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 rough provisional program:
1. The real étale topology (3-4 + n talks?):
Given a scheme X, there is a topological space R(X), the “associated real space”.
Using this one may define a topology on all schemes (the real étale topology), where
the covering families are families of étale maps which are jointly surjective on the
associated real spaces. One important result about this is that if we take the small
étale site of X, but instead of the étale topology use the (restriction of the) real
étale topology, then the topos of sheaves obtained in this way is equivalent to the
typos of sheaves on the topological space R(X). This is proved in Scheiderers book
[2] (for example). After this there are two sets of results we need. Firstly we have
general formal results about the real étale topology: finite cohomological dimension, limit theorems, proper base change, homotopy invariance, etc. All of this is in
Scheiderers book and done mostly using the first definition. Secondly we have an
investigation of the real spaces associated with (certain) rings. This goes into the
proof of the comparison result, and we will use some simple properties in the main
paper [1] directly. (For example the real space of a Henselian ring has as closed
points the real space of its residue field.) This is done nicely in [3].
A further result which we need is “Gm -invariance of real étale cohomology (i.e.
A1 {0} looks like two points to R-ét-cohomology); the proof is very similar to homotopy invariance. Also we need to extend the proper base change theorem from
sheaves to spectra (this is quite subtle).
2. Six functors (1 talk?): Basically we need to know that a “pre-motivic category”
satisfying certain properties actually has the full six functors formalism, and some
easy consequences. This can be mainly a resumé of results from Cisinski-Déglise [?].
3. Local homotopy theory (1 talk?): Any typos has an associated homotopy theory:
simplicial pre sheaves, presheaves of spectra, and of chain complexes. All with the
local model structure. The stable categories have t-structures. Geometric morphisms induce Quillen adjunctions. Etc.
4. Monoidal localisation, transfers and Milnor-Witt K-theory (1-2 talks): Monoidal
localisation can be computed as the evident colimit, under certain stability and
compactness assumptions.
Homotopy modules have transfers along all finite étale morphisms, and these
satisfy base change and projection formulas.
1
2
Finally, results about the structure of localizations of Milnor-Witt K-theory as
real étale sheaves is needed. This is contained in [4].
5. The above are preliminaries, most of which (except for the basics of real étale
topology) are treated in [1]. The proof of the main result then proceeds in three
steps:
Step 1: ρ-local homotopy modules are real étale sheaves (1 talk?):
This uses basics of the real étale topology, the transfers, the theorem of Jacobson
mentioned in (4) and some elementary sheaf theory and topology.
Steps 2 and 3 - “using the six functors formalism” (1 talk?):
By standard results from local homotopy theory, the above implies that rholocal motivic spectra are R-ét-local, over a field. Using the six functors formalism
(specifically continuity and localisation), one may extend this easily to all base
schemes (Noetherian of finite dimension). Then more local homotopy theory and
cohomological properties of the R-ét-topology easily show that there is a fully faithful functor SH(Xret ) → SH(X)[1/ρ]. The difficult bit is showing that this functor
is essentially surjective. There is a very clever argument for this in a paper of
Cisinski-Deglise. It uses in an essential way that both sides satisfy proper base
change.
6. Applications (1 talk?):
I think the most obvious thing is to study localisations of the sphere spectrum:
- away from ρ we get essentially the classical homotopy groups as homotopy sheaves
- (*) away from 2 and ρ = away from 2 and étale
- (*) rationally, the sphere is equivalent to A1 -homology
- away from ρ, A1 -homology is real homology - in particular, rationally η-locally,
the sphere is just rational unramified Witt theory (which is the same as rational
real homology)
Here the results (*) do not use anything from the previous talks, but the other
results follow from the main theorem.
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.