9
$\begingroup$

I am trying to learn ($\mathbb{P}^1$-)stable motivic ($\mathbb{A}^1$-)homotopy theory of schemes from the Cisinski-Deglise book, Triangulated Categories of Mixed Motives. In order to keep myself going during its long abstract development as well as to help see parallels with the topological stable homotopy theory I'd like to know in advance some of the concrete computational results and goals the theory is reaching for.

For example, in the topological setting calculating the stable homotopy groups of spheres has been a long-running motivating theme. Are there similar computational goals animating the development in the algebraic-geometric context (as opposed to those concerned with having a satisfactory theory with certain features, e.g., the six-functor formalism, spectra for cohomology theories etc. which are wonderfully explained in the preface to the book?) I'm specially interested in those that are of arithmetical nature.

Improve this question
$\endgroup$
4
  • 2
    $\begingroup$ An obvious one that comes to mind is the Bloch–Kato conjecture (now a theorem of Voevodsky), which drove the development of $\mathbf A^1$-homotopy theory. Or are you asking specifically about the stable variant? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 26, 2020 at 5:17
  • $\begingroup$ I think independence of $\ell$ is a rather good topic to have in mind: a goal of the six operations in the motivic world is to promote many cohomological features of $\ell$-adic sheaves to the motivic world. Anything which is trace-like (Euler characteristics, zeta functions) falls in this category. A nice paper to read on these kind of features is Olsson's link.springer.com/article/10.1007/s00229-015-0765-3 (or math.berkeley.edu/~molsson/Chernandmotives4.0.pdf for free access). $\endgroup$
    – D.-C. Cisinski
    Commented Jul 31, 2020 at 21:07
  • $\begingroup$ I also wrote a survey which revists this kind of things here: mathematik.uni-regensburg.de/cisinski/traces.pdf (but many of this topic can be found in various remarks in my joint papers with Déglise). $\endgroup$
    – D.-C. Cisinski
    Commented Jul 31, 2020 at 21:07
  • $\begingroup$ Such $\ell$-independence of traces is still a subject of research: this paper of Hiroki Kato arxiv.org/pdf/1904.02324.pdf for instance. However, the mere existence of motivic sheaves and fo their $\ell$-adic realizations make the independence of $\ell$ part of the aforementioned paper trivial. $\endgroup$
    – D.-C. Cisinski
    Commented Jul 31, 2020 at 21:32

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .