Motives Seminar: Real realizations of motivic homotopy theory
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} → 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
[1] has shown that the resulting category is the homotopy category of presheaves
of usual spectra on the “real étale site” of B. We will discuss these results and
applications in the motives seminar this semester.
Here is a program:
Lecture 1. Overview
Lectures 2-5. The real étale topology:
Lecture 2. 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. Comparison and base-change theorems [2, §15, 16]
Lecture 4. Constructible sheaves and finiteness theorems [2, §17]
Lecture 5. Relations with the Zariski topology and examples [2, §19, 20].
Lecture 6. Transfers in motivic homotopy theory [1, §4]
Lecture 7. ρ-periodic homotopy modules [1, §7]
Lecture 8. The six functor formalism [1, §5]
Lecture 9. Local homotopy theory [1, §2]
Lecture 10. Monoidal Bousfield localization [1, §6]
Lecture 11. Real étale cohomology [1, §3]
Lecture 12. The main theorem [1, §8, 9]
Lecture 13. Real realization and the η-inverted sphere spectrum [1, §10, 11]
Lecture 14. Rigidity and other applications [1, §12]+???
Here are so additional advisories for lectures 2-5: (**) and (*) indicate very important and important results, in which case the non-starred items are usually not
directly needed for the paper:
Section 0: all of 0.4 and 0.6
Section 1: Thm 1.3 (**), Corollary 1.7.1, Proposition 1.8 (*)
Section 3: everything about the rét-topology (Scheiderer always treats ét and b
simultaneously)
Section 15: Only remark (15.3.2), which is not proved in any way. A better reference may be [Delfs: Homology of Locally Semi-Algebraic Spaces, Thereorem II.5.7]
but then it has to be explained why this relates to our situation.
Section 16: Them 16.2 (**), Example 16.7.2 (**), Theorems 16.8 and 16.11
Section 17: nothing is really needed. But this is a good place to justify the proof
of example 16.7.2. Note that this requires results from the appendix.
Section 19: Theorem 19.2 plus the (straightforward) modification for Nis -¿ Zar.
Theorem 7.6. (Yes thats in section 7, for reasons which make sense in the book,
but the result is an immediate consequence of 19.2.)
Section 20: Nothing specifically needed, do as time permits.
1
2
References
[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.