Seminar zur Motiven 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 3: Oct. 29:  Marc Levine. : Recollections on real étale cohomology [B] §3

Lecture 4: Nov. 12:  Clémentine Lemarie-Rieusset. : Recollections on motivic homotopy theory [B] §4

Lecture 5. Nov. 19:  Chirantan Chowdhury : Recollections on pre-motivic categories and monoidal Bousfield localisation [B] §5-6

Lecture 6. Nov. 26:  Jan Hennig. : The theorem of Jacobsen and $\rho$-stable homotopy modules [B] §7

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