Topology Seminar Non-nilpotent elements in motivic homotopy theory
نویسنده
چکیده
Classically, the nilpotence theorem of Devinatz, Hopkins, and Smith tells us that non-nilpotent self maps on finite p-local spectra induce nonzero homomorphisms on BP -homology. Motivically, over C, this theorem fails to hold: we have a motivic analog of BP and whilst η : S1,1 → S0,0 induces zero on BP -homology, it is nonnilpotent. Recent work with Haynes Miller has led to a calculation of ηπ∗,∗(S 0,0), proving a conjecture of Guillou and Isaksen. I’ll introduce the motivic homotopy category and the motivic Adams-Novikov spectral sequence before describing this theorem. Then I’ll show that there are more periodicity operators in chromatic motivic homotopy theory than in the classical story. In particular, I will describe a new non-nilpotent self map.
منابع مشابه
New families in the homotopy of the motivic sphere spectrum
In [1] Adams constructed a non-nilpotent map v 1 : Σ S/2 −→ S/2. Using iterates of this map one constructs infinite families of elements in the stable homotopy groups of spheres, the v1-periodic elements of order 2. In this paper we work motivically over C and construct a nonnilpotent self map w 1 : Σ S/η −→ S/η. We then construct some infinite families of elements in the homotopy of the motivi...
متن کاملUnstable motivic homotopy categories in Nisnevich and cdh-topologies
One can do the motivic homotopy theory in the context of different motivic homotopy categories. One can vary the topology on the category of schemes used to define the homotopy category or one can vary the category of schemes itself considering only schemes satisfying certain conditions. The category obtained by taking smooth schemes and the Nisnevich topology seems to play a distinguished role...
متن کاملEquivariant Motivic Cohomology
The present paper will form part of the author’s PhD thesis, which will concern in part the practical computation of the motivic cohomology and the equivariant motivic cohomology of homogeneous varieties, such as Stiefel manifolds, Grassmanians, and spaces of matrices with prescribed rank conditions. Nothing proved here is particularly surprising, but it seems to the author that the spectral se...
متن کاملMotivic Hopf elements and relations
We use Cayley–Dickson algebras to produce Hopf elements η, ν, and σ in the motivic stable homotopy groups of spheres, and we prove the relations ην = 0 and νσ = 0 by geometric arguments. Along the way we develop several basic facts about the motivic stable homotopy ring.
متن کاملCohomology operations and algebraic geometry
This manuscript is based on a ten hours series of seminars I delivered in August of 2003 at the Nagoya Institute of Technology as part of the workshop on homotopy theory organized by Norihiko Minami and following the Kinosaki conference in honor of Goro Nishida. One of the most striking applications of homotopy theory in “exotic” contexes is Voevodsky’s proof of the Milnor Conjecture. This conj...
متن کامل