. ISBN: 3-540-58436-6
[GRMA] Géométrie réelle, motifs et A1-homotopie, Groupe de travail ANR HQDIAG, 2022. Workshop Notes