Inverting ρ and the real etale site

Tues., 14-16, WSC-N-U-4.05

Let [-1]:Spec k→ A1-{0} be the map corresponding to the k-point -1, and let ρ:Spec k+Gm be the extension sending the base point + to the base-point 1∈ Gm(k). If we look at real points, this is the inclusion {1,-1}→ R-{0}, which is a homotopy equivalence. Remarkably, if one inverts the map ρ in the motivic stable homotopy category SH(B) over a reasonable base-scheme B, Tom Bachmann has shown that the resulting category is the homotopy category of presheaves of usual spectra on the "real etale site" of B. We will discuss these results and applications in the motives seminar this semester. 



Lecture 1. (Tom Bachmann) Overview
Lectures 2-5. The real étale topology:
Lecture 2. (Maria Yakerson) Introduction to the real étale topology [2, General notions, §1, 3]
Include [3, Propositions II.2.1 and II.2.4]. For most of the results on limits in [2, §3], just state the main results without proof.
Lecture 3. (Andre Chatzistimatiou) Comparison and base-change theorems [2, §15, 16]
Lecture 4.(Vladimir Sosnilo) Constructible sheaves and finiteness theorems [2, §17]
Lecture 5. (Daniel Harrer) Relations with the Zariski topology and examples [2, §19, 20]
Lecture 6. (Gabriele Guzman) Transfers in motivic homotopy theory [1, § 4]
Lecture 7.(Lorenzo Mantovani) ρ-periodic homotopy modules [1, § 7]
Lecture 8. (Daniel Harrer) The six functor formalism [1, § 5]
Lecture 9. (Arpith Shanbhag) Local homotopy theory [1, § 2]
Lecture 10. (Enzo Serandon) Monoidal Bousfield localisation [1, § 6]
Lecture 11. (Marc Levine) Real étale cohomology [1, § 3]
Lecture 12. (Tom Bachmann) The main theorem [1, § 8, 9]
Lecture 13. (Maria Yakerson) Real realization and the η-inverted sphere spectrum [1, § 10, 11]
Lecture 14. Rigidity and other applications [1,§ 12]+???

Here is the program with some additional comments and suggestions for Lectures 2-5



[1] Tom Bachmann, Motivic and Real Etale Stable Homotopy Theory, arXiv:1608.08855

[2] Claus Scheiderer. Real and Etale Cohomology. Lecture Notes in Mathematics;1588. Berlin : Springer, 1994.

[3] Carlos Andradas, Ludwig Bröcker, and Jesus M Ruiz. Constructible sets in real geometry, volume 33. Springer Science & Business Media, 2012.

[4] Jeremy Jacobson. Real cohomology and the powers of the fundamental ideal. 2015.